Wheat Prices and Rainfall in Western Europe Author(S): William H

Wheat Prices and Rainfall in Western Europe Author(s): William H. Beveridge Source: Journal of the Royal Statistical Society, Vol. 85, No. 3 (May, 1922), pp. 412-475 Published by: Wiley for the Royal Statistical Society Stable URL: http://www.jstor.org/stable/2341183 . Accessed: 25/06/2014 03:22

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This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 412 [Alay,

WHEATPRICES AND RAINFALLIN WESTERNEUROPE.

By SIR WILLIAMH. BEVERIDGE.

[Read before the Royal Statistical Society, April 25, 1922, the President, Sir R. HENRYREw, K.C.B., in the Chair.]

I.-INTRODUCTION. II.-CONSTRUCTIONOF PERIODOGRAMOF WHEAT PRICE FLiuCTUATIONS. (With chart.) I lf.-INTERPRETATIONOF PERIODOGRAM. The four tests of periodicity. Thc periods found. Summary and Review (with table). IV.-COMPARISONWITH EUROPEAN RAINFALL, 1850-1921. (With chart.) V.-CONCLUSION. APPENDIX.-Harmonic Analysis of Wheat Price Fluctuations.

I.-INTRODUCTION. THE paper which I have the honour of reading to you to-night is ill substance a sequel and supplement to an article on " Weather " and Harvest Cycles," published in the Economic Journal of December, 1921. In that article (which I shall refer to here as " my former article ") I gave index-numbers of wheat prices in Western and Central Europe from 1500 to 1869 and the preliminary results of an analysis of these figures, by mathematical and arithmetical methods, with a view to discovering periodicity. I sum- marized my conclusions on this point " in three propositions of a "descending order of certainty ": First, the yield of harvests in Western and Central Europe from the middle of the sixteenth to the opening of the twentieth century has been subject to a periodic influence, or combination of such influences, tending to produce bad harvests at intervals of about 15-3 years, the first epoch falling in 1556. This proposition is about as certain as harmonic analysis can make it. Second, this period of 15*3 years, though corresponding to certain physical facts, is not a permanent one, but arises from a temporary combination of two or more shorter cycles. This proposition, though not certain, is in both of its branches highly probable.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] BEVERIDGE-Wlheat Prices antdRainfall, etc. 413

Third, the shorter cycles whose combination has given rise to the 15.-3- year period from 1556 onwards, and which are themselves more permanent than their combination. are those named above as = 4 -374 years, = 5 11 years, and (probably) = 2-74 years and = 3 71 years. This proposition is a speculation as to whose plausibility and truth different readers will take different views.* I had hoped to-night to undertake a three-fold task: first, to give a full account of the sources and construction of my index- numbers as justification of their value; second, to give the results of a complete harmonic analysis in place of the preliminary examination made for me by Mr. H. T. Curwen; third, to call attention to the economic, as distinct from the meteorological, uses of my index-numbers, and the light thrown by them on the history of prices in different countries and in Europe as a whole. Considerations of time and space have compelled me to modify this programme. In effect, I must confine myself to the second of the above tasks. The gist of my present paper is the complete harmonic analysis of my index of wheat price fluctuations for about three hundred years from 1545 to 1844, involving a test of the figures for all possible periods between two and a half and eighty- four years in length, and for many periods below two and a half years. This is followed by an attempt to relate the results of the analysis to rainfall records from 1850 to 1921. In so far as this attempt proves successful, the need for justifying my index-numbers by describing their sources and construction will no longer remain. The value of the index-numbers will be proved in the best of all ways, by showing that theories founded on them explain the facts of meteorology. I may, however, summarize and add to the statement made in my former article, as follows:- The numbers used for harmonic analysis are based ultimately on lists of prices for long periods of consecutive years in nearly fifty separate markets or districts of Western and Central Europe, and on official averages for whole countries when these become available. There has been no selection of places or of years to be included: within the area taken, every list known to me and sufficiently continuous to be used has been used substantially for the whole period covered by it, or till it was superseded by a comprehensive list for a whole country.t There has been no artificial weighting: * To avoid confusion I have omitted the letters A, B, C, and D by which I formerly identified these periods. The discovery by my further analysis of so many new periods has compelled a change of nomenclature. ? Hardly a dozen eccentric figures altogether out of many thousand- chiefly from German towns during the Seven Years' or the Thirty Years' War -have been rejected. VOL LJXXXV. PART III. 2 F

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 414 BEVERIDGE-Wheat Pricesand Raifall [May, each list has counted for one in the country from which it came, and each of the five countries or territorial divisions-Britain, Low Countries, France, North Germany, South Germany, and Austria-has counted for one. Finally, there has been no smoothing or averagingprocess which could favour the appearanceof periodicity in general or of a period of any particular length; with the one exception just to be mentioned, I have indulged in no averaging at all. The one exception is this: the index-numibersof wheat prices (as set out in Column 1 of the table appended to my former paper) showed a marked secular trend, correspondingto the depreciation of money; they grew from an average of 15 I for the decade 1500-9 to one of 222-6 for 1860-69. It was essential to eliminate this trend. Otherwise, the later observations with their high values would have had undue importance in the indication of periods: this form of weighting was most undesirable, in view of the known inconstancy of meteorological cycles. The trend was eliminated by calculating continuous 31-year averages, and showing the index-number for each year as a percentage of the average for the thirty-one years of which it was the centre (Column 2 of the table in my former article). These percentages thus record the deviation of the price for each year from the average of the neighbouringthirty-one years; they are described as the " indices " of wheat price fluctuation "; they are the numbers submitted to harmonic analysis in my former article and more completely analysed in my present one. Some distortion of the harmonics present in a sequence of observations is probably involved in any smoothing or averaging process whatever. Professor Schuster, in his original paper on the periodogram,issued a weighty warning against the use of such processesbefore applying harmonicanalysis: as a rule the harmonic analysis itself performs any justifiable smoothing effectively and without distortion. The particular process employed here is, I think, as nearly free from objection as possible. Inspection shows that all the peaks and depressions of the original figures are faith- fully reproducedin the percentages,and that the proportionsbetween neighbouring peaks or depressions are unchanged. Mathematical analysis of the precise effect of any averaging process is a difficult matter which must be left to competent mathematicians. So far as I can judge, from a formula preparedfor me by Mr. H. T. Curwen, the general effect of the particular process adopted here should be to leave harmonics up to thirty-one years of length practically untouched in amplitude as well as in phase, while those of greater

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.1 in WesternEurope. 415 length are steadily but slowly damped down, their phase remaining unaltered. This simply means that a lower intensity may be significantof a true period if it occurs at fifty years than if it occurs at twenty-five. There seems to be no selective bias in favour of particular periods. The errors due to leaving the secular trend uneliminated would almost certainly have been more serious.*

II.-CONSTRUCTION OF PERIODOGRAMOF WHEAT-PRICE FLUCTUATIONS. In my former article the harmonic analysis of the indices of wheat-price fluctuation, undertaken for me by Mr. H. T. Curwen, was carried only to the point of covering all periods of a complete number of years from two to thirty-six, and a certain number of intervening periods consisting of complete half-years. Altogether, only some fifty trial periods were examined. I pointed out that the mesh of this analysis was not close enough to throw any light upon the existence of shorter periods, such as 5x1 or 2x735, which were not nearly equal to a number of complete years or half- years. I have now made an analysis which is far more nearly exhaustive, involving the calculation of amplitudes for more than three hundred trial periods between two and eighty-four years in length. In this calculation, most of the ground already worked has been gone over again, and the results checked and corrected; the present figures should be regardedas replacing the preliminaryfigures given in my

* For an instance of an averaging process probably far less free from objection, reference may be made to Professor Poynting's paper on the Fluctuation of Wheat Prices in the Journal of the Royal Statistical Society for March, 1884. Professor Poynting gave for each year an index-number representing the average of four years (of which that year was the second) divided by the average of ten years (of which it was the fifth). The effect of this particular process, according to Professor Poynting, " is practically to destroy " all harmonics below five years, to save over half the amplitude at six years, "a greater amount up to eight years, when about five-sixths is saved, and "beyond that a continually decreasing amount, although at fifteen years still "nearly one-half is saved." This statement is actually based on an analysis by Professor Gabriel Stokes of a slightly different process (subtraction of one average from another in place of division), but the two are apparently regarded as having similar effects. Either process appears to bias the resulting figures in favour of the production of an eight-year cycle, which is the exact length of the cycle which Professor H. L. Moore, of Columbia, subsequently discovered in Professor Poynting's figures (Quarterly Journal of Economics, August, 1921). 2 F 2

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 416 BEVERIDGE-Wheat Prices and Rainfall [May,

former article.* The methods adopted in this analysis are as follows: The observations, i.e. the index-numbers of price fluctuation were arranged in m rows of p observations each (so that mtbp= N the total number of observations used), and were dealt with as follows

to tl t2~ *p * .f- i tptp+ tlp-1 *

t(m-1) p 'MP

-Sums of columns = To T1 T2 * * . Tp- 1

Then 2 2sr a= P-1TC~~ cos-c p c=O m 1) 2 -T1 'T . 2swr b=-z -sin -c p c= O met p q- a2 + b2 Nr2 300

The interval between successive observations (t) is in this case one year, and p and s are always integers, the length of the trial period under examination being P years. The number of revolutions of

this trial period in 300 observations is q(=300S) The number of observations used (N) is, of course, always an exact multiple of p, and therefore varies for different values of p. My general rule has been to take the value of N nearest to 300 in each case, more often exceeding 300 than falling short of it; this rule has sometimes been departed from, chiefly for the purpose of getting an even number for m, so that the sequence could be split into two equal halves for independent analysis. The ratio of 92 to its expectancy (by which the probability of a for any given value of 9 has to be judged) is pro- true period S

* I am glad to say that I did not in this checking process discover any important error in the amplitudes, in the whole sequence of 300 years, though the figures differ often in detail through the use of slightly different sequences of years. In one or two cases (as at eight years) the amplitudes in the separate half sequences had been insufficiently checked and have had to be altered materially.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.1 in Western Etrope. 417 portionate to Nir.* I have, therefore, discounted, so far as possible, the effect of differences in N by giving in the table the value not Nr2 of r2 but of 300. The latter quantity, described as the intensity (I), is taken as the ordinate of the periodogram, which has thus been reduced so far as possible to a uniform basis of 300 observations. Wherever, for any trial period, examination of the whole sequence of about 300 years yields a high intensity, a further examination has been made of each half of the whole sequence taken separately, in order to determine whether the apparent period has persisted in each half. The second half sequence in all cases follows immediately on the first half ; thus, at 5 * 667 (got by p = I7, S = 3) the first half sequence embraces the years 1545 to 1697, and the second half those from 1698 to 1850; at 5-7I4 (got by p 40, s = 7) the two halves are 1545-1704 and 1705-1864. The origin for the different second half sequences thus varies considerably. This plan, however, makes it easy to note the advance or regression of phase between the two halves, and thus to infer the exact length of the period. The intensities for the half sequences have been reduced to a uniform basis, as in the case of the whole sequence, r2 being multiplied by the number of years covered and divided by I50 in each case. By choosing appropriate values for p and s, the trial periods have been taken sufficiently close together to make it practically certain that every possible periodicity between two and a half and eighty-four years has been discovered. Between those limits the periodogram, for the whole sequence is continuous and the analysis exhaustive. This has involved the calculation of a very great number of amplitudes; how necessary the labour was, and how close the mesh of the analysis must be made when p/s is small in relation to N, is well illustrated by the invisibility of the very important 5 I-year period if we test at 5 0 years, and of the 3 4I5 period if we test at 3-455. Between two and two and a half years the analysis is not quite exhaustive. I have looked for certain periods near 2-2 and 2-4 years suggested to me by Mr. J. Baxendell, and I have calculated a number of amplitudes which could be obtained with little additional labour from other workings.t These are sufficient * Cp. Brunt: Combbinationof Observations, p. 201. t When p is a multiple of 4, the amplitudes for -P and P are connected S P 2- by a simple arithmetical relation enabling either to be got from the workings for the other in a few minutes.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 418 BEVERIDGE-Wheat Prices and Raitifall [May, to show the low general level of the periodogram between two and two and a half years. There may, of course, still be peaks to be found there, but I should be surprisedif this were so. The first observation is in nearly all cases that for the harvest year 1545; with this origin I have used anything up to 325 observations, ending with 1869. The harvest year is taken conventionallyas beginning at September1. The origin, therefore,from which phases are reckoned, is in nearly all cases 1544 67 (in calendar year notation). In a few cases it has been necessary or convenient to use more than 325 observations and to include for that purpose years prior to 1545. In these cases the sequence ends always in 1869, but the origin varies; to avoid confusion, the values of a and b are printed in heavy type whenever the origin is not 1544 67. The maximum phases for each period are given in two ways- "uncorrected " and " corrected." The " uncorrected" phase for any period is that derived from b formula, tan 4 =-. The, the values of a and b by the ordinary a angle 4 gives the distance of one maximum phase from the origin. Reckoned in years this is equivalent to 2 X P. since 27ror one complete revolution, representsthe length of the period. Where the length of the true period does not coincide exactly with one of the trial periods examined, the phase for the former is derived from that of the nearest trial period by a simple calculation based on the principlethat the maximum phases of the trial period and the true period will coincide at the middle of the sequence of observations, so as to secure the greatest possible measure of agreement between the true and the trial fluctuation over the whole sequence.* An illustration will make the method clear. The phase for the trial period 6 000 years is 1544 67 + 3&08 1547 75. There are fifty fluctuations of this trial period in the whole sequence of 300 observations. For a true period of 5 960 years (i.e. 040 shorter), the first maximum will occur 2 x 040 = 1 000 year after the first maximum for the whole period, i.e. at 1548 75, while the last maximum in the sequence will occur one year before that for the trial period, i.e. at 1846x75 as compared with 1847 75.

* The principle is stated by Professor Schuster in his paper on the Periodicities of Sun Spots (Transactions of the Royal Society, Series A, vol. 206, p. 72). The mathematical formula there given by him for the derivation of the true phase seems to yield the same result as the method adopted here.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] iin Western Europe. 419 The shorter fluctuation will lie symmetrically within the longer so that they coincide at their middle points. it is important to realize that even when a period found by harmonic analysis is real-that is to say, correspondsto a physical cycle, say, of rainfall-the maximum phase obtained in the analysis, whether directly or by derivation in the way described, will not necessarily coincide exactly with the actual maximum of rainfall. The formula- tan q)gives not the maximumphase of the physical a period as such, but the maximum phase of the sine curve which will fit most closely the physical period along its whole course from minimum to maximum and back again. These two phases will only coincide if the physical cycle be substantially symmetrical, i.e. if it represents a rise to and similar fall from a maximum equi- distant from the minima on each side.* If the period is unsymmetrical, e.g. if the rise is more rapid than the fall, and the physical maximum occurs before midway between the minima, the maximum phase of the sine curve as given by harmonic analysis will be later than the physical maximum, and vice versed if the fall is more rapid than the rise. There is no reason on general grounds for expecting meteorological cycles to be symmetrical. The probability is, on the whole, in favour of asymmetry, and many of these cycles are certainly

* This can be shown without higher mathematicsby a simple reductioad absurduin. For suppose a sequence of observations to contain an unsym. metrical eight-yearperiod with minima at the origin and at eight years from it and a maximumthree (in place of four) years from it. If it be possible let the formula tan p indicate this maximum, i.e. let op= .j of 27r = 1350 so that for a trial periodof eight years ab= tan 1350. Now reversethe sequence by subtracting each ordinate from a constant, say 200. In the reversed sequence there will also be an eight-years period, with maxima at 0 and 8 years and a minimumat 3 years; ' for this period should = 00 or 3600. But if the analysisis made it will not do so ; for each trial period,a and b will have exactly the same value in the reversed sequence as in the original one, but with changed signs. That is to say, if, for an eight-year trial period, b a = tan 1350 in the originalsequence, will=tan (135? + 180?)in the reversed sequence, i.e. 'p will not equal 0? or 3600. The formula a-tan O cannot give the actual maximum of an unsymmetrical eight-year period in both sequences, and, therefore, cannot do so in either. Actually, of course, in the case supposed p would be found to be about halfway between 1350 and 1800 (i.e. near 31 years) in the original sequence and between 3150 and 360? (i.e. near 7-l-years) in the reversed sequence,

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 420 BEVERIDGE-Wheat Pricesand Rainfall [May, unsymmetrical. We cannot, therefore, by harmonic analysis, determine the actual maxima of a physical cycle with complete accuracy, until we know the shape as well as the length of the cycle. We should expect only a fair and not an absolute agreement of phase between the calculated and the physical period. My present enquiry does not make it possible to determine the shapes of any of the cycles discovered, and it may well be doubted whether the shapes of weather cycles could ever be satisfactorily discovered from wheat prices. No attempt, therefore, has been made to adjust the phases as given by the formula b = tan g for a any assumed want of symmetry in the-physical cycles. There is, however, in the figures now under analysis one general factor making for asymmetry in the price cycles which does call for correction. Cycles in wheat prices are, by their nature, unsymmetrical, irrespectiveof the shape of the true meteorologicalcycle which may be their ultimate cause. A bad harvest in any year affects the price not only of that harvest, but of the subsequent harvest; it causes a relative shortage in two years at least, and not in one only. This means that any marked peak in the graph of prices is apt to be followed by another year of fairly high prices, even though the meteorologicalconditions in the second year may have been normal or favourable. It follows that if there be a physical or meteorological cycle causing bad harvests at stated periods, and if this be discovered both in meteorologicaland price records, the length of the two cycles should agree exactly, but the phase as given by the price record will tend always to be later than the phase as given by the meteorological record. The degree of retardation cannot be determined with accuracy, in the absence of definite knowledge as to the degree in which one year's harvest will influence the price of next year's harvest. All that can be said is that the retardationwill be less than half a year, and will be greater for short periods than for long ones. In the case of a long period there may be little or no retardation. The raising of prices in the second year may merely make the shape of the curve coincide more closely with that of a sine curve, and may hardly affect the phase. In the absence of precise information, I have thought it best to make the necessary correction on as simple a basis as possible, and to take it as the reciprocalof the length of the cycle, i.e. -. Thus

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions Sir A. Lowes-Dickinsoll.

SUMMARY OF BOARD OFT

Scotland. Northumberland. Durham.

Perent. Tons. Per cent. Tons. Per cent. I Tons. Per - 32,093,322 1. Tonnage of saleable coal raised ...- 31,629,033 - 11,876,210 2. Mine consumption ...... 10412 3,200,166 4-93 585,691 3 56 1,142,414 3. Miners' coal ...1...... -37 434,964 4-65 551,850 4.01 1,288,416 9 4. Tonnage disposable commercially ... .. 88-51 27,993,903 90 42 10,738,669 92-43 29,662,492 5. Miners' coal per worker,1 per annum -3*59 -10-08 -8-53

Pei 008t of Production. Per ton. Per ton. Per ton. S. d. ?C S. d. ?: 8. d. S...... 11 8.53 16,391,772 11 7 95 6,261,845 11 10 79 17,648,089 M3 6. Wages 3 7. Stores and timber ...... 2 4-85 3,364,792 2 5 57 1,323,071 2 6 16 3,728,046 8. Other costs, management, salaries, in- office and general ex- surances, repairs, 3 penses, depreciation, etc. 27.14 3,631,995 3 0 -63 1,639,144 3 2 -70 4,782,985 Fund... .. 0 1-13 131,989 0 1 10 49,400 0 1-08 133,705 0 9. Miners' Welfare 0 10. Royalties ...... 0 8-26 963,429 0 7 -66 342,681 0 9 83 1,214,620 20 11. Total costs... .. 17 5.91 24,483,977 17 10-91 9,616,141 18 6-56 27,507,445

Deduct- ( 12. Proceeds of miners' coal ...U 2 30 268,826 -78 -- 20 13. Net costs....17 3 -61 24,215,151 17 10-91 9,616,063 18 6W56 27,507,445 .... 0 7 99 0 14. Profit or loss ... ..0 4 -77 5,56,693 0 11-41 510,624 987,723 20 15. Commercial disposals.1 7 8 38 24,771,844 18 10 32 10,126,687 19 2.55 28,495,168

150,957 16. Workmen employed 121,275 54,770 ...-e et -e et -e et Pe 9 17. Men shifts worked ..... 95-34 31,965,007 91-41 13,739,746 92-15 36,157,543 52 18. Men shifts, per workman ..-...263 -57 -250-86 -239 19. Men shifts cost...... 4.66 1,561,016 8-59 1,291,064 7 85 3,079,907 20. Men shifts, per workman.- 12 -87 -23-57 -20-40 21. Saleable coal raised per man shift worked cWts - 19 79 -17-29 -17-75 22. Saleable coal raised per worker per annum tonDs 260-80 - 216-84 - 212 60 0/9/7.14 23. Earnings per man shift worked .... ? - 0/10/3.07 - 0/9/1-38 - 116/18/2 24. Earnings per worker, per annum .... ? - 135/3/3 - 14/6/7 -

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions APPENDIX III (2).

SUMMARY OF BOARD OF TRADE QUARTERLY STATISTICS FOR COAL MINES, COMPRISINGABOUT 95 PER. C]

(2) (3) (4) (5) (6) (7) Walesand Yorkshire,Notts, Derby, LaIsie ot tfs rorthumberland.orthumberland. Durham. I ~~~~SouthMonmouth. Leicester, Cannock Chase, ancasie Noshie North Wales. d Warw ick. ______I______Ian a d C e h r .N r h Wl s nent. Tons. Per cent. Tons. Per cent. Tons. Per cent. . Tons. Per cent. Tons. - 11,876,210 32,093,322 - 45,632,320 - 73,643,883 21,784,232 - 2,647,331 .93 585,691 3 *56 1,142,414 6*22 2,837,731 6*12 4,510,015 9 .53 2,075,665 10 87 287,648 *65 551,850 4-01 1,288,416 2 -24 1,023,429 2 -36 1,733,905 1 -06 232,237 2 -52 66,740 *42 10,738,669 92 *43 29,662,492 91*54 41,771,160 91 *52 67,399,963 89-41 19,476,330 86- 61 2,292,943 - 10 08 - 8853 - 489 5- 59 1- 75 4- 433 ton. Per ton. Per ton. Per ton. Per ton. Per ton. d. ? 8. d. ? s. d. ? s. d. ? s. d. ? s. d. ? 7.95 6,261,845 11 10-79 17,648,089 13 0-42 27,224,082 14 1-07 47,480,360 15 5-48 15,052,043 15 2-63 1,744,689 5.57 1,323,071 2 616 3,728,046 3 610 7,327,537 1 1183 6,692,688 2 6i57 2,480,548 3 1(61 359,359

)-63 1,639,144 3 2-70 4,782,985 3 6-89 7,464,561 2 11-14 9,868,875 4 1-43 4,011,646 2 9-25 317,696 1*10 49,400 0 1*08 133,705 0 1 09 189,304 0 1 09 306,089 0 1*12 90,624 0 1105 10,985 7*66 342,681 0 9 83 1,214,620 0 8-81 1,533,788 0 4-73 1,327,546 0 6;58 533,832 0 4-98 47,665

)391 9,616,141 18 6(56 27,507,445 20 11 31 43,739,272 19 5 86 65,675,558 22 9 18 22,168,693 21 7 62 2,480,394

78 _ - 0 1 92 333,411 0 0 93 1 262,292 0 2-18 176,492 0 2-89 27,591 )-91 9,616,063 18 6t56 27,507,445 20 9 39 43,405,861 19 4 93 65,413,266 I22 7(00 21,992,201 21 4-73 2,452,803 1I41 510,624 0 7*99 987,723 0 1*55 270,401 1 3*40 4,324,033 0 4*55 368,931 0 6t80 64,983*

)332 10,126,687 19 2 55 28,495,168 20 10(94 43,676,262 20 8 33 69,737,2991 22 11 55 22,361,132 20 9.93 2,387,820

- 54,770 - 150,957 - 209,156 - 310,381 - 132,781 15,422 ent. Per cent. Per cent. Per cent. Per cent. Per cent. *41 13,739,746 92 15 36,157,543 92 28 54,469,389 1-34 72,727,294 92 98 30,002,901 93-85 4,052,075 - 250-86 - 239*52 - 260*42 - 234*32 - 225*96 262*75 559 1,291,064 7 85 3,079,907 7-72 4,557,328 8|66 6,894,624 7-02 2,265,657 6 *15 265,548 - 23 57 20-40 - 21 79 - 2221 - 17-06 17-22

- 1729 - 17*75 - 16-76 2025 - 14-52 - 1307

- 216 84 - 212-60 - 218 17 - 237-27 - 164-06 - 171-66 - 0/9/1.38 - 0/9/7.14 - 0/9/11.95 - 0/13/0.69 - 0/10/0.40 - 0/8/7-34 - I 114/6/7 - - 116/18/2 - - 130/3/3 1 152/19/6 - 113/7/2 - 113/2/7

* Losses.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions APPENDIX III (2). iL MINES, COMPRISINGABOUT 95 PER. CENT. OF TOTAL PRODUCTION. YEAR ENDING SEPTEMBER 30, 1922.

(6) (7) (8) (9) (10) (11) North Lancasire, Stahfs,i North Wales. South Staffs and Salop. Cumberland. Bristol. Forest of Dean

Per cent. Tons. Per cent. Tons. Per cent. Tons. Per cent. Tons. Per cent. Tons. Per cent. Toi 21,784,232 2,647,331 - 1,504,881 - 1,741,715 - 298,734 7257 9-53 2,075,665 10-87 287,648 11-95 179,827 11 30 196,890 7.04 21,034 13-159d 1-06 232,237 2 52 66,740 2-80 42,214 0-28 4,882 3-48 10,401 2-92 2 89-41 19,476,330 86 61 2,292,943 85-25 1,282,840 88-42 1,539,943 89-48 267,329 83-93 604 1-75 4-33 5-87 - 0-47 _493 4

Per ton. Per ton. Per ton. Per tol. Per ton. Per ton. s. d. ? s.d. ? s. d. ? s. d. ? . d. ? s. d. ? 15 5-48 15,052,043 15 2 63 1,744,689 11 7 23 744,230 16 4 88 1,263,277 16 9 66 224,627 14 9'02 447 2 6-57 2,480,548 3 1 61 359,359 3 2 6(9 206,781 2 11-71 229,141 2 10 77 38,728 2 9-95 8t

4 1-43 4,011,646 2 9 25 317,696 4 2 86 271,853 3 5-19 264,276 4 6-51 60,718 4 3-43 13( 0 1 12 90,624 0 1 15 10,985 0 1-13 6,045 0 1-16 7,466 0 1 11 1,240 0 1.19 0 6;58 533,832 0 498 47,665 0 8-71 46,567 0 11-96 76,758 O 5 69; 6,341 0 4-36 11

22 9.18 22,168,693 21 7-62 2,480,394 19 10-62 1,275,476 23 10-90 1,840,918 24 9-74 331,654 22 3-95 671

0 2-18 176,492 0 2-89 27,591 0 0-59 3,192 0 0-38 2,491 0 002 33 0 2-92 22 7-00 21,992,201 21 4-73 2,452,803 19 10-03 1,272,284 23 10-52 1,838,427 24 9-72 331,621 22 103 669 0 4-55 368,931 0 6-80 64,983* 1 8-72 110,712* 2 10-81 223,365* 1 7-00 21,168* 0 5-56 14

22 11 55 22,361,132 20 9-93 2,387,820 18 1-31 1,161,572 20 11-71 1,615,062 23 2-72 310,453 22 6-59 684

- | 132,781 - 15,422 7,196 - 10,378 - 2,110 _ 4 Per cent. Per cent. Per cent. Per vent. P r cent. Per cent. 92 98 30,002,901 93-85 4,052,075 94*21 1,862,155 85-74 2,530,223 97*04 534,771 93-24 1,108 - t 225-96 262-753 1 258-78 -- 243*81 - 253.45 - 247 7-02 2,265,657 6(15 265,548 5-79 114,476 14-26 420,949 2-96 16,309 6-76 80 - ! 17 06 - 17 22 - 15 91 -- 40 56 - 7-73 - 17 - ] 1452 - 1307 - 16*16 - 13*77 - J 11.17 - 13

- | 164*06 - 171 66 - 209*13 - 167-83 - 141 59 - 161 - 0/10/0-40 - 0/8/7-34 - 0/7/11-92 - L0/9/11.83 - 0/8/4 81 - 0/8/0 - { 113/7/2 - 113/2/7 - 103/8/5 - 121/14/6 - 106/9/2 - 99/1,

* Losses.

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ENDING SEPTEMBER 30, 1922.

(10) (1 11) . (1 _ (13) Total. Bristol. Forestof Dean. Somerset. j Kent.

Percent. Tons. Per cent. Tons. Per cent. Tons. Per cent. Tons. Per cent. Tons. 715 - 298,734 722,861 - 718,661 - 352,283 - 224,645,496 890 7 04 21,034 1315 95,053 8"70 62,494 14-53 51,192 6-79 15,245,820 882 3'48 10,401 2'92 21,121 3 -52 25,354 1-58 5,562 2'42 5,441,075 943 89-48 267,329 83-93 606,687 87 78 630,813 83-89 295,529 90 79 203,958,601 47 _ 4-93 _4-71 _5-49 _ 3-47 _5-31

Per tOnl. Per to31. 'Per ton. |Per ton. Per ton. d.<. ? s d. ? s d. t . d. ;C s d. ?, 277 16 9 66 224,627 14 9()2 447,494 14 9j12 465,546 16 7 15 245,233 3j0813 135,193,287 141 2 10 77 38,728 2 9*95 85,817 2 3*71 72,839 3 0)*20 44,565 2 6-54 25X93,7912

276 4 6)-51 60.718 4 3-43 130,014 4 5-33 140,)182 4L 0 48 59,700 3 2-41 321643,645 466 0 1 *11 1,24 0 1.19 3,006 0 1 13 2,962 Q 1-18 1,457 0 1 10 934,1272 | ,341 4-3t 11,010 0 10'06 26,4386 0 11-85 14,587 0 7'23 6,145,262

)18 24 9 74 331,654 22 3-95 677,341 22 5'35 707,967 24 8-86 365,542 19 8'36 200,870,378

t91 0 0(02 33 0 2-92 7,377 _ - 0 1-29 1,590 0 1-27 1,083,373

127 24 9'72 331,621 22 1 03 669,964 22 5'35 707,967 | 24 7-57 363,952 19 7 09 199,787,005 M6o* 1 7 00 21,168* 0 5 56 14,040 0 5-67 14,895 { 4 11 17 72,859* 0 7'71 6,554,253 )62 23 2 72 310,453 22 6-59 684,004 22 11-02 722,862 19 8'40 291,093 20 2'80 206,341,258

;78 - 2,110 4- 4481 - 4,621 1- 1,601 1- l025,129 Per cent. Per cent. Per cent. Per cent. Pcr cent. !23 97'04 534,771 93-24 1,108,762 95'08 1,072,776 091 68 I 374,218 92 39 250,596,860 U - 253*45 - 247*44 | 232 15 - 233-74 - 244-45 149 2*96 16,309 6|76 80,430 4 92 5,540 8|32 33,939 7'61 20,636,787 i - 773 179-95 12'02 - 21 20 20-13

- J 11.17 - 13'04 - 13'40 - 18*83 - 17'93

13 - 141'59 - 161-32 - 155'52 - 220'04 - 219'14 83 - 0/8/4-81 - 0/8/0.86 - 0/8/8.15 - 0/13/1.28 - 0/10/9.48 /6 - 106/9/2 - 99/17/3 - 100/14/11 - 153/3/6 - 131/17/7

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions SUMMARY OF BOARD OF

(1) ~~~~~~~~(2) (3) Scotlanid. Northumberland. Durhami.

Per cent. Tons. Per cent. Tons. Per cent. Tons. Pe] 1. Tonnage of saleable coal raised - 36,723,060 - 13,374,566 - 37,149,652 2. Mine consumption ...... 885 3,249,778 .425 568,633 2*93 1,089,812 3. Miners' coal...... 129 474,917 4-44 593,194 3 86 1,434,490 4. Tonnage disposable commercially . . 89-86 32,998,365 91b31 12,212,739 93 21 34,625,350 t 5. Miners coal per worker, per annum tons -3 -58 -10417 -8-77

Per ton. Per ton. Per ton. Pe Cost of Production. S. d. ?e S. d. ?9 S. d. ?9 S. 6. Wages ...... 11 10.63 19,610,689 11 11*76 7,315,220 11 8-67 20,294,388 12 7. Stores and timber ...... 2 1-99 3,572,830 2 2 11 1,328,798 2 3 14 3,915,462 2 8. Other costs (management, salaries, insur- ance, repairs, office and general expenses, depreciation, etc.) ...... 2 1 88 3,558,897 2 8*19 1,637,981 3 1-23 5,370,836 3 9. Miners' W~elfareFund....0 1.11 152,568 0 1 10 55,736 0 1-07 154,806 0 10. Royalties ...... 0 7 48 1,028,499 0 6-90 351,143 0 6-94 1,001,207 0

11. Total costs ...... 16 ii 09 27,923,483 17 6 06 10,688,878 17 9 05 30,736,699 19

Deduct-

12. Proceeds of miners' coal ... ..0 2 32 319,136 -- -- 13. Net costs ...... 16 8 -77 27,604,347 117 6-06 10,688,878 17 9 05 30,736,699 19 .. ... 14. Profit or loss ...... 2 8-50 4,468,881 3 0 -76 1,871,007 2 9 31 4,806,440 1 15. Commercial disposals... ..19 52730328 20 6 82 12,559,885 20 6-36 35,543,139**21

Per cent. Per cent. Per cent. Per 17. Men-shifts worked ...... 95 86 37,822,976 91 -79 15,573,797 91-94 42,499,178 9 18. Men-shifts worked, per workman...... 285 54 -267 00 -259 89 19. Men-shifts lost...... 4 14 1,634,121 8 21 1,393,037 8-06 3,728,209 20. Men-shifts lost, per workman ....12 34 -23* 88 -22 80 21. Saleable coal raised per man shift worked cWts - 19 42 -17-18 -17 48 22. Saleable coal raised per worker per annum tons - 277-24 - 229*30 - 227-18 23. Earnings per man shift worked ... ? - 0/10/4*44 - 0/9/4.73 - 0/9/6-61 24. Earnings per worker, per annum ....I 148/1/0 125/8/3 - 124/2/1

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions APPENDIX III (3)

SUMMARY OF BOARD OF TRADE QUARTERLY RETURNS FOR COAL MINES, COMPRISING ABOUT 95 PER CEI

(2) (3) (5) j (6) (7) rthumberland. Durhaim. 1 South Wales and I Yorks, Notts, Derby, Lancs, North Staffs, North Wales. Monmouth. Cheshire. ______I______ILeicester, Cannock,Warwick. mt. Tons. Per cent. Tons. Per cent. Tons. Per cent. Tons. Per cent. Tons. Per cent. Tons. 13,374,566 37,149,65211 - 49,779,120 86,836,675 - 25,660,922 3,128,753 25 568,633 2 93 1,089,812 6-06 3,014,789 5 -52 4,789,696 8 70 2,232,157 10-34 323,554 44 593,194 3- 86 1,434,490 2 -10 1,048,416 2 -37 2,063,166 0-88 225,011 2 -47 77,178 31 12,212,739 93-21 34,625,350 91- 84 45,715,915 92 -11 79,983,813 90-42 23,203,754 87-19 2,728,021 10-17 - 8-77 4-76 6-21 - I 1-56 - 4-61 on. Per ton. Per ton. Per ton. Per ton. Per ton. d. ? s. d. ? s. d. ? s. d. ? s. d. ? s. d. ? *76 7,315,220 11 867 20,294,388 12 f141 29,602,373 11 3-41 45,127,514 13 1-67 15,243,904 13 10-07 1,887,707 -11 13,28,798 2 3 14 3,915,462 2 9 66 6,411,401 1 8 10 6,699,714 2 2 39 2,551,051 2 5 38 333,991

, 19 1,637,981 3 1 23 5,370,836 3 0 97 7,041,950 2 3 66 9,217,572 3 2-09 3,682,972 2 3 38 31 1,265 *10 55,736 0 1-07 154,806 0 1-08 207,413 0 1-08 361,067 0 1-10 106,602 0 1-14 12,988 190 1 351,143 0 6 94 1,001,207 0 8 67 1,651,098 0 4 55 1,513,863 0 5 59 540,170 0 4 55 51,737 , 06 10,688,878 17 9 05 30,736,699 19 7-79 44,914,235 15 8 80 62,919,730 19 0 84 22,124,699 19. 0 52 2,597,688

- --- - 390,718 0 102 340,329 0 174 168,113 0 285 32,509

06 10,688,878 17 9 05 30,736,699 19 5 74 44,523,517 15 7 78 62,579,401 18 11 10 21,956,586 18 9 67 2,565,179 -76 1,871,007 2 9 31 4,806,440 1 11 58 4,491,478 2 2 37 8,788,523 0 3 01 290,808 0 3 28 37,300* i 82 12,559,885 20 6 36 35,543,139 21 5 32 49,014,995 17 10 15 71,367,924 19 2 11 22,247,394 18 6 39 2,527,879 ent. - Per cent. Per cent. Per cent. Per cent. Per cent. 58,329 163,525 - 220,241 - 332,393 - 136,654 - 16,740 79 -15,573,797 91-94 42,499,178 92-37 60,278,062 90-53 85,077,373 92-35 34,785,712 93 03 4,599,929 267-00 259-89 - 27369 255-95 - 254-55 - 274-79 21 - 1,393,037 8-06 3,728,209 7-63 4,976,410 9-47 8,898,871 7-65 2,879,552 6-97 344,640 2388 22-80 - 22-60 - 26-77 - 21-07 - 20-59

- 17-18 - 17-48 - 16-52 - 20-41 - 14*75 - 13-60

- 229-30 227-18 - 226-02 - 261-25 - 187-78 - 186 90 - 0/9/4-73 0/9/6 61 - 0/9/9 86 - 0/10/7-30 - 0/8/9-17 - I 0/8/2.49 - 125/8/3 - 124/2/1 - 134/8/2 - 135/15/4 - 111/11/0 - 112/15/4

* Losses.

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MINES, COMPRISING ABOUT 95 PER CENT. OF TOTAL PRODUCTION. YEAR ENDING SEPTEMBER 30, 1923.

(6) (7) (8) (10) Lancs, North Staffs, Bristol. Forest of Dean. Cheshire. Worth Wales. South Stafford and Salop. Cuinberland.

)er cent. Tons. Per cent. Tons. Per cent. ToDs. Per cent. Tons. Per cent. Tons. Per cent. Ton 25,660,922 348,753 1,960,232 - 1,876,418 361,461 8- 70 2,232,157 10-34 3'" )3,554 9.14 179,136 8-62 161,822 6-05 21,865 10-44 87 0-88 225011 2 -47 77,178 2 -31 45,221 0 -36 6P619 2 -92 10,566 2 -29 19 90-42 23p2O3,754 87-19 2,728,021 88-55 1,735,875 91-02 1,707,977 91 -03 329,030 87 -27 728 1-56 4-61 5-77 0-63 4-77 4 er ton. Per ton. -Per ton. Per ton. Per ton Per ton. d. E S. d. 1 E s. d. E S. d. S. d. y, S. d. -82 12 5-37 453 3 1 - 67 15,243,904 13 10-07 1,887,707 9 7-77 837,324 15 2-31. 1,297,402 14 8 1 242,419 2 2-39 2,551,051 2 5-38 333,991 2 5-26 211,644 2 5-83 212,290 2 1-84 35,427 2 1-54 77

i 3 2-09 3,682,972 2 3-38 31.1,265 2 9-72 243,878 3 4-00 284,671 3 4-19 55,097 2 6-15 91, 0 1.10 106,602 0 1-14 12,988 0 1-13 8,158 0 1.09 1 7,786 0 1-10 1,507 0 1-15 3 0 5-59 540.,170 0 4-55 51,737 0 7-17 51,842 0 11-33 80,652 0 4-51 6,178 0 3-88 11

9 0-84 22,124,699 19- 0-52 1 2,597,688 15 7-05 1,352 846 22 0-56 1,882,801 20 8-46 340,628 17 6-09 637

7 0 1-74 168,113 0 2-85 32,509 0 0-39 2,807 0 0-54 3,902 0 0-06 77 0 2-35

8 11-10 21,956,586 18 9-67 2,565,179 15 6-66 1,350,039 22 0-02 1,878,899 20 8-40 340,551 17 3-74 630 0 3-01 290,808 0 3-28 3 77,300 0 6-38 46,132 1 10-29 158,578* 0 2-79 3,820* 1 5-87 54 - 9 2-11 22,247,394 18 6-39 2,527,879 15 0-28 1,303,907 20 1- 3 1,720,321 20 5 61 336,731 18 9-61 684

Oercent. Per cent. Per cent. Per cent. Per cent. Per cent. 136.654 16,740 7,840 10,564 2,213 4 92-35 34,785,712 93-03 4,599,929 93-81 2,197,791 88-32 2,751,213 95-93 615,55f) 96-18 1,135 254-55 274-79 280-33 - 260-43 278-16 281, 7-65 2,879,552 6-97 344,640 6-19 145,044 11-68 363,721 4-07 26,142 3-82 45 21-07 20-59 18-50 - 34-43 11-81

14-75 13-60 17-84 13-64 - 11-74 14,

187-78 186-90 250-03 177-62 163-34 207, 0/8/9-17 0/8/2-49 0/7/7-44 0/9/5-18 0/7/10-52 0/7/11 111/11/0 112/15/4 106/16/0 122/16/3 109/10/11 112/1

Losses.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions PINGSEPTEMBER 30, 1923.

(10) (11) (12) (13) Total. Bristol. Forest of Dean. Somerset. Kent.

Per cent. Tons. Per cent. Tons. Per cent. Tons. Per cent. Tons. Per cent. Tons. 8 361,461 - 834,594 - 958,025 499,710 __ 259,143,188 2 6.05 21,865 10-44 87,100 7 88 75,448 11 43 57,136 6 -12 15,850,926 9 2-92 10,566 2 29 19,127 2 72 26,066 1*29 6,431 2 33 6,030,402 7 91 03 329,030 87-27 728,367 89-40 856,511 87 28 436,143 91-55 237,261,860 4.77 - 4.75 5-28 3-53 - 552

Per ton. Per ton. Per ton. Per ton. s. d. ? s. d. ? s. d. ? s. d. ? s. d. ? 2 14 8 82 242,419 12 5-37 453,308 12 2-43 522,575 14 3-53 311,711 12 0-39 142,746,534 0 2 1-84 35,427 2 1-54 77,516 1 10 51 80,325 2 2 14 47,509 2 1-77 25,477,958

1 3 4-19 55,097 2 6-15 91,506 2 710 111,005 2 3-52 50,007 2 802 31,657,637 6 0 1-10 1,507 0 1-15 3,477 0 1-12 4,016 0 1'14 2,078 0 1 09 1,078,202 2 0 4 51 6,178 0 3 88 11,771 0 8-38 29,912 0 8-68 15,781 0 6(41 6,333,853

1 20 846 340,628 17 609 637,578 17 554 747,833 19 701 427,086 17 5*68 207,294,184

2 0 0-06 77 0 2 35 7,110 - -- 0 0 82 1,492 0 1 28 1,266,193

9 20 8 40 340,551 17 3.74 630,468 17 5-54 747,833 19 6 19 425,594 17 440 206,027,991 8* 0 2-79 3,820* 1 5-87 54,234 1 8-63 73,6127 0 7-46 13,564* 2 0-87 24,585,604

1 20 5 61t 336,731 18 9161 684,702 19 2 17 821,460 18'10 73 412,030 19 5 27 230,613,595 - ___I______'_I _

Per cent. Per cent. Per cent. Per cent. Per cent. 1,091,741 4 - 2,213 - 4,029 - 4,934 - 1,820 - 3 95-93 615,559 96318 1,135,259 94-94 1,394,093 91-80 496,415 92-17 289,22'i,357 264-92 - 1 278-16 281-77 - 282 55 - 272-75 1 4 07 26,142 3-82 45,045 5 06 74,292 8-20 44,327 7-83 24,553,411 11-81 - 11'18 - 1506 - 24-36 - 22-49 17-92 11-74 - 14-70 - 1374 20-13 - 237-37 - 163-34 - 207*15 - 194 17 - 274-57 - - - 0/9/10.45 8 - 0/7/10-52 - 0/7/11.83 - 0/7/596 0/12/6.70 3 109/10/11 - 112/10/3 - 105/18/3 - 171/5/5 - 130/15/0

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in WesternEurope. 427

The Periods Found. The periodogram shows a number of peaks of high intensity. The greatest, near I5.250, reaches 76 I7, while seven others (at or near 5*IO0, 5*667, 9*750, II*000, I2*840, I7*500, 20*000) exceed 30 oo. There are other peaks which, though lower in themselves, stand out conspicuouslyfrom their surroundings; such are those at or near 3'4I5, 35-000 or 36000, 54o000. There are yet others which will be found to call for consideration. The best plan will be to go over the periodogram, from the beginning at two years (q7- 150) to the end at eighty-four years (q - 3 6). As might be expected from the nature of the observations used the general level of the first part of the periodogram(from q-= I50 to q = iio) is very low. The average at the fifty odd intensities calculated in this part is barely i0oo. There is no definite peak. The periods near 2'2o and 2 40 years suggested by Mr. Baxendell do not appear, or are represented at best by quite insignificant rises; there are some indications of the latter period in the second half sequence, but there is nothing that can be called important evidence. It was hardly to be expected that annual data with a definite connection between each pair of successive years would throw much light on periods of less than three years. This expectation is fulfilled. The fact that a definite peak does show just after q = 110 is passed, at 2735 years, is all the more significant. 2-735 years.-The shortest of the periods inferred by my arithmetical analysis in the Economic Journal was one of 274 years with a maximum phase (in the present notation) at 1848-43. The first definite peak on the periodogramis at 2 *735; with a maximum phase at 1848 *48. The agreement between the two results is astonishingly close; unfortunately, it does not prove the reality of the period, but only that mathematics and arithmetic have read the same facts in the same way. The peak at 2 735, though it stands out well, being about six times the average of the neighbouring fifty intensities, is low in itself (7 *82), and the periodall but disappears in the second half sequence. So far as it remains, the phases in the two half sequences agree. There is a probable but not certain change of sign in both a and b. The corrected phase is 1848 *11. As was stated in my former article there is a good deal of meteorologicalevidence for a cycle of just undertwo and three-quarter

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 428 BEVERIDGE-Whteat Prices and Rainfall [May,

years in lengthh* For this reason, and having regard to (a) the prominence of the peak itself in relation to its surroundings; (b) the close agreement between my mathematical and arith- ruetical results, I have provisionally retained the period in my general table, though I have not used it in the final comparison with rainfall. 3*415 years.-The periodograin rises to 15*53 at 3*412 and 15-84 at 3-417; the actual peak is taken as lying between these points at 3-415. This is confirmed by analysis for the half sequences, which shows the period as active in each half (though more emphatic in the first half), and as greater than 3-412 and smaller than 3-417. Both a and b almost certainly change sign in the immediate neighbourhood.t (b 2-240 320 The phase for 3 417 is given = + 307 tan 3240 as 3 07 years from the origin 1541P67, i.e. at 1544*74, yielding a subsequent maximum in 1848 84. For 3*415 years the corresponding maxima are 1544184 and 1848-74 (uncorrected). The corrected maximum is 1848 *45. A period of something over three and one-third years is stated by Mr. J. Baxendell to have been prominent formerly in the rainfall of various parts of England, but recently to have become very feeble. Periods of about three and a half years have been traced by a good many writers in various meteorological records, but have generally been identified by them with one-third of the sunspot cycle of eleven years. 4-415 years.-The periodogram rises to 13-ii at 4-412 years and i6 48 at 4 417. The actual peak presumably lies between these points and probably nearer to the later one. This is confirmed by comparison of phases for 4-417 in the separate halves. The period shows a good deal more strongly in the first than in the second half, but there is a substantial intensity in each case. The phases in the two halves show some irregularity. Both a and b change sign in this neighbourhood (between q = 68-9 and q = 67.9). b The phase for 4-417 is given (by + 3 847 =19_) tan 770 a + - 866 -tn701'

* To the references there given may be added Sir Napier Shaw's paper on " An Apparent Periodicity in the Yield of Wheat in Eastern England; 1885-1905 " (Proc. Roy. Soc., p. 1906, Series A). t b changes certainly, and a having fallen from + 3-108 at q = 87-5 to (027 at q = 87-1, almost certainly does so.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions SCALE OF INTENS[TIES 80

0 150 140 130 120 110 Periods (in years). e

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I elIr'

120 110 100 90 80

C:. P-4ul1

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions DF WHEAT PRICE FLUCYUATIONS (1545 to about 1844).

80 70 60 50 40 - hO co 0 e P-4 _ ell CO co - o I* co a I* 0 i 1W U) U) a b X

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 50 40 30 20 10 0 O t-0b O O 0o or0 0O koED O0 o O Qg O0 coO

X0 0 _0 > > CI 0 0 0

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in WesternEurope. 429 as *95 from the origin 1544 67, i.e. at 1545 62, yielding a subsequent maximum at 1850 37. For 4 415 years the corresponding phase is *67 earlier, i.e. at 1850-30 (uncorrected). The corrected phase is 1850 07. A period of almost exactly this length has been found by Mr. Baxendell in the rainfall at Southport and Bolton. He puts the length at 4*42 with the last maximum as falling in May or June, 1921. This would bring another maximum to about 1850* 60. There is, therefore, good agreement of phase as well as length with my results. The period now shown by harmonic analysis agrees closely, but not exactly, in length with that of 4*374 years which I deduced by arithmetical methods in my former article. Obviously the two methods were getting at the same facts. The period, as now determined, remains an almost exact sub-multiple of 30 6, and thus a factor in the productionof the 15j-year cycle (7 x 4 - 415 = 30 *905). The difference, however, of less than 4 years (or i per cent. of the period)is quite sufficientin the course of seventy odd revolutions to bring the phases out of relation to one another, and to destroy the connection which I suggested before in phase as well as length between my period and that of 4 38 years discovered by Professor Schuster in the sunspots. The last maximum of this sunspot period would have fallen in 1918* 90, almost at the minimum of the rainfall period as now determined. This is not in itself conclusive against a connection between the two. On the whole, however, any such connection must now be regarded as highly improbable. 5 100 years.-The periodogramshows a well-defined and very high peak (42 34) at 5@100 years, with large intensities (34 05 and 57. 09) in each half-sequence. Both a and b change sign between q= 58 9 and q - 585. * The first maximum phase is b _ +2-980 given (by - - + 2 9__ = tan 270 33') as *39 years from the origin a + 5-712 1544 67, i.e. at 1545*06, yielding a subsequentuncorrected maximum at 1845 96. The corrected maximum is 1845 76. If there were no true period here, harmonic analysis would indeed be a sorry guide. Fortunately, there is no room for doubt. A period of just this length and closely agreeing in phase has been found independently not in one but in three or more other records,

* This does not appearin the table as printed because the amplitudefor 5.125 (q = 58 5) there shown is calculated with a different origin. If the sanmeorigin is retained b becomesnegative at q = 58 5. VOL. LXXXV. PART III. 2 a

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 430 BEVERIDGE-Wheat Prices and Rainfall [May, by Mr. Baxendell in the direction of the wind and in the rainfall at Southport and elsewhere, and by Captain Brunt in the temperature at Greenwich. It is inconceivable that in each of four distinct analyses the same period should appear by chance or unless there was the same reality behind all four appearances. The exact length assigned by Captain Brunt to the period in the Greenwichtemperatures is 5 09 years, with a maximum phase at the end of April, 1843, i.e. 1843-33, so that the minimum would be at 1845.88.* The next maximum of rainfall is given by Mr. Baxendell as due in February, 1923, i.e. 1923 12, which would correspond to 1846-62. The phase given by the wheat prices coincides almost exactly with the minimum of temperature in Captain Brunt's analysis, and is earlier than the maximum of rainfall as found by Mr. Baxendell. In respect of this period, harmonic analysis confirms completely the conclusionreached in my earlierarticle by arithmetical methods. I then named a period of 5.11 years, with a maximum phase near 1709 00, in crop years, i.e. 1708 67 in calendaryears. The periodogram now shows a period of 5S10 years, one of whose maxima would fall at 1708 16.

For the next three and a half years, to about 8 5 (q = 35), the periodogramhas a succession of peaks whose interpretation is not altogether easy. One of these, near 5 417, is discussed below in connection with the 11-year period; of the rest, only the four highest, near 5 667, 5 933, 7.417 and 8 091, seem to- deserve independent consideration. Most of the others-as at 5 800, 6 - 200, 7-000 and 7-200-are markedly lower, or are found only in one half of the sequence. 5-671 years.-At 5-667 the intensity is high (32 72), and the period is found equally strongly and with continuity of phase in each of the two halves. Though the calculations made do not show an actual change of sign in both a and b, they suggest it as highly probable. Comparison of phases in the two halves

* Captain Brunt's cycle is not symmetrical,i.e. the actual maximum of temperaturedoes not occur exactly half-way between two minima but about six months later. The minimum, however, is much more strongly marked than the maximum,and the assumptionmade above that the maximumphase as given by harmonicanalysis will be about half-waybetween the actual minima appearsbo be justified. There is also a secondaryminimum occurringabout two years after the main one.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in WesternEurope 431 suggests a length for the period slightly in excess of 55667 ; the figureadopted is 5 671. The maximum phase for 5'667 is given b 1- 966_ -_ = ___96 = tan 3390 41') as 5 35 years from the origin (by a + 5-312 1544-67, i.e. 1550 02, yielding a subsequent maximum in 1850*35. The correspondingphase for 5 - 671 is 1850 46. The correctedphase is 1850 28. There is no demonstratedparallel to this period in meteorological records. Many writers, however, have found, or have thought they had found, cycles of half the sunspot period of eleven years; the most precise of those in recent years is perhaps G. Hellmann, who, in the rainfall of Europe from 1851 or 1905, traced a periodicity of between five and six years.* 5 960 years.-At 'a933 the intensity is 23'63 for the whole sequence, and 29*48 and 33*07 for the separate halves. The progression of phases between the two halves, both here and at 6*000 years, suggests a period of about 5*960 years, with an intensity presumablywell over 30. Changeof both signs is probable but not certain and may be prevented by interference. The phase 121 for 6 -000 years is given (by b _+ = tan 181? 46') at 3 08 a + 3-503 years from the origin 1544* 67, i.e. at 1547* 75, yielding a subsequent maximum at 1847* 75. For 5 *960 the corresponding phase is 1846 75. The corrected phase is 1846 58. A six-year period, or rather one of just under six years affecting barometric pressure, has recently been advocated by J. Schneider. He claims that it occurs in the pressure records at certain stations in Europe and in Batavia, and connects it with the movement of the moon's nodes. Traces of a six-year period were found also by Simon Newcomb in 1908, in the temperature of the tropics.t The period is on the borderland; on the whole, in view of its markedcontinuity and agreementof phase in the two half-sequences, T am inclined to believe in its reality. It has, at any rate, better

* Schwankungen der Niederschliige, by G. Hellmann (in publications of Prussian Meteorological Institute, 1909). Cp. also Helland Hansen and Fridtjof Nansen: Temperaturochwankungendes Nordatlantiochen Ozeans und in der Atmosphdre (1917) for many references to the " sub-division " of the eleven. year period. t J. Schneider: Die Meyer-Seemannscbe Luftdruckschwankung im Lichte einer sechsjihrigen Mondperiode (Annalen der Hydrographie und maritimen in mneteorologie,1918, January). Schneider cites Angenheister as his authority for the Batavia period. Newcomb's paper is in the Transactions of the American Philosophical Society for 1908 (A Search for Fluctuations in the Sun's Thermal Radiation through their Influence on Terrestrial Temperature). 2 G 2

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 432 BEVERIDGE-Wheat Prices and Rainfall [May, claims than the 7 417 and 8050 year l)eriods Wvhichare next to be described. 7-417 years.-The periodogram rises above 20 again at 7-417, reaching 2I * 72, and having good intensities in each separate sequence (290o6 and 29@ 54). Both a and b change sign between q= 40 0 and q = 40 9. The phases in the two halves, however, are not continuous; there is either serious interference by another period, or change in the length of the period, or an abrupt shift of phase. The determination of phase in these circumstances presents a difficulty. Using the figures for the second half-sequence only, we get an uncorrected maximum in 1845 35; if the whole sequence be used, the date is 1846-00. The corrected phases are 1845-22 and 1845 87 respectively. The nearest direct meteorological parallel that I have been able to find is the period of about 7.3 years, which seems to occur in the growth of trees in Arizona.* On the other hand, the period is almost exactly twice the length of the period 3-71 years, which many observers have found in the sun and in terrestrial meteorology.t In the EconomicJournal I named a cycle of 3 71 years as one of the four shorter ones suggested by my arithmetical analysis ; alternate maxima of this cycle would coincide with the maxima of the 7 *417 period. On the whole, I incline provisionally to regard this period as real, though its abnormalities prevent me from using it in my final comparison with rainfall. 8X050 years.-The periodogram rises at 8-ooo to i8-67 and at 8-o9I to 23-23 ; the peak is assumed to lie between these points, at 8 050. Both a and b change sign between q = 37 - 5 and q = 36 - 6. The period owes its whole strength to the second half (intensity for 8091 =42 98) ; in the first half it is feeble (9 52), but shows fair agreement of phase. The first maximum for 8 091 is given (by

=- 4- 15 = tan 3470 12') at 7 *76 years from the origin 1513 *67, i.e. at 1521 43 yielding a subsequent maximum in 1845 23. For 8 050 the corresponding phase is 18445 33. The corrected phase is 1844 21. An eight-year cycle has been found by different writers in a variety of meteorological and other records. Professor F. H. Bigelow in 1901 mentioned a cycle of this length as occurring in barometric pressure over the United States. Dr. Otto Petterson, I believe, has

* A. E. Douglass: Climatic Cycles and Tree Growth (1919). t References are given in my earlier article, p. 442.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in Western Europe. 433 found it, also as a pressure cycle, in Europe. J. Maurer in 1918 showed it, very clearly, in the winter pressure over the Alps between 1865 and 1918. Finally, Professor H. L. Moore, finding it first in the rainfall of the Ohio Valley and subsequently in crop yields and prices on both sides of the Atlantic, is apparently inclined to treat it as the dominant cycle in meteorology, and has recently sought to explain it by reference to the well-known eight-year interval between transits of Venus. In Europe at -least the cycle clearly has not anything like the importance attributed to it by Professor Moore. The evidence offered by him of its operation in Europe is indeed far from convincing. My own figures, while they support the view that such a period exists, show also that it became important only in modern times, and was inoperative during the greater part of the seventeenth century. On the other hand, it seems to have gained strength steadily for the last 200 years. Its provisional acceptance as a real though subordinate cycle seems to be justified.*

* The following table gives the amplitudes and phases of an eigbt-year trial period in each of five sequences of 64 years from 1545 to 1864

Sequence*. r2. 0. I. 1545-1608...... 21*58 .... 1250 II. 1609-1672 ... 2 72 .... 236? III. 1673-1736 ...... 27 *44 .... 147? IV. 1737-1800 ...... 40*16 .... 1260 V. 1801-1864 ...... 78 68 .... 2040

It will be seon that the period has been important only since the middle of the eighteenth century, and that for the greater part of the seventeenth it was inoperative. The phase has moved backwards and forwards in the neighbourhood of 1700. The eight-year period of pressure in the Alps is described by J. Maurer in the Meteorologi8cheZeitschrift for 1918, the minima being at 1870-1, 1878-9, 1885-6, 1894-5, 1903-4, 1910-11 and 1916-17 (?), and the maxima at 1865-6, 1873-4, 1881-2, 1889-90, 1897-8, 1904-5, 1912-13; by other records the series of maxima is carried back to 1818-19. The phase agrees sufficiently well with that of the eight-year cycle in wheat prices (assuming a maximum in pressure to be represented by a minimum of prices and vice ver8d), but the length of the period seems to be just below rather than just above eight years. Unfortunately the writer gives no figures, but only plotted points and a freehand graph, the drawing of which is not above criticism. Professor Moore's writings on this cycle are contained in a short book (Economic Cycle8: Their Law and Cause, published in 1914) and in a series of articles in the Journal of the Royal Stati8tical Society for May, 1919, and May, 1920, the Political Science Quarterlyfor June, 1920, and the Quarterly Journal of Economics for February, 1921, August, 1921, and November, 1921. The

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 434 BEVERIDGE-Wheat Prices and Rainfall [May, 9 *750 years.-At 9 *750 years is a well-definedpeak (33 89), and each half-sequence yields a high intensity (38.44 and 29 72) with continuity of phase. Both a and b change sign between q!= 30 b 3-3560 The phase is given (by - +_ - = tan 3210 25') and q 31. a + 4-462 at 8-71 years from the origin 1544-67, i.e. at 1553-38, yielding a subsequent maximum at 1845. 88. The corrected phase is 1845 78. No certain meteorologicalparallel to this period can be traced. One or two writershave traced a period of just over ten years in the rainfall. More important and better fitting my requirements is the period of about 9 5 years which Captain Brunt in 1919 found indicated in the Greenwich temperatures. The evidence of my periodogram is clear and consistent, and stronger than in several other cases where confirmationby meteorological records compels belief. Provisionally, therefore, I have felt bound to treat this period as real, though differing from other periods as apparently not influencingthe rainfall. 11000 and 5*4 years.-With these periods we come to one of the classic mysteries of cosmical meteorology. My periodogram shows at 11a000 years the large amplitude of 33 84. Both a and b change sign in the immediate neighbourhood. There seems at first sight to be no question that the eleven-year cycle, which has dominated the sunspots and the minds of meteorologists for so last of these contains a summary of evidence and an explanation of the cycle, by reference to the planet Venus. Much the strongest evidence to my mind is the analysis of the Ohio rainfall in the book; the number of years covered- from 1839 to 1910-is small, but over the whole of that sequence the eight- year cycle -stands out clearly. Professor Moore finds the same period in the crop yields of France and the United Kingdom, but the number of observations (39 years) is altogether too short for profitable harmonic analysis. The only long record used by him-English wheat prices from 1760 to 1875-is, as has been stated above, biassed in favour of an eight-year cycle. The analysis of Sauerbeck's index-numbers from 1818 to 1913 can only be made to show an eight-year period by the heroic expedient of taking the average between two peaks at 8*7 and 7*4 years; the actual amplitude at 8 0 years is conspicuously small. If I understand Professor Moore rightly, he identifies a high yield of crops with a heavy rainfall in Western Europe as well as in America; this is contrary to experience. For the reference to Professor Bigelow's paper (Report of the Chief of the United States Weather Bureau 1900, 1901) I am indebted to one of Professor Moore's articles, and for the reference to Dr. Petterson's paper to an article by J. Schneiderin the Annalen der Hydrographieund MaritimeMeteorologie for1918, Heft I.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in Western Europe. 435 long, must be accepted as one of the principalfactors ill the harvests and the weather of Western Europe.* At the test of continuity, however, the eleven-year cycle in wheat prices breaks down. Its intensity in the first 154 years (1545-1698) is 96 oi; in the next 154 years (1699-1852) it is only 3 47. The importance of the period is seen to be due entirely to its vigour before 1700. From thence to the middle of the nineteenth century it is almost inoperative. Its history can be followed in detail in the following table, showing amplitudes and phases for five successive groups of sixty-six years each from 1500 to 1829, with a sixth, slightly overlapping group, 1804-1869.

Intensities and Phases of certain Periods in successive groups of sixty-six years.

11.000 Year Period. 5 500 Year Period. Group. Intensity. Phase. Intensity. Phase.

I 1500-1565 .... 56*68 3380 103*53 400 II 1566-1631 .... 70*83 2690 25*68 1900 III 1632-1697 .... 213 88 2280 0*81 2960 I T 1698-1763 .... 13*63 140 58*65 2970 V 1764-1829 .... 29-48 2450 134*80 1770 VA 1804-1869 .... 5.00 560 76*84 850

The amplitude, which is considerable in the first group, rises yet further in the second, and is very great indeed in the third, ending with the seventeenth century. In the next group, 1698-1763, the cycle has gone; a small amplitude remains, but the phase has changed completely. Group V witnesses a small recovery and a return towards the normal phase. There is a regression of phase in the first three groups indicating a length just under eleven years; in agreement with this, the amplitude at 10-8 years over the whole sequence 1545 to 1852 is found to be much greater than that at 11 2. If we insist on the best period to fit the figures we get something like IO093. The sunspot period, on the other hand, was determined by ProfessorSchuster in 1906 and by Mr. Newcomb in 1901 as definitely

* The eleven-year cycle is so well known that I forbear to give references. Apparently it was first named by Schwabe in 1844, as occurring in the sunspots, and was traced in terrestrial magnetism by Sabine in 1852, and by Baxendell (senior) in various meteorological data in 1864.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 436 BEVERIDGE- Wheat Prices and Rainfall [May,

more than eleven years; in the middle of the nineteenth century, io * 8 was a favourite figure, and helped to lead Jevons to untenable conclusions. Latterly Professor Turner favours a period of fluctuating length. It is safer to avoid refinements and to speak of an eleven-year cycle. Yet in spite of or because of these peculiarities the period near eleven years remains one of the most interesting features of our periodogram, and one whose connection with sunspot activity seems highly probable. To begin with, the maximum phases in my periodogram and as given by Professor Schuster for the sunspots are in agreement. b 4-4260 The first maximum for 11 *0 is given (by - = 849 =- tan 2270 55') at 6 96 years from the origin 1544l 67, i.e. at 1551 62, which yields a subsequent maximum at 1870 e 62 (uncorrected) or 1870 - 53 (corrected). Professor Schuster gives 1870 4 as the maximum phase of his period (including the harmonics) or 1872 0 if the fundamental period be considered alone. In the next place, the eleven-year sunspot cycle shows the same alternation of great activity and insignificance. Professor Schuster, in his analysis covering 150 years from 1750 to 1800, examined separately the first seventy-five years (1750-1825) and the next seventy-five (1826-1900). In the latter, the eleven-year period came out in full strength; in the former, it was all but invisible. These seventy-five years fall within the time when the period was equally invisible in wheat prices. In the third place there seems to be a curious arithmetical relation between the fluctuating length of the sunspot period and another cycle suggested by my periodogram, to which a length of 5 423 years may be assigned, but which it is prudent to specify less exactly as 5 4 years. This cycle, in direct contrast to the 11 000-year cycle, is almost invisible before 1700 and very strong after it. The contrast is brought out still more clearly by the table for five successive groups of sixty-six years. As the amplitude for 11 0 years rises to a great height and then falls, that for 5.5 (representing 5 4) falls and then rises. The culmination of the eleven-year period between 1632 and 1697 corresponds to a total disappearance of any amplitude for 5.5 years. Now 5-4 is definitely not half of 11 0, still less of 11 125, the figure ultimately reached by Professor Schuster for his period. But it is all but exactly half the length of the actual sunspot period throughout the eighteenth and half the nineteenth centuries. In giving the sunspot maxima from 1615 to 1894 Professor Scbuster

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in Western Europe. 437 calls attention to the fact that from 1675 to 1848 the average length of the sunspot period was consistently about 10 8 (more accurately it was 10-82). It was only the long intervals between 1625 and 1675, and after 1848, that brought up the average length over all to more than eleven years. In other words, over 170 years fairly corresponding with the reign of the 5-4 period in wheat prices, the average length of the sunspot period is exactly twice that and well under eleven years. Before that, when the eleven-year period is dominant in wheat prices, and after, when it is dominant in sunspots, come the longer intervals that bring up the average. My earliest figures of all point to yet one more movement of this alternation with the 5 4 period dominant and the 11 0-year period recessive in the first half of the sixteenth century. Whether there is physical reality behind this apparelit relation of the 11 0-year and the 5 4-year periods, and if so of what nature, I cannot pretend to say. For practical purposes it is clearly unsafe to treat either period as operative at the present time. 12 050 years.-There are fairly high intensities at 12 000 years (23 .30) and at 12 143 years (2I * 66), and there is certainly a change of both signs between q = 25 7 and q - 24 7. The intensities alike for 12 000 years and for 12 143 years are good in each half- sequence. The progression of phase for the two periods agrees in indicating a period slightly in excess of 12 143. On the other hand, the intensity at 12@000 is slightly higher, and so would suggest a true length slightly under 12 - 070. The most consistent interpretation of the periodogram is for a period of about 12 050 years with a sudden change of phase at some point in its history, amounting to a year or more. For 12 - 050, the uncorrected phase, derived from 12 000 in the second half-sequence, would be 1841 98 and the corrected 1841- 90. It is possible, however, that this period is continuous, but is subject in the periodogram to interference by the much stronger neighbouring period at 12-840. There may be a meteorological parallel to this in the periodicity of twelve years traced by Balfour Stewart in 1880 in the rainfall of England and Paris.* Balfour Stewart's paper does not make it possible to determine the phase of his period, so that the parallel must be regarded as open to question. On the whole I have thought that the period just merited inclusion in my list. 12-840 years.-On the evidence of the periodogram the period between i2 8oo and I3o000 years ranks second only to that near

* Proceedings of Manchester Literary and Philosophical Society (February 24, 1880), on the " Long Period Inequality in Rainfall," by Balfour Stewart.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 438 BEVERIDGE-Wheat Prices and Rainfall [May,

I5'250. Its intensity alike in the whole sequence (46'oo +) and in each half (44.82 + and 72' i6 +) is greater than that of the established 5'1 year period, and it does not show the anomalies which mear the period near I7'500 years described below. Both a and b change sign between q 23' 1 and q = 24'0. The exact length of the period is open to argument: clearly it lies between I2' 8oo and I2*875. For the length taken (I2'840) the maximum phase (derived from that for I3'000) is 1840'27 (corrected). The corrected phase is 1840'19. I have not myself discovered any meteorologicalparallel to this period, but Mr. Baxendell informs me that a cycle of about this length has been traced by ProfessorTurner in various meteorological data (1915). Moreover,he identifies the periodicity traced in 1880 *byBalfour Stewart in the rainfall at Padua and Milan with this period of mine rather than with the shorter one of just over 12 years mentioned above. In any case, the strength of the period in wheat prices compels provisional acceptance. 15'225 years.-My investigation began two and a half years ago with the appearance, in statistics of exports and of barometric pressure,of a cycle which I named as " between I5 *2 and I5 *4 years " in length. In Mr. Curwen's preliminary analysis much the most prominent feature was a period of about fifteen and one-third years, with a decisively high intensity and close agreement of phase with the cycle which I had named. In the fuller analysis now presented, involving the use of a slightly different sequence, this period is still the leading feature, with an intensity exceeding 76' i6, but a slightly shorter length, between I5'200 and I5.250 is suggested. Both a and b change sign between q = 19'7 and q = 205. The uncorrected phase for i5'250 is 1556'03, yielding a subsequent maximum in 1846'28. For i5 '225 the corresponding phase is 1846-03; the corrected phase is 1845'97. It is impossibleto doubt that this striking feature of the periodogram correspondsto some physical facts, and, in spite of the failure of meteorologists to discover an independent cycle of about fifteen years in weather records, such a cycle may exist. But I am inclined to attribute certainly the importance, and possibly the very existence, of the peak at I5Z 250 in my periodogram, not to any single cycle of that length, but to a combination of smaller cycles of lengths which are all close sub-multiples of I5'250 or its double. Of the seven cycles under eight years of length found in my present analysis, two (5 I00, 7.4I7) are close sub-multiples of 15, and four more (2*735, 3*415, 4-415, 5*960) are close sub-multiples of 30 or 31. With the single exception of 5'67I,

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in WesternEurope. 439 all the seven shortest cycles found by me have exact multiples between 29-7 and 30-9; the average of these six multiples is 30-37, which is almost exactly twice the main period of I5 225. In my former article I explained the period near I5*3 by the meeting of three minor cycles (2*735, 4*374, 5'II) near 30*6. Harmonic analysis has yielded many new short cycles, but, with one exception, every one of them is a close sub-multiple of the same interval. For this reason, while I think that meteorologists may still find a single period near I5 3 years, I have not used any such period in my calculations below, but have treated it as wholly the result of combination among lesser cycles. 17*-400years.-At I7'500 and I7'333 are intensities surpassed only by those near I5 250; and the natural course is to assume a real period between these points-say, at I7 400. The test of changing signs is satisfied. But examination of the two half- sequences yields puzzling results. For I7*500 we get high intensities in both halves (69.34 from 1545 to 1684, and 55 53 from 1685 to 1824). For I7.333 we get an extremely high intensity in the first half (I36- i9 from 1545 to 1700), followed by practical disappearance of the period in the next half (II-94 from 1701 to 1856). The possibility of arithmetical error has, I think, been excluded. The sequencefor I7 * 333, it will be seen, covers thirty-two years (1824-56) not used for I7 500; the explanation of the discrepancy between the intensities must apparently be that in those years the cycle disappeared or changed its phase completely. In any case, while it seems most improbable that intensities as high as those near 17 500 do not represent some physical fact of periodicity, uncertainty as to the precise length of the period (including, of course, its phase), and as to its persistence, has led to its omission from my final calculation. 19 900 years.-The intensity at 20000 years is high both in the whole sequence (37p88) and in each half (5o007, 23 97). Both a and b change sign between q = 14 3 and q = 15 2. The period is seen by examinationof neighbouringintensities, and of progression of phases, to lie between 20o000 and I9 750. The actual length shown, i9 9oo, accords fairly well with all the indications. The uncorrectedphase, derived from that for 20o000 years, is 1852 95; the corrected phase is 1852 90. No meteorological parallel has been found; but the evidence of the periodogramis strong. 35-500 years.-The periodogram rises to a well-defined peak between 35 000 (23.ii) and 36ooo (23' 29), indicating a period of about thirty-five and a half years. Both a and b change sign

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 440 BEVERIDGE-Wheat Prices and Rainfall [May, between q- 8 1 and q - 8 8. The uncorrectedphase for thirty-six + years.1 is given (by-= ? 851 -tan 108? 50') as i o 88 years from a ~~- 1- 654 the origin 1544a67, i.e. at 1555 55, yielding a subsequent maximum in 1843-55. For 35 5 years the correspondingphase is two years earlier, at 1841 55. The corrected phase is 1841 52. This result is of peculiarinterest and importance. Unquestionably we get here the well-known thirty-five-year cycle discovered by Dr. Eduard BrUcknerin 1890, as causing a regular alternation of dry and warm periods with wet and cold ones.* Dr. BrUckner traced his cycle, in the first instance, in records of temperature, rainfall, and barometric pressure during the past two centuries, and arrived at a length of thirty-six years, the centres of the first and last wet periods falling in 1700 and 1880 respectively. Examina- tion of records of hard winters, of the dates of the wine harvest, and of the freezing and thawing of Russian rivers, led him to fix ultimately on a smaller length, viz., 34'8 ? *7 years. If the true length of the cycle derived by me from wheat prices were thirty-six years, maxima would fall in 1699 and 1879-that is to say, all but exactly in the years named by Dr. Bruckner. For the actual length taken by me, 35.5 years, the maxima fell at 1699 02 and 1877 02. This length is just within Dr. BrUckner'slimit of error; consideration of earlier famines-in particular 1315-6-suggests that the true length may be less than 35.5 years, but cannot be greater. The agreement between the two results could hardly be closer, and is conclusive as to the reality and importance of the BrUckner cycle. Dr. Brucknerused throughout averaged figures for several years at a time-usually five; he deprecated assumptions of excessive mathematical accuracy in meteorologicalphenomena, and regarded 34 8 + .7 years as an average only; the actual distance between any two successive groups of wet years or of dry years might differ very considerably from the average. My own impression is that the BrUckner cycle itself is a good deal more regular than its discoverer supposed; the irregularities arise from the operation of other cycles of which he took no account. It is only necessary to add that since Dr. Brucknerwrote, his cycle has shown itself once more unmistakably in the dry years of the middle nineties and in the downpour which turned Flanders into mud during the years after 1912.

* Klimrnschwankungenseit 1700 (published in " Geographische Abhand- lungen," vol. iv, part 2, 1890, Vienna).

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.1 in WesternEurope. 441 54 00 years.-The group of high intensities between fifty-two and fifty-six years indicates a period in this neighbourhood. The highest intensity is actually at fifty-four years and this has a fairly central position in the whole band of high intensities. It is therefore taken provisionally as the length of the period. Both a and b change between q 5 and q = 6. The uncorrected phase is 1860-46 and the corrected one 1860-44. There appears to be a meteorological parallel in a period found by Mr. Baxendell in English rainfall and wind direction. In any case the intensity of the period in wheat prices is unquestionably significant. It is greater absolutely than that for the well- established BrUckner period, and having regard to its position in the periodogram is relatively even more noticeable. 68 000 years.-The periodogram rises to a peak for the last time at sixty-eight years, a and b changing signs within the required limits on both sides of this point. The intensity is only I3 58, and it is impossible to apply the test of continuity. The uncorrected phase is 1591*44, yielding a later maximum at 1863x44; the corrected phase is 1863x43. For so long a period an intensity even of I3 6o is probably significant. Though I have not, myself, found any record of such a period in meteorology, Mr. Baxendell informs me that there is certainly a period near sixty-seven years in air pressure. The period, moreover, is almost exactly one-fourth of the 271 years interval, which I described in my former article, and agrees closely in phase with the most striking instance of that interval- the series of years 1044, 1315, and 1586. The sixty-eight-year period (based on figures from 1522 onwards) has maxima in 1047, 1319 and 1591. It may be added that one of the best attested and most accurately dated of early famine visitations of Europe, that which preceded the famous Edict of Diocletian in A.D. 300, coincides exactly with an earlier maximum of this same period (A.D. 299). Three of the four next longest periods (omitting the uncertain I7'4 and I5.225) would have been at or near their maximum at this same date; The fifty-four-year period in the phase assigned to it has maxima in 1590, 1320, 1050 and 296. The i91i9 and I2*84 periods both have maxima in 300. On the whole the provisional acceptance of the sixty-eight-year period seems to be justified. It may be convenient here, in concluding the list of periods found, to refer to the 271 year interval mentioned above. Just as nearly all the short periods now shown by harmonic analysis are found to

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 442 BEVERIDGE-Wheat Pricesand Rainfall [May, be close sub-multiplesof the thirty to thirty-one years named by me before this analysis was begun, so many of the new periods, whether large or small, are found to have a common multiple near 271 years. These include 2*735, 5100, I2*840, 54*o, 68'o and the elusive 5 *4. This is a striking result; it is the more striking because the 271 year interval was largely deduced by me from famines occurringbefore 1545.

Summary and Review (with Table). The results of the foregoing survey are summed up in the table headed " Apparent Periods in Wheat Prices." I have there set out the intensities (in the whole sequence and in each half) and the phases for nineteen periods suggested by my analysis; I have added a brief reference to meteorologicalrecords in which the same periods have been found. In doing so, I have for convenience of referenceidentified the periods by letters of the alphabet, beginning with A for the shortest and running up to V. This involves a change of nomenclature from that adopted in my former article, when I used the letters A, B, C, and D to describe the four periods then suggested; the risk of present confusion is, I think, outweighed by the advantage of the change. I have left letters unoccupied at A, B and D, because from Mr. Baxendell's researches it seems highly probable that one or more additional periods exist and should ultimately be found between two and three and a half years. The immediate criticism that cannot fail to be made on this table is that there are altogether too many periods to be believed. In reply I certainly do not wish to assert that to every one of the apparentperiods shown in the table there correspondsan independent physical cycle. The position is rather as follows: (1) As to some of the periods in the table there is no room for doubt. In this class are G (5.Ioo years) and T (35.500 years). For the first of these we have a high intensity in the whole sequence and in each half-sequence of wheat prices, confirmedby discovery of the same period (showing agreeement of length and phase) by harmonic analysis of meteorological records. The second has long been admitted as one of the best established cycles in meteorology; its appearance with striking agreement of phase in my analysis makes assurancedoubly sure. For these two cycles the independent evidence both of wheat prices and of meteorology is strong and their agreement is conclusive. (2) For a second group of periods the evidence of wheat prices

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in WesternEurope. 443 is as strong as or even strongerthan in the first group, but confirma- tion from meteorologicalrecords is, up to the present, weaker or is lacking. These includelH(567I), M(9*750), P(I2*840), Q (I5*225), S (19.900), U (540ooo) and possibly V (68-ooo). It will be noted that with one exception (5.67I) for which there is in fact a good deal of rather vague evidence, all these are relatively long periods the comparatively small number of years for which good meteorological records are available makes it anything but surprising that the evidence for long periods should be weaker. It is difficult, without questioning the whole method of harmonic analysis, to doubt that there is some physical reality-whether it be a single cycle or a combination of cycles-behind these appearances of periodicity in wheat prices, just as there is behind those of the first group. (3) For a third group of periods there is some, but not first-rate, evidence both in wheat prices and in meteorology. Such are E (3.4T5), F (4 4I5), J (5.96o), and L (8o-50). The existence and persistence of physical cycles correspondingto each of these periods is highly probable, though perhaps not certain. (4) In a fourth group, which includes C (2 * 735), H (5 * 423), K (7 4I7), N (IIo000), 0 (I2.050), and R (I7.400), we meet with inconstancy of action, sudden changes of phase or other puzzling features. These periods will certainly repay further study; they can neither be accepted nor rejected definitely: they must be held in suspense. To them ought to be added perhaps a period indicated by the peak at twenty-four years. Finally, it may be asked whether there is really anything surprising in the assertion that our weather is governed by many different cycles. Why should we look for one or two alone? Thereis nothing in nature to justify this attitude. The movement of every astronomical body through space represents a compound of many cycles. The movement of a planet, even in relation to the Sun, is also the complex result of many influences; only because one of these influences-which used to be called the attraction of the sun-was predominant could we think of this motion or revolution in an exact ellipse about the sun. But astronomers knew better; if they had not allowed for other influences, i.e. for other cycles in the movement of the planet, due to small perturbing forces, they would have erred in their predictions. The trouble about the search for weather cycles in the past is not that it has been unfruitful, but rather that each observer found a different cycle and between them they found too many. Many of them of course were over-sanguine or ill-equipped; many of

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 444 BEVERIDGE-Wheat Prices and Rainfall [May,

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This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 446 BEVERIDGE-Wheat Prices -andRainfall [May, them, however, were not. Is it not at least as likely that many of these observers were right, as that they were all wrong ? Is it not possible that the complexities of our weather are due to its being governed, not by chance, but by a considerable variety of regular influences which sometimes assist and sometimes neutralise each other ? I shall endeavour now to present you with a result proving that this is not merely possible but certain.

IV.-COMPARISON WITHEUROPEAN RAINFALL, 1850-1921. On the evidence already submitted it is, I believe, impossible to doubt the reality, importance and persistence for long intervals of time of a considerablenumber of meteorologicalcycles (probably not less than twelve in all) affecting wheat prices. This is, however, so remarkable a result that any a prior argument, such as that used up to the present, is bound to be unconvincing. The ultimate proof of any period is in prediction, and to the lay mind lies in that alone. I do not propose, to-night, to make a prediction of the weather for the next twenty years, or to invite you to wait here to see if the prediction is fulfilled. It is, however, possible to do something not quite as convincing, but of the same nature, without prolonging this meeting unduly. With very few exceptions the numbers used in my analysis relate to the time prior to 1850, for which accurate meteorological records are scanty. Between 1850 and the present time we have a stretch of seventy years for which such records are abundant. If the cycles derived from my analysis of wheat prices before 1850 are real and persistent, they must have been operating after 1850. I propose to test their reality in the severest possible way by constructing a curve to represent so far as possible what should have been their combined effect between 1850 and 1923, and comparingthis curve with meteorologicalrecords for that period. It would clearly not be justifiable to use for this purpose, without further evidence, every one of the nineteen cycles included in my table. The six cycles which I have put into my last group (C, H, K, N, 0, and R) are too unstable or too uncertain in respect of their precise phase or length. One (Q) represents probably a combination of other cycles rather than an independent cycle. Another (M), though there is little or no doubt as to its reality, is left out of account in the first instance for the reasons given below. The whole of the eleven remaining cycles (E, F, G, I, J, L,

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.3 in WesternEurope. 447 P, S, T, U, V) have been combined by a simple graphic method. Each of them has been represented by a symmetrical curve of the same amplitude, and convex below so that the maxima have the form of cusps. These curves have been plotted, with the exact lengths and corrected phases yielded by my harmonic analysis, in rows above one another on a horizontal scale covering the time from 1850- 25 to 1923 75; their ordinates have been measured and added up by a divider for two dates, April 1 ( 25) and October1 ( 75) in each year. The mean of the sums of the ordinates for the two dates in each year has been taken as the figure for that year; these means give the graph shown in the upper half of Chart B*, which, for convenience, I shall describe as the " synthetic curve." How is this synthetic curve to be interpreted ? With what meteorologicalrecord is it. to be compared? The success or failure of a harvest depends, of course, upon a variety of factors, and the importance of different factors-rain, heat, cold, drought, storms, sunshine or the lack of it-varies with the country, the crop and the season of the year. The data now under consideration refer to wheat and to Western Europe alone; with those limitations it is clear that if we are looking for a single factor which is uniformly adverse to a good harvest, or whose absence is uniformly beneficial, we shall get nearest to finding it in rain. The ability of wheat in Western Europe to withstand the severest drought has been well illustrated by the experience of 1921.t If, therefore, we are to find a parallel to the curve just constructed in any single meteorological element, we should find it in the rainfall. The peaks of the synthetic curve should correspond to times of heavy rain; the depressionsto times of drought. In the lower half of Chart B I have drawn accordingly a curve representing the rainfall of each year from 1850 to 1920 over an area correspondingas nearly as possible with that covered by my table of wheat prices. This rainfall curve has no pretence to finality; my paper has been prepared under conditions of great pressure, and I have used the most readily available statistics. The chart is based on returns for ten stations in the British Isles, six in France and Belgium, and eight in Germany and Austria. Each of the three groups is shown separately; I have drawn also a single line representingthe unweighted mean of the three groups.1 * Facing p. 452. t See SupplementaryNote A. + The recordsfor six British stations (Greenwich,Stonyhurst, Seathwaite, Rothesay, Culloden, Edinburgh)and all the other stations (Dijon, Pouilly, Paris, Bar-le-duc, Geneva, Brussels; and Gfitersloh, Berlin, Konigsberg, Gbrlitz, Trier, Stuttgart, Vienna, Hermannstadt)are taken from the naer-r 2 H 2

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 448 BEVERIDGE-Wheat Pricesand Rainfall [May, Inability to obtain readily recent records for German and Austrian stations has made it impossible to draw this line for the last fifteen years. If we comparethe upper curve, compoundedfrom the wheat price cycles, with the average rainfall for all three groups, certain striking similarities appear. Each curve, starting from a maximum at 1852, falls to a very low minimum in 1857, which represents the remarkable and general drought of that year. There follow in each curve marked depressionsat or near 1864, 1870, 1874, 1883-4, 1887, 1897-8 and 1904. The agreement of the two curves at their peaks, representing heavy rainfall, is not quite so good, yet can hardly be questioned. At 1852, 1866, 1872, 1882, they agree exactly; at 1877 they agree except that the upper curve culminates a little earlier. After 1905 the German figures are scanty, and the violently fluctuating French figures represent five stations only. If we consider only the British figures, we get, in the synthetic and the rainfall curves, a remarkable correspondenceof peaks at 1906, 1912, 1916, 1920, and finally the great drought of 1921, accurately and emphatically demanded by the cycles of wheat prices from 1545 to 1850. There are, of course, discrepancies between the curves. The rainy years, 1860 and 1903, do not show on the synthetic curve, and 1882 is not nearly as prominent as it should be; the depression at 1893 is barely visible on the synthetic curve, and is followed there by a deeperdepression at 1895 which does not appear in the average for the three European groups, though it does in the British group taken by itself. These and other differencesin the relative importance of the various peaks and depressionssuggest-as is, of course, the case-that the different cycles have not been appropriately weighted in the construction of the synthetic curve; they have quite deliberately not been weighted at all.* Taking these and other sources of minor error into account there can, I think, be no reasonable doubt of a physical connection

by G. Hellmann already cited ; the figures for each year representing the rainfall of that year as a percentage of the mean for fifty years, 1851 to 1900. The figures for the four other British stations (London, Haverfordwest, Glengyle, Belfast) are taken from a table at pp. 214-5, of Briti4h Rainfall by M. de Carle S. Salter (1921); the rainfall for each year being given as a percentage of the mean for sixty years, 1860 to 1919. * Curiously enough both 1860 and 1903 had their due prominence and 1893 appeared as a marked depression in earlier drawings of the synthetic curve made when I was using slightly different phases and different periods (including some such as C, K, and N and Mr. Baxendell's period of 2 39 years, which I later felt bound to put on one side).

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922-.] in WesternEurope. 449 between the two curves; the coincidencesare altogether too marked to be random. We have not achieved, but we are in sight of the possibility of achieving, the construction by a synthesis of cycles based on price recordsfrom 1545 to 1850, of the rainfall curve from 1850 to 1920. The synthetic curve here presented must not be judged as itself an attempt at such a reconstructionof the rainfall. It would have been easy by appropriate shiftings of phase and selection of periods to have secured a better fit; experiments on these lines should form the next step in the investigation. In the first instance, however, I wished to follow implicitly the indications of my own analysis. The synthetic curve might have been drawn in 1850*; except in so far as the omission of the period was dictated to me by consideration of later-facts, it stands exactly as it would have been drawn in 1850 by any man who had worked on the figures as I have, and had been rash enough to believe in them. Suicha man would have successfully predicted every great drought from 1857 to 1921; he would have erred seriously only in expecting 1898 to be a good deal drier than it was and he would have under- estimated the severe but rather local drought of 1893. Even if he had included M, he would not have destroyed the correspondence of his predictionand the facts, but he would have diminishedit. The omitted cycle, M, has been left out because its influence on the fit of the two curves is found to be almost uniformly adverse. This may mean that it has now gone out of existence, or that it operates entirely through some factor other than the rain. Oddly enough, it is distinguishedin the periodogramby peculiarregularity of amplitude and phase. Its inclusion in the synthetic curve would not indeed so far alter that curve as to destroy the agreementof the main features (its maxima like those of almost every other cycle almost uniformly avoid the big drought years) but clearly, if it is a rainfall cycle at all, it is either very unsymmetrical or of small importanceat the present time.t The agreement between two curves is so striking that it could not possibly be due to chance. The test shows the general correct- ness of the hypothesis just made, namely, that some or all of the periods shown by analysis of wheat prices from 1545 to about 1850 are real, and have been operative as rainfall cycles from 1850 to 1921. The agreement is not, of course, perfect in all respects;

* Actually some of the sequences used extend for a few years beyond 1850. This slight anachronismcan, I hope, be overlooked. t See SupplementaryNote C for anotherpossible explanation.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 450 BEVERIDGER-WheatPrices and Rainfall [May, such agreement as is shown does not prove that the hypothesis is Correctin all its details. One or more of the periods used for the construction of the upper curve may never have been a real one, may have gone out of action since 1850 or, although real and still operative, may affect only meteorological factors other than rain. In all these ways, the hypothesis perhaps may be incorrect. In some other ways, it certainly is incorrect. In drawing the curves in the way described I have had to assume that the cycles were all of equal importance, that they were all symmetrical, and that they could be represented by curves of a particular shape. It is quite certain that they are not all of equal importance; it is certain that some of them are not symmetrical and possible that none are. It is most improbable that any or all of them, in fact, coincide exactly with the shape assumed. Again, my hypothesis has treated each of these cycles as absolutely regular in the sense of being equally important at each return to a maximum. It is quite improbablethat this is the case. Finally, while it is possible that some of the cycles used in the construction of the upper curve should be excluded, it is equally possible that some other cycles should be added. Some of those shown in the periodogram, but apparently inactive between 1700 and 1850 such as C or N, may have become active again in the latter part or after the end of this interval; other quite new periods such as those of 2 20+, 2 39+ and 3 09+ years found by Mr. Baxendell, may have arisen.* The close agreement between the two curves of prediction and of fulfilment, in spite of these inevitable errors in the hypothesis, is all the more convincing. The chart that I have just drawn,with the evidence previously submitted, answers once for all the question whether persistent meteorological cycles exist and whether they

* So far as I can judge, the inclusion of the two longer of these new periods would materially improve the fit of the synthetic and the actual curve. Indeed at one stage of my investigations, before I knew the phases assigned by Mr. Baxendell to these two periods, I deduced the phases required to bring a synthetic curve containing them into the closest possible agreement with the actual rainfall, and these proved to be almost exactly the phases subsequently named to me by Mr. Baxendell. I should be surprised if most or all of these three periods did not prove to be real. The general course of the synthetic curve would not be affected if in its construction the separate cycles were represented by some simple curve other than the imaginary form actually used, e.g. by sine curves or by straight lines, slanting up and down. But there is no reason for expecting nature to work by sine curves even though harmonic analysis does; the imaginary curve actually used gives on the whole a slightly better fit.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in WesternEurope. 451 can be discovered. They do exist and some or all of them have been discovered. They have been discovered in nearly all cases not by myself but by other observersbefore or independentlyof my own enquiry. The value of the harmonicanalysis of wheat priceshas lain, first, in afford- ing an independent line of investigation, and, second, in making it possible to bring into account so long a series of years as to separate cycles which in shorter sequences might have been confused and have concealed each other. It has also had the value of showing that periods which could be traced in meteorological records for fifty or one hundredyears at most, have in fact existed for centuries, and so far as can be seen are permanent.

V.-CONCLUSION. This paper has already exceeded all due limits, yet it leaves, as I mean still to leave, the two most interesting questions of all unanswered. The first of these questions is: What will the weather be next year and in the following years ? To that I can only reply, I do not know for certain, and therefore say nothing at all. In a free country no man can be compelled to prophesy against his will. I have rather to take this opportunity of withdrawing any such prophecy I am supposed to have made, and any comparison of what may be expected in 1923 with what happened in 1315-16. In some respects harmonic analysis has closely confirmed the arithmetical results of my former article. In addition to the dominant periodicity near fifteen years, three out of my four cycles derived arithmetically appear in the periodogram; the fourth (as to which I then expressed most doubt) does not appear, though its double does. But in the demonstration of so many other periods (more than anyone could have ventured to advocate as the result of any subjective method of enquiry) harmonic analysis has gone far beyond what I suggested before. The relative importance of the different cycles, and so their combined result, are altogether altered. If the upper curve on my chart is continued backwards to 1315-16 it indicates, as it should do, abnormally bad weather and harvests in those years. If it is continued onwardsfrom 1923, as it can be continued by anyone with a little patience and a pair of dividers, it suggests no extraordinarymeteorological happenings for the next six or seven years at least. But anyone who, in this way, continues the curve, does so at his own risk. In my former article I wrote that if I had then had to prophesy,

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 452 BEVERIDGE-Wheat Prices and Rainfall [May, I should have foretold heavy rain and bad harvests in 1923, but that the time for prophecy had not come as there was still time for enquiry. I have now to say that my own further enquiries, while in many respects confirming my former ones, in more important respects differfrom them and do not lead to at all the same conclusion as to the general meteorological conditions of the near future. I hope that what, in spite of all precautions, was wrongly taken by many people as a definite prophecy of mine for 1923, will now be regarded as withdrawn and not replaced by anything else; 1923 may prove wet, but if it does I shall claim no credit for prophecy; I hope, not very hopefully, to escape blame if it proves dry. I believe that in view of the persistence which it now seems right to attribute to certain periodicitiesin the weather, trustworthy prophecy of the weather will, in due course, become possible. It is not possible on the facts as I have given them; it will become possible, if at all, only after detailed investigation has shown the nature, the shape, the relative importance, and, above all, the local variations of each of the cycles, of which we can now say little more than that they exist and cause periodic changes in the rainfall, or to put it in another way, only after we have managed, from the cycles found, to reconstruct not the main features only, but all the significant features of the weather in some past sequence of years. That, I hope, can be done as soon as properly equipped enquirers are preparedto devote sufficient thought and time to the purpose. The second question is, What are the causes of the cycles now found to exist in our weather? I can make no attempt here even to suggest the beginning of an answer, even to point the direction in which we should look for causes. Somewhere or other in the solar system there are periodic movements affecting our weather and crops, ten or twenty or more in number, far more regular than had ever been believed, possibly approaching in some cases the regularity and persistence of free orbital motion, subject in other cases to sudden birth and death. These movements may be of one type, or of several types; they may be in the sun or the planets or the moon; in the earth or in the air or water upon its surface. Something, perhaps, can still be learnt from economic records by further analysis of grain prices and study of famines. An accurate and documented chronology of famines before 1500 might help materially to determine the life and the exact lengths of the various cycles. Systematic testing of the periods for discontinuity since 1500, by showing when particularcycles arose or changed their phase, might give a clue to their nature. So might considerationof

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions WHEAT PRICE CYCLES AND EUROPEAN RAINFALL,

1850 1860 1870 1880 1890 1900

I i~ _ _ _ _ I __ _ _ I I.I

WHEAT PRICE CYCLES SCY-YNTHVE0TIC CURVE.

RAINFALL IN WESTERN AND CENTRAL EuRoPE.

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Francea Belg9;ar|

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The "SYNTEETIC CURVYE"is constructedby additionof eleven cycles appearingin wheat prices from 1545 to U. V, in the Table of Periods. The "4RAINFALL " Curves are based on ten stations in the British Isles, six: in Fru The rainfall for each year being given as a percentage of the average for 1851-1900, or in some cases 1860-l9]L9. the mean of these three groups of stations.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions WHEAT PRICE CYCLES AND EUROPEAN RAINFALL, 1850-1922

1870 1880 1890 1900 1910 1920

_ _ _ _ l _ __ _ . _ _ _ _ l _ _ _ I _ _

Rainfallin Brish Isles Francea Belgiarr A &~~~~~~GrmngaM Austr;&

stations. Instructedby additionof eleven cycles appearingin wheat prices from 1545 to about 1850, and identifiedas E, F. , I,LJ, L, P. S. T. Le "4RAINFALL " Curvesare basedon ten stationsin the BritishIsles, six: in Franceand Belgium, and eight in Germanyand Austria. Onas a percentageof the averagefor 1851-1900,or in some cases 1860-19]L9. The Curvefor Westernanld Central Europe represents

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in WesternEurope. M those cases in which periods seem to have changed their length. The importantG periodis a leadinginstance, with a length apparently between 5 oQ and 5*IO in the first half of my sequence (1545 to 1694) and between5*IO and 5*II in the secondhalf (1695 to 1854). If this increase is real, is it a continuing one ? If so, we get an inference of motion subject to friction, like that of a comet through the solar atmosphere, or like the rotation of the earth under the retardinginfluence of the tides. Is the increase a temporary phase, a growth within narrow limits to be succeeded by a correspondinglimited decrease? If so, it may suggest free orbital motion subject to perturbations. Again, much might be learnt by analysing prices not for Western and Central Europe as a whole, but for separate countries, so as to discover differencesin the local importance of particular cycles, and still more in their local phases. Finally, something might be learnt by consideringprices not of wheat alone, but of other grains which react differently to meteorologicalconditions. The sources used for the constructionof my wheat price index contain sufficientmaterials to provide separate indices also for barley, for rye and for oats. For the most part, the answer to our second question-as to the causes and physical nature of the cycles shown in wheat prices- must be sought in other fields, above all in the comparative study of meteorologicalrecords for all parts of the world. The economic data used here relate only to Western and Central Europe, and inferences from those data apply to that region alone. Some of the cycles certainly influence other regions. The BrUcknercycle is known to affect all parts of the world with a few well-defined exceptions; the outstanding Indian famines such as those of 1345, 1555, 1629-30, 1660-1, 1769-70, 1876-8 are all associated with abnormal weather in Europe. But it is possible that some cycles are confined to particular areas; it is more than probable that others, though affecting widely distant regions, do not affect them simultaneously or in the same sense. In this study of similarities and differenceslies probably the way to the heart of the problem. But I cannot enter on this way; it is for astronomersand physicists. Even my own economic enquiry, so far as I can see, is at an end. And even this paper has its end at last.

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APPENDIX. Harmonic Analysis of Wheat Price Fluctuations. Note.-The first observationrelates to the harvest year 1545 except where a and b are given in heavy type.

Number q I I I2 Period of 2 (First (Second Pe18. Years. (3008) a. b. (N(a2+b2)) Half of Half of N. P \300 Sequence). Sequence).

2*000 300 150*0 + *11 -*01 2*049 336 146*4 - *40 - *09 *19 2*054 304 146*1 + *48 - *72 *77 2*061 340 145*6 + *38 - *57 *54 2*069 300 145*0 + '25 + .63 *46 - 2*074 336 144 6 - *61 + *51 *71 - 2-080 312 144*2 + *92 - *50 1*14 - 2*087 288 143*7 - *52 - *11 *27 - 2*095 308 143*2 - *91 + *90 1*69 - - 2*105 320 142*5 + *90 + .07 *86 - 2*112 288 142*0 + *90 + *80 1*38 - 2*133 320 140'0 + *89 + *15 *84 - 2*154 308 139-3 + *48 + *23 *29 - 2*182 288 137-5 +1-32 - *59 1.99 4.53 *48 2*200 308 136*3 - *13 - *60 *39 *19 *71 2*222 320 135*0 - *32 - *62 *52 *46 2*83 2*261 312 132*2 + *50 - *22 *31 2*286 320 131*2 - *38 - *85 *93 2*316 308 129-5 +1 39 -1*05 3*11 - 2*333 308 128*6 - .10 - *205 *08 2 353 320 127-5 + *90 + *07 *86 2*364 312 126*9 - *12 - *63 *43 - 2*370 320 126*6 + .05 - *28 *08 - 2*375 304 126'3 + *29 - *43 *27 2.61 2*38 2*381 300 126*0 - *19 -1*22 1*53 - 2*385 310 125 8 -1*00 - *89 1*86 1*52 1*86 2*391 330 125-5 -1 39 - .54 2K18 *76 4.34 2 395 309 125*3 - *72 + *60 *90 _ 2*400 312 125*0 + *34 + *68 *60 *56 3*03 2-412 328 124-4 - *08 - *65 *47- 2*417 348 124 1 + .63 + *57 *69 .90 1*07 2 435 336 123*2 + *44 + *01 *22 - 2*452 304 122*3 -1*40 - *51 2*23 2*462 320 121*9 - *25 +1 49 2*44 _ 2*476 312 121*2 - *38 + *35 *27 2*483 288 120*8 - *07 + *74 *53 2*500 320 120*0 - *24 +1'19 1*56 2*512 324 119.4 + *86 + *39 *97 2.516 312 119*2 + *45 + *24 *26 2*529 301 118*6 - *19 _ 31 *13 2*545 336 117*9 -1.39 - *81 2*89 2*555 322 117 4 + *38 + *50 *42 2*571 306 116*7 +1 25 + *55 1.91 _ 2*588 308 115*9 + *30 + *43 *28 2-600 312 115*4 +1 02 - *39 1*25 2*615 306 114*7 - *75 - .24 *63 2*625 294 114*3 - *45 +1.36 2-01 2*643 296 113*5 + *95 - *62 1*27

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in WesternEurope. 455 HarmonicAnalysis of Wheat Price Fluctuations-contd.

Number qI= I 1 12 Period of 300s . b. (N (a2+b)) (First (Second p18. Years. = (O8~(. 1. (~~Y Half of Half of N. \p 300 Sequence). Sequence).

2-667 312 112-5 - *92 +1.20 2-38 2-687 301 111 6 +1 23 - *02 1 52 2*692 315 111-4 - *04 + *23 *06 _ 2 706 322 110 9 - *27 +1 33 1 97 2-714 304 110-5 + *83 +1-17 2-10 2-727 300 110 0 + *86 +1.46 2 87 2-733 287 109-8 +2 05 +1419 6-16 13-27 1 09 2 735 279 109 7 +2 44 +1 23 782 - 2-737 312 109-6 +2-23 +1.00 6-22 13-97 1-52 2-741 296 109-4 +2-43 + *25 5-86 2 750 308 109 1 + *90 - *84 1 55 2-762 348 108-6 - *57 - 04 .37 2 769 324 108 3 +1 49 + *23 2 28 _ 2-778 325 108-0 +1 20 - *92 2-48 2-800 336 107-1 -1.01 - .19 1-18 1-62 1.01 2 818 310 106-5 + *55 +1?07 1 49 2-833 323 105-9 + *78 - *10 *67 2-846 296 105-4 + *41 + *42 *34 2 857 320 105 0 + *96 + *21 1 03 2-875 322 104-3 + *35 + *14 *15 2-888 312 103-9 +1 51 + *26 2-43 2-895 330 103 6 - *69 -41 57 3 21 2-909 320 103-1 + *70 -1.11 1-84 2-933 308 102-3 - *04 + -39 *16 2-947 336 101-8 - .93 -1 19 2-57 2-960 296 101-4 - *00 -1415 1-30

3 000 300 100-0 - *29 .-39 *23 1*29 *03 3 040 304 98'7 + *09 + *75 *58 3 077 320 97*5 + *05 +1-18 1.50 3-111 336 96-4 + *91 - *44 1-15 3 143 308 95 5 +2-01 + *23 4-20 3-167 304 94.7 + *46 -1-05 1-33 3-200 320 93-8 + *43 + .95 1?16 3-217 296 93-2 +1 25 + .00 1.55 _ 3-250 312 92-3 -1 22 - *47 1?80 3-273 324 91-7 - *55 +1?18 1-82 _ 3-286 322 91-3 - *11 + .99 1?07 3 304 304 90-8 + *13 + *75 .59 _ 3.333 320 90 0 + *90 +1-58 3.54 3-364 296 89-2 +1.76 + *98 4 00 3.375 324 88-9 + *55 + *92 1 24 3-385 308 88-6 + *35 +1?03 1 21 3 400 323 88-2 +1-12 +2-37 7-41 16-76 8-02 3*407 276 88-1 +2-98 +2-81 14-90 3-412 348 87 9 +1 27 -3-98 15-53 23-55 10-48 3-417 328 87-8 +3 08 -2-24 15-84 21-25 11-56 3-429 288 87-5 +3-11 -1 40 11-16 21-22 1169 3*444 310 87*1 + *09 - -99 1 03 11 58 8 87 3 455 304 86 8 + *55 + *29 .39 3-462 315 86-7 +1 57 +1 02 4-87 3 500 308 85-7 +1 20 - .94 2.38 7-21 1 50 3-524 296 85-1 +1-41 -1-18 3-31 _

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HarmonicAnalysis of WheatPrice Fluctuations-contd.

Number q I I1 12 Period of (First (Second P/S. Years. = (3008) a. b. (N(a2+b) Half of Half of N. p 300/* Sequence). Sequence).

3 538 322 84*8 + *50 -1.45 2 53 _ 3 556 320 84.4 + *02 - *43 20 _ 3*571 325 84-0 + *80 - -69 1 21 3*600 324 83-3 -1 03 + *82 1*88 3*619 304 82*9 +1d18 +1-23 2-94 3*636 320 82*5 +1d14 + *13 1*39 3-643 306 82*3 - *16 + *27 *10 - 3*667 308 81 8 -2*14 -1*07 5*87 - - 3*679 309 81*5 + *34 -190 3*83 - - 3 692 288 81 3 +1 28 - *22 1 63 - - 3-700 296 81-1 + *90 - *59 1 18- - 3*714 312 80 8 +1.15 +1b78 4 65 _ 3*727 287 80 5 - *45 -1 65 2*72 _ 3*750 315 80 0 + *64 - *06 *44 _ 3-778 306 79 4 -1'17 - *68 1'86 - 3*800 304 78 9 +1460 + *80 3*24 _ 3*833 322 78*3 -1*12 -1*63 4*17 _ 3*857 324 77 8 +1 63 + *45 3*08 _ 3*888 280 77*1 - *15 + *66 *43 _ 3-895 296 77*0 - *66 +1o00 1 42 - 3*923 306 76-5 + .64 -1 61 3*06 - - 34962 309 75*7 - *67 +1V74 3 59 - 4-000 300 75-0 +1-47 -1*13 33646 4*077 318 73*6 + *57 - *26 *41 3 01 *28 4*111 296 73*0 +1-13 -1 70 4*13 _ 4 143 290 72 4 - *50 + *23 *30 _ 4 167 325 72 0 +1 21 + *32 1 70 4*173 322 7149 + *66 -1 46 2 77 - - 4*200 294 71 4 - .99 - *41 1*02 _ - 4*250 323 70-6 + *50 -2 73 8*32 - - 4-286 300 70 0 - *65 + *79 1*04 _ 4.333 312 69 2 -1*50 -1*30 4*10 - 4.353 296 68 9 -2 85 - *24 8*05 _ 4*364 288 68 7 -2 98 + *75 9 07 _ - 4.375 315 68 6 -2 47 + *87 7*19 _ - 4 385 342 68 4 - 50 +2 55 7 72 _ - 4.400 308 68 2 -1*38 +3 27 12 89 25417 7 18 4*412 300 68 0 + *08 +3 62 13*11 36 51 4.45 4*417 318 67*9 + *87 +3 85 16 48 29*16 7 06 4-429 310 67 7 +1b80 +2 41 9*32 _ - 4*444 320 67 5 +2415 + *83 5*66 35.76 16*46 4*471 304 67*1 + *91 + .79 1*48 - 4*500 306 66 7 +1-87 + *72 4*09 14 71 12 82 4-571 320 65*6 - *21 + *04 *22 4*600 322 65 2 - *08 +1 24 1*65 - - 4*667 336 64.3 + 19 + .93 100 - - 4*750 304 63*2 - *12 +2 28 5 28 _ 4-800 288 62 5 +2 44 +1 08 6*84 8 34 5 61 4*857 306 61*8 -1*06 -1*30 2*89 - - 4*888 312 61-4 -1*80 +2411 8*00 _ 4.933 296 60 8 +1457 +1458 4*91 -

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HarmonicAnalysis of WheatPrice Fluctuations-contd.

Number q I- Ii '2 Period of =300 2 (First (Second P/S. Years. a. b. (N (a2+b2) Half of Half of N. p 300 Sequence). Sequence).

5*000 300 60.0 +1*85 +1o00 4*30 11*84 4.94 5-067 304 59*2 - *05 +3-98 16-09 - _ 5*091 336 58*9 - *73 +5 55 35*05 46*59 33*20 5 100 306 58*8 +0571 +2 98 42*34 34*05 57 08 5*111 322 58*7 +5-70 + *29 34*91 5 125 328 58*5 +3-97 +2 90 26*38 25*81 29 99 5*143 324 58*3 +2 46 +2-46 13*09 - - 5*200 312 57-7 + .02 + -30 .10 - _ 5 250 294 57*1 +1-74 +1-92 6*56 - 5*333 320 56*3 + *71 -4.46 21*72 19 24 45 05 5*400 324 55*6 +1-04 +3-71 16*06 1 07 66*99 5*415 325 55.4 +4-27 +1l90 23 66 5*429 304 55*3 +4-72 - *28 22*61 3-14 65*69 5*455 300 55*0 +1-37 -3.73 15 76 5*500 308 54*6 -1-04 +1 49 3;39 4*80 30*91 5 555 300 54*0 +2 40 - *68 6*23 5*600 336 53*6 + *46 +1-21 1*88 13*87 12 67 5*667 306 52*9 +5 31 -1*97 32*72 33*06 32*58 5 692 296 52*7 +2-05 -3*91 19*18 24*78 23*80 5*714 320 52*5 + *35 -2.13 4.97 39-80 17*81 5-750 322 52*2 +1-39 - *33 2.18 - 5*800 290 51*7 +3-55 -2*75 19*47 5*846 304 51*3 + o00 -2.29 5.35 5'933 356 50*6 +4 37 + *91 23*63 29*48 33*07 6*000 300 50*0 -3*50 - *12 12 29 14 67 23*41 6'111 330 49*1 - -79 -f190 4*66- 6*143 301 48 8 + .74 -2*96 9*32 6*167 296 48*6 - *22 -2*94 8 56 6*200 310 48*4 -2*02 -3*38 16*02 48*74 5*63 6*250 325 48*0 -3'23 - .11 11.30 37*24 2*77 6*286 308 47*7 -1*72 - *59 3.41 6 333 304 47*4 -1'52 +1-29 4*02 6 400 320 46*9 + *80 +2-74 8-71 6*500 312 46*2 + *69 - *73 .94 6.52 *27 6-571 322 45*7 +1 49 - *77 3*02 - 6*667 320 45*0 + *25 + 21 .11 - 6*727 296 44*6 + *08 - *13 *02 - 6*750 324 44*4 - *20 -1*66 3.01 6*800 306 44*1 + *23 - *65 *48 6*909 304 43.4 + *58 +2-56 7 00 _ 6*933 312 43.3 +1-68 +2-01 7*15

7*000 308 42*8 +3 10 -2 17 14*74 22*57 13*38 7*143 300 42*0 +1-83 -1*86 6*79 _ 7*200 324 41'7 + *54 -3.93 16*96 19*99 3.74 7.333 308 40*9 +1-52 -2*81 10*46 20*64 10*41 7*400 296 40*5 -2*33 -2*72 12*65 7*417 356 40*4 +1-50 -4-01 21*72 29*06 29*54 7*429 312 40*4 -3*80 -1*49 17-28 30*47 24*31 7.500 315 40*0 + *17 +1S50 2*40 _ 7*600 304 39*5 -2*33 -1*37 7.43 9*40 13*91 7*667 322 39*1 -1*46 -2*61 9.57 - -

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HairmonicAnalysis of WheatPrice Fluctuations-contd.

Number q I C I 12 Period Yaf.=(300S)of b. (N (a2+22 b2)) (Firsta f (Second(eC rwiSdP1. Years. (OS a. ( (+)'.Half of Hailf of N p _ 300 Sequence). Sequence).

7-750 310 38 7 +1-38 - .39 2.13 7'857 330 38-2 - *50 + *28 *36

8-000 312 37.5 -3-96 +1 34 18-67 3.97 40.11 8-091 356 37'1 +4-32 - *98 23-23 9'52 42'98 8-200 287 36-6 +1'62 - *64 2-90 _ 8 222 296 36-5 + *19 - *56 *34 8-333 325 36-0 + *21 + *91 *95 *59 6 23 8-500 323 35-3 + *17 +3 19 10-41 8-667 312 34-6 +2-51 -1 01 7.59 8-800 308 34'1 +2-97 + *83 9.77

9.000 306 33-3 -1A51 - *57 2-65 13-81 10'02 9'200 322 32'6 - '16 -1'56 2'65 _ 9'333 336 32'2 - *74 + *64 1'08 _ 9'500 304 31'6 +1'08 +1'07 2'26 8'11 1'57 9'667 290 31'0 +5'03 + *37 24-55 44'25 17'51 9'750 312 30'8 +4'46 -3'56 33'89 38'44 29'72 9'818 324 30'6 +1'21 -4.94 27.90 - -

10'000 320 30'0 -1'19 - *83 2'25 13'90 7'92 10'200 306 29'4 + *86 - *22 *80 3.99 13'23 10'250 328 29'3 - '69 +1O10 1'84 10'400 312 28'9 +1'88 -1'65 6'52 - - 10.500 294 28'6 +2'46 -1'82 9'19 42'66 2'45 10'750 301 27'9 +1'47 -3'13 11'98 _ 10'800 324 27'8 +1 00 -4.75 25'48 74'83 4'26 11'000 308 27-3 -3'85 -4'26 33'84 96-01 3-47 11'200 336 26'8 -2'48 + *55 7'24 35'08 2 12 11.500 322 26'1 -1'32 - *66 2'34 11'667 280 25'7 + *46 +1'42 2'07 *43 9'42 12'000 312 25'0 -2'47 -4'04 23'30 35'10 31'73 12'143 340 24'7 - '22 -4.37 21'66 26'58 17'44 12'333 296 24'3 -2'44 +2'74 11'43 _ 12'500 325 24'0 -1'22 +2'63 9'13 _ - 12'667 304 23'7 +2'28 +5'19 32'58 29'88 35-43 12'800 320 23'4 +5'70 +3'26 46-01 - 12'875 309 23'3 +6'46 + *77 43'58 - - 13'000 312 23'1 +4'26 -4'32 38'23 44'83 72'17 13'333 320 22'5 + '40 + '37 *32 31'88 34'65 13'500 324 22'2 +2'56 -2'09 11'79 _ 13'667 328 21'9 +3'49 -1.34 15.28 _ 14'000 308 21'4 +1.15 -1.00 2.38 8'75 38'0 14'500 290 20'7 -3-78 - *18 13'82 14'31 125'48 14'667 308 20'5 -1'50 +4'23 20-69 _ 15'000 300 20'0 +6'32 -2'66 46'83 70'38 141'48 15'200 304 19'7 +1'19 -8'52 75'04 86'11 111'03 15'250 305 19'7 - *28 -8'65 76'17 15'286 321 19'6 -2'35 -7'15 60'62 15'333 322 19'6 -3'89 -6'55 62'29 15.500 310 19-4 -6'92 -2'02 59'11 75'92 45'30 16'000 320 18'8 -1'46 +4'52 24'02 68'42 7.33 16'667 300 18'0 +5'21 - '39 27.33

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] in WesternEurope. 459

HarmonicAnalysis of WheatPrice Fluctuations-contd.,

Number q I = It 1 2 Period of (First (Second p/s. Years. (300s) a. b. (x(a+bY))11 a2 b2 Half of Half of N. P 300 Sequence). Sequence).

17-000 306 17 6 +2-56 -6-35 47-84 104-45 13*70 17-333 312 17 3 -3 04 -6.65 54.55 136-19 11.94 17-500 280 17 - 1 -6 18 -4.45 54 12 69-35 55.53 18-000 306 16-7 -4.40 +1-25 21.29 70 45 12-49 18-500 296 16-3 -1 -46 +2-25 7 10 19*000 304 15 8 +1o00 - *23 1o07 1 47 10 71 19-750 316 15-2 -4-73 -1-59 26-25 20 000 320 15 0 -5- 71 +1 69 37 88 50 07 23 97 21 000 294 14-3 + *78 +2-61 7-28 45-05 15 18 22-000 308 13 6 +1 87 +1-58 6-18 18'98 5 91 23-000 322 13-0 -2-45 -1-43 8-61 8'93 10-38 24-000 288 12-5 + .45 +5 19 26 10 7-27 60-59 24-667 296 12-2 +4-31 +1 99 22-21 25-000 325 12-0 +3-86 - *19 14-94 27.01 30-29 26-000 312 11 5 +1 23 -1 34 3.43 46-80 16-73 27-000 324 11*1 + .50 - *33 *38 16-01 11*18 28 000 308 10-7 - *49 + *68 *72 - 29 000 290 10-3 +1 08 -2.12 5 46 14 11 14 83 30-000 300 10 0 -1 53 -2 34 7 81 4.75 11.68 31'000 310 9-7 -1-98 + .13 4-06 7.99 28.74 32 000 320 9 4 - *37 + *51 *42 18 77 22 80 33 000 330 9*1 + *96 - *78 1*68 28-35 28-91 34 000 306 8-8 -3 00 -2-15 13 90 35-000 280 8-6 -4.64 +1-79 23-11 6-49 53-72 36-000 288 8-3 -1-65 +4-85 23-29 15-64 46-98 37-000 296 8 1 +2-08 +3-92 19-47 18-99 48-23 38-000 304 7.9 +2 99 + *56 9.37 14 35 33-23 40-000 320 7-5 -1 44 - *63 2-63 - - 41-000 328 7.4 -1493 + .93 5.01 _ 42-000 294 7 1 + *93 +3-02 9.75 - - 44-000 308 6-8 +3-00 - .14 9-27 - - 45-000 315 6-7 +1 69 -1 99 7 14 _ 46-000 322 6-5 + *16 -2-27 5-58 _ 48-000 288 6-3 - *76 - *09 *56 _ - 50*000 300 6o0 +1 83 +2 19 8 14 _ 52-000 312 5-8 +4-77 - *57 24.03 _ 53-000 318 5-7 +4 22 -2.60 26-08 _ 54-000 324 5 6 +2-84 -4-01 26-09 _ 55*000 330 5.5 +3-54 -3 30 25-82 - 56-000 336 5.4 +3-31 -2-36 18-47 - 58-000 290 5-2 +3-89 +1-49 16-82 - 60o000 300 5*0 -3-08 - *93 10-32 - - 62'000 310 4 8 -1 62 + .39 2-88 - 64-000 320 4 7 - *78 + *13 *66 _ 66-000 330 4.5 - *56 - *56 *69 _ 68 000 340 4.4 +2 90 -1i88 13 58 - 70-000 280 4*3 - *69 - *16 .47 _ 74-000 296 4.1 -1 20 + *82 2 07 - 76-000 304 3*9 - *66 +1 17 1 83 78-000 312 3-8 + *58 +1 26 2-00 _ 80-000 320 3*7 + *77 + *82 1 34 - 84-000 336 3-6 + *26 + *69 *62 -

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 460 Discussion [Mlay,

DiscUSSION ON SIR WILLIAM BEVERIDGE'S PAPER. MR. G. UDNY YULE,in opening the discussion, said this was an extraordinarily difficult paper adequately to discuss after only a few days' consideration. It represented a very detailed analysis, based on a mass of material, and really to appreciate it would require, not a few days but months of hard work on the original data given in the Economic Journal. Before proceeding to deal with the few points that had struck him he wished to congratulate the author on the completion of this stage of his labours, and the Society on having obtained so important a contribution to their Journal; it was a paper which, he felt sure, would be referred to for many years as fundamental in its subject. They would share the author's regret that he could not include a full account of his index-number of wheat prices in Europe since 1500, but obviously considerationsof space prevented this. The subject would require a paper to itself. The first point that he had to raise concerned the cycle of approximately15 *2 years period which, in the periodogram,appeared to be by far the most outstanding period observed. He was exceedingly puzzled by the author's very unparental treatment of this particular cycle. It was Sir William Beveridge himself who first suggested its existence, on evidence which, he had to confess, had seemed to him at the time quite insufficient, in a note in the EconomicJournal in March,1920 ; the data used in that note referred to the export trade. The first analysis of the present data, published in the same Journal for December last, not only confirmed the existence of the period, but showed it to be predominant, i.e. the period giving the greatest amplitude observed, thus completely routing his previous skepticism. The more detailed analysis now given did not essentially change the position, it only suggested a slight adjustment in the length of the period, from some 15 3 or 15 4 years to about 15 2. This was not a material alteration; yet both in the note of December last and in the present discussion Sir William Beveridge seemed rather inclined to ignore the period, or to endeavour to explain it away. He did not understand the reasonsgiven for this course. One possiblereason, he thought, might be put aside at once; the failureto find such a period in other data, so far as at present analysed, ought not to affect their judgment. They were discussing an analysis of wheat prices, not meteorological data, and they could not assume that a period found in the first should necessarily also be found in the second. Judging froin some sentences in the paper, he thought that the author would probably agree with him in this. Putting that point aside, the author said on p. 438 " I am inclined to attribute certainly the importanceand, possibly, the very existence of the peak at 15-250 in my periodogram,not to any single cycle of that length, but to a combination of smaller cycles of lengths

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] on Sir William Beveridge'sPaper. 461 which are all close sub-multiples of 15 250 or its double." He did not follow this reasoning. The superposition of several harmonic curves of various periods could not yield a harmonic curve of another a Ad longer period. Such a superposition would, it was true, give ' beats,"' like the beats heard when two notes that were nearly but not quite in tune were sounded together; and the highest maxima in those beats would occur when all the component maxima were in phase or nearly so, and therefore at intervals corresponding to the least common multiple of the component periods. But the analysis of the compound curve would not give a harmonic component of the period indicated by the interval between greatest maxima. The existence of a certain period between highest maxima (or lowest minima) and the existence of a harmonic component of the same period were, it seemed to him, two quite distinct things which ought to be clearly distinguished-that was one reason why he a little mistrusted eye-judgment based solely on inspection of charts. In his note in the Economic Journal the author had emphasized another point on which he had not touched in the present paper, but which might have been in his mind, and which seemed to the speaker to be of great importance: the point that every recurrence of the maximum in the weather-cycle might not show in the crop because it would not always occur at a time of year when it would seriously affect the growth of the crop. He had tried to consider the possible effect of that kind of interference and-speaking with a good deal of hesitation, because the matter might be very complex -it seemed to him again that the main effect would be, with a cycle of non-integral period, to produce " beats "-not an apparent cycle of different period. So that-in this case also it did not seem to him that the 15 *2 period in the crop could be explained in terms of shorter periods occurring in the weather. It seemed to him very difficult to suppose that it was not a real period, even if there were no analogy in meteorology; he was inclined to use here the words used by the author (p. 429) concerning another period: "If there were no true period here, harmonic analysis would indeed be a sorry guide." At the same time a test which he had made of the general importance of all the periods " which are close sub-multiples of 30 or 31 years " led to a very disappointing result. He had correlated Sir William, Beveridge's " index of fluctuation " for year n with that for year n + 61, obtaining 250 pairs of observations from the years 1500-1810; he recognised that he should rather have taken 1540-1850, but had no time to do the work again. As 61 so nearly included as sub-multiples so many of the more important periods he hoped there would be a fair correlation of somewhere between 0Q3 and 0 4. The actual correlation was only 0 08. He did not quite understand the result, but thought it must be due to the fact that many of the approximate sub-multiples were not precise. It seemed to him an odd feature of the periodogram that there were a number of periods which were roughly, but not precisely, VOL. LXXXV. PART III. 2 i

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 462 Discussion [May, sub-multiples of each other. He hoped the author would devote a little further attention to the 15-2 year period, and especially that he would evaluate the amplitudeand phase, if he had not already done so, for the two half-sequencesof 150 years each. That brought him to another point, with regard to the footnote on p. 425, respecting the test of agreement between the two half- sequences. He thought that the note was a little misleading. It was quite true that the chance of obtaining an intensity not less than I by analysis of a random series was the same as the chance of obtaining intensities not less than I from each of the two halves of the same series. But if there were real continuity of a cycle they found (approximate) agreementof the amplitudes from the two halves, and also (approximate) agreementof the two phases. The chance of getting any assigned degree of agreementbetween the two halves of a random series in these respects would obviously be the answer to quite a different question from that considered in the note and the chance of getting close agreement would be small. His third point concerned the short period of 2-2 years. This looked to him as if it might be identical with the period that gave the curious up-and-down alternations in infantile mortality, on which he had commented in a recent paper read before the Society, and for which he gave the approximate period of 32/15 or 32/17 years, Dr. Brownlee identifying it with a periodicity in measles. He would like, as in the paper referred to, to emphasize that with annual data only you cannot identify the period with certainty. Sir William Beveridge's period might be 2 2 years as he stated, i.e. the year might be 10/22 of the period; but if the year was 12/22 of the period, or the period 1-6 years, the same figures would be obtained. The period might be either 2-2 or 1-833. One small point in conclusion. He was rather surprised at the approximate agreement in phase found between some of the wheatprice cycles and the rainfall cycles. He would not have looked for agreement, but rather for a considerablelag, not merely for the reasons given by the author, but because wheat was so deep-rooted a crop, and a crop consequently which could utilise not only the rain of the year in which it was grown but also the rain of one or two previous years, as was shown by the experience of wheat growing in arid districts. He had to apologise for remarks which were very sporadic. It only remained for him formally to move the vote of thanks for a most interesting paper.

Mr. A. W. FLUX, Honorary Secretary,then read a letter from Sir Napier Shaw, who regrettedvery much being unable to be present: " Referring to the synthetic curve dealt with in the latter part of the paper," Sir Napier wrote, " I am not very clear about the comparison of the synthetic curve of wheat prices in Chart B with the rainfall of Western Europe. A curve made up of discontinuous values derived from harmonic components is set up, at 5 in. to a

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] on Sir WilliamsBeveridge's Paper. 463 year, and datum points, e.g. 1920-25, marked; against that is set a rainfall curve, each year apparently representedby a single figure. If I rightly understand,the figure for 1951 is put at theend. of 1951 on the price curve. I think if rainfall for a year is brought into a single point the correspondingtime-point should be the middle of the year, not the beginning or the end. The matter is a trifling one, but it wants dealing with in the legend on the diagram. " The agreementsand disagreementswhich Sir WilliamBeveridge adduces should be very interesting to the Society; his 'prediction ' of a famine in 1315-16 is very dramatic. It seems to clinch the demonstration of the reality of these determinations by cycle in a peculiarly convincing way. I am not myself disposed to attach so much importanceto salient points. If values are periodic,any point should be just as periodicas its neighbour. I am, however, disposed to find a special interest in the subject at the point where Sir William Beveridge breaks off. When he says that his part is finished and it now remains for physicists or others to find the cause, I do not quite agree. What does a cycle mean ? Has the property affected no voice in the determinationof its own periods ? Sir William seems to think that it is perfectly passive and takes the mould of some periodic cause just as wax takes the impressionof a seal; but I do not think that is true of any one of the quantities concerned, viz.: wheat prices, wheat yield, rainfall, warmth, sunspots, etc. Each one of these may have something corresponding with a natural period of its own, due to what I may call the reaction of the environ- ment to violent disturbance. " For example, wheat prices. A disturbanceof the market such as the famine of 1315-16 does not leave the market of 1316-17 in the same position to deal with the new wheat-crop as if 1315-16 had not occurred. To take an extreme case, if the famine of 1315-16 had been so severe that there was only one wheat grower but many consumers left for 1316-17, wheat prices would be high; on the other hand, if the survivors had been all wheat growers and there were no other consumersthere would have been a slump in prices. What I mean is that, quite apart from any further disturbancedue to further shocks, recovery from severe economic shock of any kind is gradual and probably oscillating,with a tendency to some natural period. " In the same way with wheat yields. A wet late summer and autumn spoils the yields of two consecutive years; if the visitation is very severe we have two years of very bad harvests. Then, even if all the seasons following the bad season were normal, it is not likely that the yields would instantly become normal; the conditions of the ground and the seed are not normal. What is the natural period and the natural process of recovery from a severe visitation that changes the fundamental condition of life of the plant? Again, there is probably a natural period covering some years. "Thirdly, the rainfall. Any violent disturbance of the normal sequenceof rainfallis due to some seriousdeformation of the general 2 I 2

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 464 Discussion [May,

circulation of the atmosphere lasting for months, and if the conditions which produced the disturbance are removed and normal conditions supervene the general circulation will not recover all at once and forthwith; it also will take time and probably show some oscillations extending over years before it recovers normality. " If these various quantities have natural periods such as I men- tiorn, then the display of periodicity becomes as much a question of resonance in the recipient as of forcing vibration in the operator, and I do not think that we are likely to find the true cause of the variations in wheat prices or in rainfall by simply looking for the stamp that bears the image of the seal. I look upon all such things as wheat prices, wheat yields, rainfall, temperature, sunspots, etc., as belonging to systems which have their own elasticity and resilience. Nothing is simply and passively non-resistant or non-reactive, and consequently each must have its natural period of recovery from disturbance. " So we might ask the economists, what is the natural period and process of recovery from a violent displacement ? Is it oscillatory or simply secular ? And in like manner we might seek the natural periods of recovery from deformation in the case of the wheat crop, the rainfall, the temperature and sunspots. Then we may regard the experience of the wheat prices as the result of a series of deforming 'punches' from the variation of wheat supply; they again to a series of ' punches ' from the rainfall curve and so on; then also Sir William Beveridge's periods may be the expression of a synchronism between natural periods. In this way we can understand how the vitality of periodicities may fluctuate. It seems to me a much more promising field of inquiry than looking in the heavens or on the earth for a stamp that fits the curve of wheat prices. " I have written at this length because, unfortunately, I cannot be at the meeting on the 25th. I am to be on the way from Paris to Rome on that day to attend an assembly of the International Research Union. The last few paragraphs of this letter may perhaps serve as a contribution to the discussion, or at least express what I should like to say therein.'"

Mr. A. W. FLUX: I thought it more appropriate to read what was written by Sir Napier Shaw, who would have been a much more appropriate person than myself to second the vote of thanks, than to begin with any observations of my own, because I had not anticipated being in this position until a few hours ago; not that I did not look, and look with very great interest, at Sir William Beveridge's paper, because it intrigued me very much indeed. It reminded me of a period about the time of my early connection with this Society, when I should have welcomed enormously any apparatus that would discover periods underlying irregular movements. Sir William Beveridge has, in his paper, been able to utilise the results of some mathematical work that lies between that

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] on Sir William Beveridge'sPaper. 465

time and this, and I should have gone as eagerly at the work then as he has done now. The work that he has done is extremely painstaking work, and merits our very gravest attention. It seems to contain the promise of a very great deal. I think he has posed to us a series of problems in a form in which they are amenable to thorough examination, and for that reason I think we should feel very greatly indebted to him for all the work that underlies a paper of this character, which is an enormous amount of work. But I remain unconvinced that the business of the world proceeds in a series of regularly returning cycles. I do not think the paper has succeeded in proving that to us; and though unquestionably there are a number of important cyclical events in the course of the life of the community, it does seem to me as though the acceptance of what we might call the whole doctrine of the paper would require us to believe that if we could only discover the cycles, there are enough of them, and the dominant ones have such a degree of permanence, that we should be able to see ahead the general outline or trend of events, if not their details. We ought, indeed, to be able to foresee the minute details if there is a true cyclic variation underlying everything. If in fact we are subject to such a series of whirls, we become passive instruments in the hands of a remorse- less nature. I do not believe that this is the case, and while I admire the work that has been put into it, and admire the perspi- cacity that shows up throughout the paper, it has not convinced me that the series of phenomena of wheat prices, rain, and so on',. are completely determined by a series of underlying cyclical changes. One asks one's self what lies behind the whole of this theory of harmonic analysis. Unless I am mistaken it is a very ancient theory, as Fourier's theorem shows us that any series of variations you place, however irregular, may be represented by a suitable series of harmonics. You may shut your eyes and, with a piece of chalk make a scrawl across the blackboard, and it is theoretically possible to determine a series of cyclical curves which, duly related to one another in phase and amplitude, when summed up will reproduce that rash scrawl perfectly. But does this prove that when we have done it those cycles mean very much ? There are indications at various points of Sir William's paper that some of the periods to which he draws attention are not persistent, and that some are very marked during a certain term, and become but of little importance at other times. There is, it would appear, something else lying behind them than the cyclical movements in particular periods to which he has drawn attention. Even in determining those there is another phase of the thing which troubles my mind even more than the mathematical analysis and the interpretation of it-that is the data; for the longer I handle statistics the greater the importance I feel to be attached to the nature of the material on which we operate. There could hardly be, I think, a greater admirer than myself of the mathematical methods of analysis applied to statistical material; but if we are going to apply so refined a method of analysis we want

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 466 Discussion [May, to know a good deal about, and to feel a considerableamount of reliance on, our data. It is useless trying to put a fine edge upon a piece of soft steel. If you want to make a useful razor, you must have a hard piece of steel before you start grinding. Now I admit it is a matter of choice, and, as time would not allow of dealing with both matters, one had to be selected; but I sincerely regret that Sir William, before he demonstratedthe result of this series of mathematical operations, did not first put us in a position to consider what is the nature of the data available, and whether the material is a hard piece of material at which we may cut or which we may polish if we will, or is a soft material which, if we put our thumb into it, will readily retain the impress of the thumb. What kind of confidenceare we entitled to place in these records of wheat prices ? Sir William said that, in accordance with Sir Arthur Schuster's warning,he had not adopted any averagingmethod before presenting the data. But has lie not averaged for a year throughout ? He has one price record for each year, and the periods that he is consideringare periods where fractions of a year, as Mr. Udny Yule pointed out, are of very considerableimportance. What the effect of averaging wheat prices for a year, or taking some record of a wheat price some time in the year as representingthe average prices for that year, may have upon the original graph that we begin to analyse, is a matter that appears to need some consideration. Then, in combining the movements of prices at different places not connected with one another, the simple averaging of prices on a certain number of markets scattered more or less at random round Western Europe-what effect is that going to produce ? That is something of a puzzle to me. It seems to me I would have liked to consider the nature of the material that we are to analyse with a very great deal of care before proposing to place a large degree of confidence in the results of the handling of the material, even with the most trustworthy piece of analytical machinery. I will not speak longer than to deal with one further point. Sir William suggests that certain periods which he has discovered are the dominating periods in this series of wheat prices. If they are, I would suggest that we can determine it in one way, and I do not know that we can really determine it in any other way. We want to compounda curve like the synthetic curve he has shown us. That curve is not a curve of wheat prices as they existed between 1850 and 1923, but a curve continuing the record of pre-1850 fluctuations. There would rot, perhaps, be much interest in comparingthat with an actual analysis of prices since 1850, because the railway train and steamship have so considerably affected the movement of wheat prices in Western Europe over that period, that actual prices since 1850 might not be very significant with reference to the persistence and influence of the causes that give rise to the cyclical fluctuations determined over an earlier period. I gather from one remark that fell from Sir William in the course

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] on Sir William Beveridge'sPaper. 467 of his exposition that he has actually carried that synthetic curve back. Now what I think really interesting to know is, when the variations made up of these selected cyclical movements, taken together, are deducted from the original record, what is left of the movement? There may be something left which varies in a degree comparablewith the original, and the question of whether the influence of certain cyclical movements that may really exist and really persist is dominant might be judged from this residual record. Apart from the cyclical variations, some large and irregular influence might even offset the cyclical maximum and reduce it to the general average or even something below the general average. That is the difficulty in using these cycles, when determined, for any purpose of prediction, unless we know that we have got them all, and that no disturbance not represented in them is likely to overwhelm the result of the cyclical influences. If that were the case, how much have we advanced when we have determined the cycles ? Of course, one of the difficulties of the test I am now suggesting is, that the shape of the cycle is not known,* and that might affect rather seriously the practical work of determining the residual. Possibly Sir William has made the test and can tell us the result of it; if so, I should like to put that question to him, because I think the answer would be instructive at any rate, even if it has not the importance which I personally would be disposed to attach to it. There are a few other points which, if there were another half- hour before us, I should have liked to mention; but I will content myself by seconding very heartily the vote of thanks to Sir William.

The PRESIDENT announced that they had received a letter from Dr. Brownlee. who unfortunately was not able to be present.

Dr. GREENWOOD read the letter referredto " As far as I understand harmonicanalysis, it is quite impossible "for a period of I5 225 years to arise from a combination of lesser "periods. This you can-easily satisfy yourself is the case by taking "a simple example. Suppose i, p, i repeated four times against I, nmm, i repeated three times: when the harmonic analysis is "made there is no trace of a period of twelve, although graphically "it is apparent. This seems to me a fundamental blot on the "paper. " There is another blot which appears in the paragraph which begins at the foot of page 441, where without any examination a

* With reference to the shape of the cyclical fluctuation, it would appear proper to take together any periodic terni and terms for periods which are sub-multiples of that of the term in question. The combination of such terms, with appropriate amplitudes and phases, would yield a cyclical movement of unchanged period hut of modified form. Possibly the number of periods to be considered could be modified, and with advantage, by treating together any period and its higher harmonics.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 468 Discussioni [May, "common multiple of the different periods is sought. After the "second harmonic it is very difficult to make sure whether any "sub-multiple period found is anything more than a mathematical "fiction."

Mr. PERCY WALLIS said he wished to say a word with regard to the problem. When he came to that meeting he was feelingthat it was a question of prices, and he now felt that it was a question of rainfall. But what he wanted to ask was this. He supposed it was taken on price, not because the rainfall had any direct influence on the price, but that it had an indirect one through the crop per acre, and he supposed there was no record of the crop per acre during the period. Were they to assume that the next seventy years the chart would be exactly the same as the one shown on the paper, because it sounded as if it were going to repeat continuously ? With Mr. Flux, he certainly felt that the results of all the inquiries he had been able to make into past records of prices seemed to be a, very inadequate basis on which to form any judgment. An examination of the more recent periods showed such very wide fluctuations from totally different causes, that he could not feel that they could estimate the crop to the acre from the price. He had been looking at the United States wheat prices from 1866 to the present time, and he found there that for exactly the same crop to the acre the price had been 1 25 dollars per bushel and 0 75 in the middle series, and 1 15 at the end of the period. It seemed to him that if they were judging from the price, they would get a very wrong conception of what the crop to the acre was ; and apparently that was what had been taken in that question, because he could not feel that the rainfall had any direct influence on the price except through the crop to the acre.

Mr. H. W. MACROSTYsaid he would like to endorse the necessity for their having before them the data upon which Sir William Beveridge's very interesting paper had been based. He could not pretend that he knew anything about the method of analysis which Sir William had adopted, but as a working statistician he claimed to know something about the collection and comparisonof data and about the difficulty of ascertaining data for any commodity which were reliable to-day and comparable over a period. With that daily experience ever present in his mind he wanted to be very sure indeed of the data to which this instrument of harmonic analysis had been applied before he placed any confidence in the results of it. He wanted to know the full series of prices at all the various centres which Sir William indicated in one of his earlier papers in the Economic Journal. When the authenticity of those prices had been determined, and when they had become quite sure that the reporterseven at the same place over the same years were quoting the same thing in the same way-a habit which was not too frequent in their modernmarkets to-day, as he knew to his cost-then

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] on Sir William Beveridge'sPaper. 469 he wanted to know how far over the earlier period of Sir William's examination it was permissible to take a number of detached markets with very little, if any, communicationbetween them, and average up the prices which were found so as to construct an average price for Western Europe. Frankly, he did not believe that any averages over those'early centuries, compounded in that way, were of any value either for mathematical analysis or for descriptive purposes. That, however, was merely his personal opinion, and he thought they wanted a close examination of the data by economic historians and those statisticians who were skilled in the recordsof agricultural prices. Similarly, he thought they wanted-and he wished the Society could get it-the co-operationof the MeteorologicalSociety in'fhe matter, and an examination of Sir William's comparison of the cycles which he had derived with the meteorological cycles. He particularly wanted the meteorologists' opinion as to whether an average rainfall for the year, or an average temperature for the year, meant anything at all with regard to the effects of weather on crops and of crops on prices.

Sir THOMAS MIDDLETON said he wished to thank Sir William Beveridge for the exceedingly interesting paper which he had given them. He could not agree with the criticisms which Mr. Flux had passed on weather cycles, when one had before them the remarkable synthetic curve which Sir William had put on the screen. What Sir William predicted in 1850 was a good or a bad harvest for some sixty or seventy years ahead; and if they took his predictions, for example, for the years 1916 and 1917, they got on his curve precisely what happened. They had in these years general conditions independently altogether of the war, which if they had occurred before 1850 would have resulted in high prices. Those results were connected primarilywith the rainfall and with certain other features of the years 1916 and 1917 which were associated with the rainfall. ' He had read throughthe paperwith great interest ; one particular paragraph had arrested his attention, and that was Sir William's mention on page 431 with regard to his period I (5.67I years). With reference to that period, Sir William said that there was no demonstrated parallel in meteorologicalrecords. Now this was a period in which he happened to be particularlyinterested, and this morninghe thought he would spend a few minutes in seeing whether any parallel could be shown to exist. He thought he could find a parallel; but after he had spent not a few minutes, but a couple of hours, he came to the conclusion that he must be one of those " over sanguine and ill equipped " persons to whom Sir William referred. But it might illustrate a point that had been mentioned once or twice in the discussion if he gave them one or two figures which he had got out. This period of 5.67I years happened to connect two actual periods in which there was a physical cause for high

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 470 Discussion [May, prices in wheat, namely, continuous heavy rainfall. There was a period in the autumn of 1918 when they had two months or more of continuous rainfall. That period correspondedalmost exactly in its meteorological features with a similar period which he had previously noted as occurring in 1799. In 1799 heavy rain began about the 21st or 22nd of July, and in 1918 it began about the 31st of August, so that the interval was 119X09years. Sir William's period of 567I years went twenty-one times into this interval. This period also fitted in with the bad harvests of 1816, 1839 and 1879. Therefore this morning he thought he would look into the 5 67I year period a little more closely, as he had not examined it carefully before. He took the period and compared it with the rainfall for the months in the years between 1799 and 1918; but he had been unable to get the rainfall for 1805 and 1810. If one began in the year I799*56 (about 21st July) and added on three times 5-67I, they got i8I6-57, i.e. July, 1816. The two months', July-August, rainfall at Greenwich came out at 24 inches over the normal. If they went on to i822 24, there was in April, I .2 inches over the normal. Then for i827 9 they had in December, 1827, and January, 1828, 3-67 inches over the normal. Then the period 5.67i broke down, but re-appeared in April-May, 1856, July-August, 1867, December-January, 1878-79, April-May, 1907, and September- October, 1918, but it broke down in about half the number of cases. There were two consecutive wet months in several of the expected years, but they did not come at exactly the right time. He thought that the discrepancies did not necessarily show that there was no period; but that there was a period with something else operating which masked the effect. His time was exhausted, but what he wished to suggest was that if one took meteorological data and examined them in detail, a number of those cycles to which Sir William Beveridge had drawn attention might be found to have physical bases. He considered that Sir William had established a case for periodicity in weather.

The PRESIDENT said that at that late hour he would not attempt to comment on the paper at any length. It had interested him extraordinarily,although he had to confess that until that evening he had not had an opportunity of reading it with any care. As Mr. Yule had remarked, considerable study of it was needed to qualify one to make any observations about it. He agreed entirely with Mr. Yule that the proceedings of the Society were enriched by the paper, which was a contribution towards a subject of extreme importance and interest, not only to statisticians and economists, but also to agriculturists. There was one observation he would venture to make which might possibly expose his own failure to grasp the full significance of the paper. As he understood from Sir William Beveridge's interesting description the curve of wheat prices from 1850 was founded on the records of wheat prices for about three hundred

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] on Sir WilliamnBeveridge's Paper. 471 years, and might have been constructed seventy years ago if the data had then been collected and analvsed as Sir William Beveridge had now done. In fact, in 1850 the curve might have been constructed as a forecast of the variations in the level of wheat prices which a farmermight look forwardto. It would not have purported to give him any information about the actual prices he might expect, but he would have been justified in regardingit as an indication of the ups-and-downsof the market. In other words, at certain periods he would expect prices to be rising and at other periods to be falling. For example, he. would have looked at the indications for the early " nineties" and would have seen that a downward movement was to be anticipated, and he would have found when the time came that the indication given by the curve was quite correct- for in 1890-3 the price of wheat fell to an abnormally low level- actually reaching the lowest level of the century. But in looking to the end of the curve the farmer would find that at the end of the second decade of the present century wheat prices might also be expected to be on a descending scale. The farmer studying the curve in 1850 would hardly expect that the wheat prices in the period 1915-20 would be rising by leaps and bounds until in 1920 they reacheda level higher than had been known for about a century. No doubt the fault would lie in the farmer in not reading the curve correctly, but if the cycles did not truly indicate the tendencies of the markets they were of somewhat limited interest to the practical man. The lower curve constructed from rainfall records confirmed the indications of the wheat prices curve, if in fact rainfall was the main factor in determining prices. But was not that one of the questions at issue ? There were other points on which he would have liked a little more explanation, but he would not further detain them and would now put the vote of thanks to Sir William Beveridge for his most interesting paper. The vote of thanks was carried unanimously.

Sir WILLIAM BEVERIDGE in reply said: I have first of all to thank you for your very kind attention while I have tried to compress into a paper material long enough for a small book, for the vote of thanks which you have just passed, and for the very valuable comments and criticism of those who have spoken. I have certainly nothing to quarrelwith in those criticisms, though I. think I can give a good answer to nearly all of them. I should like to begin my reply by referring to the speech of your President. I am grateful to him for the opportunity of making clear the nature of my synthetic curve*; in my paper I obviously did not make this as clear as it should have been. The synthetic curve, though it is based on cycles which have ultimately been

* Sea supplementarynote A.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 472 Discission [May,

derived from an analysis of wheat prices before 1850, has nothing to do with prices during the period 1850 to 1921 which it covers, and is not in any way to be taken as a prediction of the course of prices then. In the first place, European wheat prices since 1850 have been influenced by two factors-the development of the credit cycle and opening up of America-which were absent or relatively weak before then. In the second place the phases of the economic cycles derived from wheat prices have been variously, though slightly, corrected (as is explained in a part of my paper which I could not read) to turn them into meteorological cycles. In the third place, for reasons set out at length in my former article, it is not to be expected that even a real rainfall cycle should influence the harvest at every revolution ; sometimes it may be practically inoperative; it shows in the periodogram only because of the long series of years and of revolutions included. The synthetic curve is meteorological, not economic; it is given as a prediction that might have been made in 1850 of the European rainfall of the next seventy years, but certainly not of the harvests and still less of prices. A farmer who read it in the way suggested by Sir Henry Rew would have certainly come to grief. To Mr. Yule as the mover of the vote of thanks and as having raised a number of important points, I am particularly grateful. One of those points-as to the duplication of amplitudes in harmonic analysis and the possibility that a high amplitude at 2 2 years may really represent either a period of that length or one of about i*83 years, I would like to reserve -for consideration later.* As to the I 53 year cycle Mr. Yule says that when I first put it -forward I appeared to him to be doing so on quite inadequate grounds. But I think Mr. Yule (and others) rather over-estimated the assurance with which I then put it forward; I-did not say that the existence of the cycle was certain or even probable, but only that it was worth enquiring into; in that at least I seem to have been proved right. Mr. Yule now criticizes my " unparental " attitude in questioning the independent reality of the period, after it has demonstrated itself so strongly in the harmonic analysis of wheat prices. Well, I am in rather a difficulty about that period. My unparental attitude is due to modesty. I should naturally like the period to be real, and I think that I have myself found it in barometric records. I have worked out the amplitudes for the first and secondhalf sequences as Mr. Yule suggests and will give them in the table of amplitudes as finally printed; they show a strong periodicity in each half near I 5 years. But no competent meteorologist seems to have found such a periodicity either in the barometric pressures or anywhere else. Further, the famine records before 1550, so far as I have studied them, are more consistent with the suggestion that the I5'3 period results from a combination of other cycles than with the view that it is independent.

* See supplementarynote B.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] on Sir William Beveridge'sPaper. 473 This brings me to the theoretical point raised both by Mr. Yule and by Dr. Brownlee. Of course a number of separate harmonics in a set of observations do not, simply because they all have some common multiple, build up a distinct harmonic component of the length of that multiple. To suppose that they did would be to make an elementary error as to the nature of harmonic analysis; it would, as Dr. Brownlee says, be a " fundamental blot " on my paper. But I never meant this; it is not an error that could be made by anyone, however ignorant of mathematical theory, who had 'ever done any practical harmonic analysis. Three separate rainfall cycles of 3, 5, and 71 years will not, just because they have a common multiple of 15 years, cause a cycle of that length to show itself in a periodogram of rainfall. My figures, however, relate directly to wheat prices and indirectly to wheat harvests + the relation of the harvest to the rainfall or any other meteorological condition is not simple. Three rainfall cycles of 3, 5, and 7j years may produce a true I5-year cycle in the harvestsbecause it is only excess of rain, not rain as such that destroys the harvest. Each of the shorter cycles by itself may show strongly in tho rainfall but little if at all in the record of wheat prices; their combination may dominate the wheat prices but be invisible in the rainfall. There is no inconsistency between my suggestion and the arguments of Dr. Brownlee and Mr. Yule; there is only a misunderstandingas to what I meant by a combination of cycles. For this misunderstanding I am probably myself mainly responsible. I had dealt at length in my former article with the different way in which cycles and their combinations might show in the meteorological and in economic records. Having this in mind, I did not repeat the argument or specificallyrefer to it in my later paper. Dr. Brownlee's other criticism relating to the 27I years' interval is also, I think, founded on a misunderstanding, and one for which I do not feel the same responsibility. Ile appears to think that I believe in the existence of a cycle of 271 years showing itself at its 4th, 5th, 2 1st, and further harmonics in the various periods named by me. My former article, however, made it quite clear that the 271 years represented not a true period but merely a " conjunction interval " happening to be the common multiple of a number of true periods, and therefore leading to the production of important maxima of prices or famines. Returning to Mr. Yule's speech, the reason given by him for attaching importance to finding the same period in both the first and second halves of a sequence taken separately, is a very valuable contribution. The finding of the same period in the two halves with agreement of phase as well as of length involves, of course, a great strengthening of the argument for the reality of the period.* With reference to Mr. Yule's final point, when he expressed some surprise at the comparatively close agreement between the phases

* See supplementary note C.

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 474 Discussion [May, of the rainfall cycles and the wheat price cycles, I ought to call attention to the fact that in determining the phases of my cycles, I put the origin at the beginning and not at the middle of the first harvest year to which the observations relate. On the whole, having regard to the nature of the data this seemed the most reasonable origin to take. If, however, anyone else prefers to treat the middle of the first harvest year as the origin, he would then get a lag of something like half a year between the two sets of data. He would find the synthetic curve always about half a year later than the actual rainfall curve. Dr. Shaw's comments are of great interest. Unfortunately he had to make them upon an earlier draft of the paper and of the chart at the end of it. Since he made these comments the chart has been revised so as to show the dates more accurately. It has also had to be revised owing to the discovery of an arithmetical error, and the revision of the synthetic curve, while it still leaves the years 1315-16 as exceedingly wet years, does not give them at present the overwhelmingimportance which they had in my earlier draft. Revision of the text of my paper has also been necessary. I am sorry not to be able to deal with all the other speakers. I must content myself with answering so far as possible the general criticism put forwardby Mr. Flux and Mr. Macrosty. The first relates to the nature of my data. I would naturally have liked very much to begin my paper with a full account of the sources of my index-number, and a justification of its value on a prior grounds. I did in fact begin my first draft in this way, but the retention of this would have made the paper, which is already much too long, altogether monstrous. I can only say now briefly that the material is in fact a great deal better, and the disturbing influences of wars, inflations, &c., is very much less than almost anyone would suppose before looking at the figures. The statistics I have used have in fact in all cases been deliberately drawn up for comparative purposes, generally in order to fix a corn rent from year to year. For each place the price given for any year does yield a fair comparison with the years in its neighbourhood on either side. Further, the value of the figures is strikingly shown if one calculates the index-numberas I have done for each of the five or six main divisions of the whole area (Britain, Low Countries, France, &c.). The general agreement between the divisional curves at a time when there could have been relatively little trade and no common market price strongly confirmsthe value of the figures, and the identity of the main cause affecting the prices. I certainly hope at some later time to give a full account of my index-number. I venture to think, however, that its character is really put beyond suspicion by looking at its results. It is inconceivable that if my statistics were, as Mr. Flux says, " soft " and not hard enough to stand the process of harmonic analysis, they would under that process yield exactly the same periods as those

This content downloaded from 185.44.77.40 on Wed, 25 Jun 2014 03:22:47 AM All use subject to JSTOR Terms and Conditions 1922.] ontSir William Beveridge'sPaper. 475 which have been found by the same process in other records. To take one cycle alone, my figures show that if in wheat prices there is any periodicity at all, there is a periodicity of about 5 I years having a particular maximum phase. Captain Brunt, using a totally different set of figures, but the same objective method of harmonic analysis in absolute independence, says in effect that if there is any continuous periodicity in Greenwichtemperatures from 1850 to 1910, it is a period of about 5 I years, with a phase which is found to agree very closely with mine. Mr. Baxendell again independently found the same period with the same phase. What conceivable possibility is there that my figures could yield this sort of result by chance ? Exactly the same argument of almost the same strength arises fromnthe 35-year period and others. The last point I wish to make arises from what Mr. Flux said. I certainly do not wish to assert that the whole of the weather can be reducedto a series of cycles. The cycles which I have found may, I think, ultimately be found to account for 30 per cent., or 50 per cent., or possibly even 75 per cent. of the weather. To this extent the rainfall may be reduced to mathematics. But I do not in the least know what this proportionis, and I know still less what may be the law governing the balance. Until we know it is impossible to prophesy. That is the reason why I do not yet prophesy.

SUPPLEMENTARY NOTES. Furnishedby Sir William Beveridgeafter the meetikV. The following notes deal with one or two special points on which investigations suggested by the discussion of my paper have been carried a little farther.

Note A.-Synthetic Curve and Rainfall. Since my paper was written I have seen Mr. R. H. Hooker's paper on the Weather and the Crops in Eastern England, 1885-1921 (printed in the Quarterly Journal of the Royal Meteorological Society, April, 1922). From this, the important fact emerges that in the area under review all the significant correla- tions between the yield of wheat and the rainfall are negative. In Mr. Hooker's own words " wheat never seems to want more rain than it gets," and again, wheat " never seems really to require an average rainfall at any period." This is in complete accord with the hypothesis on which the synthetic curve has been constructed and compared with rainfall, a maximum of prices, i.e. a poor crop of wheat being associated with excess of rain and never with deficiency. No other crop examined by Mr. Hooker has this peculiarity; no other crop would therefore serve equally well for my harmonic analysis. The coefficient of correlation between the synthetic curve and the curve of actual rainfall in Europe from 1851 to 1905 (as drawn in the chart) is *384 1 - r2~ and the probable error (.674 X r) is .077. For the first differences, the correlation coefficient is .435 and the probable error .074. Thus the correlation coefficient is in one case five, in the other case six times the probable error. This would, I think, be accepted by most statisticians as satisfactory evidence of a close connection between the two curves.

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