The Determination of Longitude in Landsurveying.” by ROBERTHENRY BURNSIDE DOWSES, Assoc

316 DOWNES ON LONGITUDE IN LAXD SUIWETISG. [Selected

(Paper No. 3027.) The Determination of Longitude in LandSurveying.” By ROBERTHENRY BURNSIDE DOWSES, Assoc. M. Inst. C.E. THISPaper is presented as a sequel tothe Author’s former communication on “Practical Astronomy as applied inLand Surveying.”’ It is notgenerally possible for a surveyor in the field to obtain accurate determinationsof longitude unless furnished with more powerful instruments than is usually the case, except in geodetic camps; still, by the methods here given, he may with care obtain results near the truth, the error being only instrumental. There are threesuch methods for obtaining longitudes. Method 1. By Telrgrcrph or by Chronometer.-In either case it is necessary to obtain the truemean time of the place with accuracy, by means of an observation of the sun or z star; then, if fitted with a field telegraph connected with some knownlongitude, the true mean time of that place is obtained bytelegraph and carefully compared withthe observed true mean time atthe observer’s place, andthe difference between thesetimes is the difference of longitude required. With a reliable chronometer set totrue Greenwich mean time, or that of anyother known observatory, the difference between the times of the place and the time of the chronometer must be noted, when the difference of longitude is directly deduced. Thisis the simplest method where a camp is furnisl~ed with eitherof these appliances, which is comparatively rarely the case. Method 2. By Lunar Distances.-This observation is one requiring great care, accurately adjusted instruments, and some littleskill to obtain good results;and the calculations are somewhat laborious. The observation consists of measuringthe angular distance between the moon and certain tabulated celestial bodies (given in the Ephemeris of the Nautical Almanack for the month), and also the altitude of one or both of the bodies, or, in the case of the true mean time a.t. the observer’s place being

Minutes of Proceedings ht.C.E., vol. cviii. p. 230.

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. Pnper3.1 DOWNESLOXGITUDEON IN LANDSURVEYING. 317 known, of the distanceonly. Inthe first case two or three instrumental observers are required, but in the second only one. That one is, however, dependent on thetrue time in amuch greater degree, and much extralabour isinvolved inthe computation. Next, by comparing the distance so obtained, after reduction to true distance with the tabulated lunar distances for the meridian of Greenwich, as given in the Nautical Almanack (whence the Greenwich time of theinstant of observation is computed, andby comparison withthe true mean time of the observer’s station),the longitude of the place is found.Hence, if the true mean time of the place is not known, it is necessary that the Iatitude should be known, or that a true azimuth should be observed, the latter course beingunusual and inconvenient. The usual conditions are that the latitude is known, and one or both altitudes or the true mean time are observed. The accuracy of the observationdepends upon the exact measurement of the lunar distance; a smallerror in the observed altitudewill not much affect the calculation, but an error of l’in the distance will cause an error of nearly 30’ in arc of longitude. For this measurement it isnecessary to have a good and accurately adjusted sextant, which shouldbe used with the long telescope. It is best when possible to observe at least one altitude-that of the moon by pre- ference, if its position admits of it, because the correction for the moon’s parallax in altitude and the moon’s semi-diameter depend upon the apparent altitude-for an error of 1”in time represents a in arc, which is theprincipal trouble when no altitudeis observed. The observation is easier when the altitude of the moon is less than that of the sun, or greater than that of a star or a planet.The moon’s enlightenedlimb should bebrought into contact with the nearer limb of the sun or with the centre of a star or planet. In taking the altitude of the moon with a sextant, the upper or lower limb must be brought down to the horizon. In the computation the corrections necessary to all the data to be used, both from observation and the NauticalAlmanack, depend- ing upon the features observed, are first applied. The moon’s semi-diameter is obtained as follows:-Take the semi-diameter for Greenwich mean noon or midnight (Ephemeris, pp. 111 of the Nautical Almanack) and apply to it the correction for the assumed difference of longitude; next apply the positive correction for augmentation (Chambers’ Tables, p. 432), and the result is the moon’s augmented semi-diameter. The correction for augmentation is dependant upon apparentaltitude, but it is practicallythe same whetherthe altitude be true or apparent,

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. 318 DOWNES ON LONGITUDE IN LAND SURVEYING. [Sclocted still, in thecase of no altitude being observed, it may be left until after the apparent altitudehas been calculated. The correction for the moon’s parallax in altitude is dependant upon apparent altitude; hence, if no altitude be observed, it will be necessary to defer the reduction until after the apparent altitude has beencalculated. To thehorizontal parallax obtained from the Nautical Almanack (Ephemeris, pp. 111) apply the correction for the assumed difference of longitude; the equatorial horizontal parallax is a negative quantity dependant on latitude (Chambers’ Tables, p. 427), and theresult is reduced horizontal parallax, which must be reduced to seconds and multiplied by the cosine of the moon’s apparent altitude (corrected for refraction) when, after reduction tominutes and seconds, it is theparallax in altitude. To obtain the true altitudeof the moon’s centre, if the altitude of a limb of the moonbe observed, from this observed altitude subtract the “dip” (if the sextant is used for this part of the observation), also theindex error ( T), if any,and apply the moon’s augmented semi-diameter(as above), andthe result is the moon’s apparent altitude; from thissubtract the refraction (Chambers’ Tables, p. 428) and add the parallax in altitude (as above). If no altitude be observed, thisquantity must be computed, as explained below. In determining the apparent distance, to the observed distance, after correctingfor index error, if auy, apply themoon’s augmented semi-diameter (previously obtained), and, if the sun be the companion body observed, the sun’s semi-diameter (Nautical Almanack, Ephemeris, pp. 111). If the altitude of the second body, sun or star, be observed, the apparent altitude and true altitude are deduced in the ordinary manner. If a planet be used, toits observed altitudemust be applied a correction for parallax in altitude,for which is required a Table that does not appear in the books usually in the hands of a surveyor. If one or more altitudes have to be calculated owing to their not having been observed, the moon’s right ascension and declination must be taken from the NauticalAlmanack (Ephemeris, pp. V) and corrected for the assumed difference of longitude. Also the right ascension and declination of the companion body must be taken from the Almanack,and, if necessary, corrected for the assumed difference of longitude. Having now right ascension of the moon and of the companion body, subtract the less from the greater and the difference is the

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. Pfbpe15.l DOWNEB ON LONGITUDE IN LAND SUBVEYIIG. 319 hour angle between them. Next, having the latitude of the place, the true altitude (if the moon’s altitude has been observed) and declination of the moon, the hour angle between that body and the meridian of the place can be computed by the extra meridian computation for time. Thehour angle thus obtained, on being applied to the hour angle between the meridians sf the bodies, will give the hour angle between the meridian of the place and that of the companion body. Then with the hour angle, latitude and declination of the companion body, its true altitude can be computed, and from thisresult the apparent altitude must be deduced by theconverse of the usual correction. In the eventof no altitude havingbeen obtained byobservation, both altitudes must be calculated from the true mean time at the place, which must be very accurately known. The procedure is as follows :-From the true mean time of the place at the instant of observation must be deduced the siderealtime, or right ascension of the meridian of the place by the conversion of the interval of mean time sinceprevious mean noon, intothe corresponding interval of sidereal time, and to this must be applied the sidereal time at previous mean noon, the result being the right ascension of the meridian of the place. Having the right ascension of the . place, and that of each of the bodies observed, the hour angle of each of the bodies may be readily deduced, and then the true and apparentaltitudes canbe obtained (as related above) for the companion body. This method of computing the altitudes shows the meaning of‘ eachstep, butthe following concise tabulated form for the computation, given in Norie’s epitome,’ which is mathematically the same, may be noted :-Having obtained the hour angle (HA) of the body, as mentioned above, then

cos HA X cot lat ( -10.0 in logs) = tan Arc I. Arc IN polar distance = Arc 11, if HA be less than 90°, or Arc I + polar distance = Arc 11, if HA be greater than 90°, that is, whenthe hour angle is less than G hours, takethe difference between Arc I and the polar distance to obtain Arc 11, but if greater take the sum sine true alt = sin lat X sec ( - 10.0 in logs) Arc I X cos Arc 11, and then the apparent altitude isdeduced as before.

“A Complete Epitome of Practical Navigation,” by J. W. Norie.

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. 320 DOWNES OS LOKGITUDE IN LdND SURVEYING. [Selected The mathematical principle of the computation of the true distance is as follows :-In Kg. 1, Z is the zenithof the observer ; S’ and M’ the apparent places of the two bodies, and S and M the true places of the same; Z S’ and Z M are the apparent co-altitudes, already obtained by observation or calculation ; and Z S, Z M are the true eo-altitudes similarlyobtained (M’ appears below M, because the amountof the moon’s parallax exceeds the refraction due toaltitude); OHR isthe horizon and 0 is the place of observation; S’M‘ is theapparent distance, also observed, and from this data it is required to obtain S M, the true distance, the procedure beingas follows:-With the given sides, Z S‘, Z M‘ (apparent co-altitudes) and S’ M’ (apparent distance) compute the angle S’ Z M‘ = angle S Z M. Then in the triangle Z S M having the angle S ZM, and the sides Z S, Z M (true co-altitudes), the angles at S and M may be calcu- Fig. 1. lated,or more strictly,their sum and difference, whence the angles can be deduced, and then all the information necessary to find S M, thetrue distance, is obtained for the true value to be computed accordingly. Norie gives the tabular form 1 on the followingpage for this ’ calculation, which, whilst mathematically identical with thefore- I \ going, is concise, andmaysave F------______&______h time in applyingit. The truedistance must be compared with the tabulated distancesfor the meridian of Greenwich, givenin the Nautical Almanack(Ephemeris, pp. XI11 to pp. SVIIIj. As it isnecessary to have a Nautical Almanack for this cperation, and as there is givenin that publication a most complete and lucid explanation (Explanation to Tables) of the remainder of the computation, it is unnecessary to repeat it. As the longitude of the place of observation is of course not known until the computations are completed, it is necessary to assume a longitude as closely as possible, under the circumstances, forthe reduction of the elementsobtained from theNautical Almanack, which are all for the meridian of Greenwich. Should

~ * ‘‘A Complete Epitome of Pmctical Nayigation,” by J. V. Norie.

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. Papere.] DOWNES ON LONGITUDE m LAND SURVEYING. 321 the computationshow this assumption to have been much in error, it will be requisite to begin the computation afresh with a new assumption of longitude, based on the first calculation,

Apparent distance Moon’s Apparentaltitude . ... log sec - 10 Second body Apparent altitude . ... log sec - 10

Half sum . .. . logcosine Apparent distance Difference . ... logcosine Moon’s altitudeTrue . .. . logcosine Second body Truealtitude . ... logcosine

2) Sum 2) Sum of logs Half sum = Arc I AIC It log aine Sum . . logcosine Difference . logcosine 2) Sum of logs

Half True distance . i. log sine 2 True distance required

re-working the reduction of the elements on this new basis ; and it may occasionally happenthat this process will have to-tbe repeated a third time. In thereduction when the distance Fig. 2. but no altitude is observed, theonly different feature, besides the additional labour of calculatingthe altitudes, is that of computing the moon’s apparent altitude corrected for refraction, which is dependant on the moon’s parallax in ,, altitude.The latter, in this case, has to becomputed from thehorizontal parallax with the true altitude, instead of with the apparent altitude, as when c the altitude of the moon is observed. Fig. 2 will explain the problem ; Z is the zenith of the observer ; 0 theobserver; C thecentre of theearth ; the moon on the sensible horizon, and m the corrected and reducedhorizontal [THE INST. C.E. VOL. CXXXIII.] Y

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. 322 DOWNES ON LONGITUDE IN LANDSURVEYING. [Selected

parallax; N isthe moon inthe position observed, and n the parallax in altitude; C M, C N represent the moon’s distances from the centre of the earth (assumed to be equal), and N 0 M is the apparent altitude corrected for refraction (which is the quantity first to be obtained) ; *Y Q C isthe mean radius of theearth, 3,956 miles. The horizontal parallax is obtained from the Almanack and changed into reduced horizontal parallax, as previously explained, which is m in the figure; 0 C N is the true co-altitude and has already been computed at this stage. Then :- 3956 cosec m = C M = C N, or the moon’s distance, 3956 - CN 180 - OCN CON-n next = tan 3Y56 + C N tan 2 2’ 180-OCN_____ CON-n and p2---= 90’ + N 0 M. 2 + Then, by subtracting 90’ from theresult, NOM, theapparent altitude correctedfor refraction is obtained ; nexttake the refractiondue to N 0 M from theTable (Chambers’Tables, p. 428) and N 0 M + refraction = the moon’s apparent altitude, requiredfor use in the reduction of apparentdistance to true distance, as detailed above. This computation is also necessary if one altitude-not that of the moon-be observed. Probably there are quickermethods of achieving this result, butnone more readily apparent in procedure. A full example of the computation, with no altitude observed, is given in AppendixI. Method 3. By Moon Culminating Stare.-This is a verysimple and accurate method of obtaining longitude, and when it can be ntilized has many advantages over the lunar method. Longitude is difficult toobtain with accuracy in any case, except with powerful observatory instruments; but this method, though not free from sources of error, will in general give closer results than thelunar method underthe conditions of a survey camp. It possesses thefollowing advantages over the lunar distance observation:-Itis more easy of manipulation, for whilstthe culminating method ismerely the observation of a couple of meridian transits with a theodolite in the ordinary way, the lunar distance methodinvolves the use of a hand instrument.The reduction is comparatively simple and short. The operation can be conducted by a single observer with a stop-watch. The time- piece used need not be regulated by any particular meridian, as it is only the lengthof the interval between transits that isobserved. There is practically no sourse of error from refraction or ordinary

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. Papers.] DOWNES ON LONGITUDE IN LAND SURVEYING. 323 irregularities of theinstrument. On theother hand, atrue meridian is necessary ; second, the culmination method cannot be employed in daylight, and the observer is tied down to certain definite lnoments which for at least half the lunar month occur at very inconvenient hours, unlike the distance method which can be employed almost at any time that the moon is visible; third, most of the stars that have to be used are of small magnitudes, and are difficult,to find sometimes even when searched for with the aid of thelatitude and the true mean time;fourth, the culmination method usually occupies a longer time, and there is the chance always of being stoppedfor thenight by clouds at acritical moment. The difficulty of precise reduction of the lunar elements is common toboth methods, butthe culmination. method is dependant on the right ascension alone. The principle upon which the culmination method is founded, is that whilst the right ascension of a star remains practically oonstant for, at least, one day, that of the moon alters rapidly. Certain stars, thepositions of which in theheavens are approximate to that of the moon, are selected and tabulated in the Nautical Almanack underthe title of Moon CulminatingStars, that is, starsculminating at about the same timeand declination as the moon. Inthe Table opposite to each! staris given its apparentright ascension anddeclination; in the same Table is also given the right ascension of the first or second limb of the moon (distinguished by the numbers I and I1 respectively), attheir upper and lower culminations(distinguished by the letters U and L respectively).These are all reduced tothe meridian of Greenwich. The difference between the rightascension of the selected limb of the moon, and that of any star given in the Table for the same day, is the difference in sidereal time between their respective transitsat the meridian of Greenwich. Butas the moon istravelling whilst the star is stationary in right ascension, this difference of timeis not the same a shorttime before orafter that transit. Now on observingthe respective transits of these bodies on some meridianother than that of Greenwich, a different interval of time between themwill be obtained, and the difference of these two intervals will represent the distance in rightascension through which themoon has passed in travelling from the one meridian to the other. Then, by ascer- taining the rateof change in rightascension of the moon, the time occupied bythat body inchanging its position tothe extent observed can be computed, andthat time is the difference of longitude between the two meridians. Y2

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. 3 24 DOWNES ON LONGITUDE IN LAND SURVEYING. [Selectea The method of operation is as follows :-Set up the theodolite on the true north and south line andobserve the meridian passage of the earlier of the two bodies, the staror the moon’s limb, as the case may be, and at the instant of transit start the stop-watch. Next, on the transit of the second body, stop the watch the instant of transit. This will complete the observation, and the interva) of time so obtained is the quantity tobe compared with the difference of the right ascensions given in theTable. As before stated, the stars for :this purpose are often of smalk magnitude and not easy to distinguish ; hence it is advisable to first calculate the apparent altitude and time of transit, and with this information to set the instrument on to the position of the star’s culmination. If thetrue time is knownnearly, itwill enable the observer to select the right star with certainty, which is otherwise often impossible, owing to the close proximity of other small stars. For this reason it is decidedly advisable to have the true mean time and latitude of the place obtained prior to this observation ; when, by the application of the latitude to the declination, the true altitude, and with the addition of the refraction,, the apparent altitude,is obtained. To this altitude the instrurnenb should be set, and in order to guard againstpossible inaccuracy, the observer should begin to watch for the star a few minutes before the calculated time of transit.The true time is, however, nob required in the reductionof t.he observation. For reducing the observationvarious methods embodying the same principles may be employed, but probably the following will be found as simpleas any arriving t,he at same degree of accuracy :- Convert the observed interval of mean time into the corresponding interval of sidereal time, and apply it to the right ascension of thestar observed, according as thetransit of the moon is before or after that of the star, the result is the right ascensiolr of the moon’s limb at the meridian of the place. To this apply thesidereal time of the semi-diameterfor theday, according as the first or second limb of the moon has been observed ; and the result, A, is the right ascension of the moon’s centre at the meridian of the place. Nexttake from theTable the right ascension of the limb observed, and to it apply the sidereal time of the semi-diameter as before, which will give the rightascension of the moon’s centre at transit at Greenwich; this is the sidereal time of the moon’s transitat Greenwich. Add 24 hours, if necessary, and subtract the sidereal time at previous mean noon (Ephemeris, pp. 11),which gives the intervalof sidereal time since previous mean noon. Convert this into mean time, and the result,

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. Papers.] DOWNES ON LONGITUDE IN LAXD SURVEYING. 325 B, should be the same as the mean time of transit given in the Ephemeris, pp. IV, but to seconds and decimals of seconds instead of to the nearest six seconds only. Extract from the Ephemeris for the month the right ascension of the moon, for the hour having the nearestpreceding right ascen- tsion to that of theresult A. Add to the right ascension given forthis hour,half thevariation due to 10 minutes, andthe correspondingtime is 5 minutesafter the hour. Ifthis does notbring the right ascension up to thequantity A, takeout ;the difference of the variations due to 10 minutes for the hour named and that following it; an6 dividing the difference by 6, add & tothe variation in 10 minutes for theearlier of the two hours, andadd the result to the right ascension already -obtained,which will then correspond with 15 minutes past the hour. Bepeatthis operation untilthe resulting right ascension lis within a distance of the amount A, that will be overlapped by %he variation of the succeeding 10 minutes, and then proceed to deduce the restby proportion, thus :- Variation in the 10 Amount of right ascen- I minutes next in order ):I sion still to be made up 1 = 10 minutes : Time occupied in making up that difference. Add this result to the time already obtained, and the total sum .of thetime will represent the true mean timeat Greenwich, whenthe right ascension of the moon's centre is equal to the quantity h. Take the difference between this last obtained mean time and the quantity B; the result is the difference in mean time between .the meridian of the place and that of Greenwich.Convert this into sidereal time in theusual manner, and the result is the >required difference of longitude in hours. An example of the method is given in Appendix 11. In longitudes considerably removea from that of Greenwich stars of greater magnitude than those of stars given in the Table of moon culminating stars may frequentlybe used for this method, to obviate thedifficulty of identification existing in thecase of stars .of lesser magnitude. The computation is the same in suchcase, only %hat the right ascension of the star used must be corrected to its proper place for the date of observation (when necessary) in the .mannerset forth in the explanation to Tables inthe Nautical Almanack. The Paper is accompanied by the F'gs. which are reproduced Gn the text. [APPENDIXES.

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. 326 DOWNES ON LOXGITUDE IN LAKD WRVETISG. [Selected

APPENDIXES.

APPENDIX I.

EXAMPLEOF ‘‘ LCNA~DISTAXCE ” OBSERVATION WITH NO ALTITUDEORSERVED. June 25th, 1890, at 9h. 59m. 1e.73~.true meantime at place, in latitude 15O 42‘ N., longitude (assumed) 62O 30’ E., the following lunar observation wae taken between thc planet Mars and thc moon. Observed distauce Mars’ ccntre to moon’s farlimb 53O 56‘ 40”. Indexerror 1’ 4” to subtract.Required the true longitude. I I, H. 1. S. 0 Moon’s right asceu-) 3o Moon’sdeclination . 2 20 43.2 N. sion .... Corrected for longi- Corrected for longi-\ - 22.i7 tude (assumed) .> +a 35.2 tude (assumecl) .I --

-__- Corrected_~.~~~~~ declinatiou 2 ?R 18.4 Correctedright as-) 12 55.09 90 0 0 cension ... -- N. P. D. .... --S7 36 41.6 , ,I H. X. S. Moon’shorizontal pad-) 55 25 Mars’ rightasceu- 1542 28.55 lax...... sion ....). Corrected for longitudc Corrected for longi- -7.6 (assumed). ... +10 j -- 55 35 Corrected rightas-\ 15 42 20.03 Beducedequatorial ho-} -o.s censlon ...I rizontalparallax . . -- -- 0 , I, Reducedhorizontal pa-} 55 34.2 Mars’ declination . 22 40 42.6 S rallax ..... Corrected for longi- -- tude (assumed) . ] -2.9 Moon’s semi-diametercannot be -- computed untilaltitude is ob- Corrected &clin-} 22 4o 39 ,7 tained. ation tained. .... Mars’ horizontalparallax = 17”, 90 0 0 parallax in altitude = - 11.5”. N. P. D. ...--112 40 39 7 H. If. S. True mean time of observation = interval of mean] g 59 18.7,3 timesince previous mean noon ..... Acceleration ...... + 1 35.45 Interval of siderealtime since previous mean noon 10 0 5i. 18 Sidereal time of previousmean noon from Almanack G 14 15.41 Siderealtime of observation = right ascension of) 1G 15.59 meridian ...... Right ascension of Mars ...... 15 42 20.93 Hour angle of Mars ...... 0 32 54 66 = in arc 8’ 13’ 40”. hTorie’s Epitome of Navigatiou.

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. Papers.] 90WNES ON LONGITUDE IN LAND SURVEYING. 3 27

H. M. 8. Right ascc~nsionof rncrillian as above .... 16 15 15’59 , moon ...... 12 29 55.09 Hour auglc of moon ...... 3 45 20.5 = in arc 56’ 20’ b”.

To COXPUTETHE ALTITUDES.

0 , I, Mars’ hour angle mrst Y 13 40 COS 9.995507 Latitudc .... 15 42 0 cot-l0 0.551159 sin9.432329

Arc1 ..... 74 8 42 tau 10.51686ti sec-l00,563513 Mars’ N. P. D. .. 112 4040 Arc I1 .... 38 31 58 cos 9.893346 -- Mars’ truealtitude . 50 47 11’5 sin9.889188 -- (less 10) Mars’ parallax in altitude ....} -11.5 -- 0 , I, 50 47 00 Thcreforellars’true) 47 Refraction ... altitude +46 ...I -__Apparentaltitude . 50 47 46 Mars’apparent alti-} 5o 47 46 tude ....

~~~ ~ ~

0 , t, Moon’shouranglewcst 56 20 8 cos 9.743767 Latitude .... 15 42 0 cot-l0 0.55115Y sin 9.432329

Arc I ..... 63 6 39 tau 10.294926 sec-l0 0.344608 Moon N. P. D. .. 87 36 42 Arc 11. .... 24 30 3 cos 9.959020 Moon’struealtitude . 32 59 13 sin9.735957 (less 10)

0 I ,I P cosec H. P. = 3956 cosec 55‘ 34” = 244761 24476190 0 0 0 I I, 3956 32 59 13 240805 tan 61” 29’ 364” = 6042 35.5 - _.- 248717 61 29 36.5 240805 57 0 47 248717 180 0 0 122 1212 -- 90 0 0 2) 122 59 13

&Toon’s apparentaltitude . . 33 12 12 { correctedrefraction. for 61 29366 Refraction ...... 0 1 30 Moon’s apparentaltitude . . 32 13 42 :. :. True altitude 32O 59’ 13”. Apparent altitude 32O 13’42”.

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. 328 DOWNES ON LONGITUDE IN LAND SURVEYING. [Selected

, I, Moon's semi-diameter ...... 15 7.6 Corrected for longitude(assumed) ..... +2*8 Augmented for altitude ...... 8.3 Moon's augmentedsemi-diameter .....15 18'7

TO COIPUTE THE TRUEDISTANCE.

0 I ,I Observed distance ... 53 56 40 Index errorIndex ..... 14 -- 5355 36 Moon's augmented diameter...... 0 15 19 observed to far limb :. subtract.

Apparentdistance ... 53 40 17 Mars' apparentaltitude . . 50 47 46 sec-l0 0'1992267 Moon's apparentaltitude . 32 13 42 sec-l0 0.0726658

2)136 41 45

68 20 53 cos 9.5669819 -4pparent distance ... 53 40 17

14 40 36 cos 9.9855930

Mars' truealtitude ... 50 47 11 cos 9.8008637 Moon's truealtitude ... 32 59 13 c08 9.9236556

2)83 46 24 2) 19.5489927

Arc I ...... 41 53 12 Arc I1 ...... S630 38 sin 9.7744963

Sum ...... 78 23 50 cos 9.3034669 -- Difference ...... 5 22 34 cos 9.9980854

2) 19-3015523

26 34 56 sin 9.6507761 2 --

53 9 52 = true distance.

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. f%perS.] DOWNES ON LONGITUDE IN LANDSURVEYING. 329

TO COMPARE WITH GREEXWICHDISTANCE AND DEDUCETHE LONGITUDE. 0 I l1

GreenwichticalAlmanack) distance (from.... Nau-) 54 36 58 P. L. 2850 at 25d. 3h. Om. Os. True distance at place ... 53--952 l 27 G P. L. 3152 - 0 I ,I Approximateinterval ... -0302 = 54.52 47 Correction for second difference from NauticalAlmanack ---1 Meantime at Greenwichlunar by ...... 5 47 53.5 Acceleration ...... 1-57.15 Interval of sidereal time since Greenwich previous mean) 48 50.6B noon ...... Siderealtime of previous mean noon ...... G 14 18.41 Right ascension (of Greenwich) at instant of observation 12 3 906 Sidereal time, or right ascension of meridian, as above . --16 15 15.59 Difference of longitude to east ...... 12 4 6.53 15 Longitudein arc ...... 63 1 37.95 Therefore longitude required 4h. 12m. 6.58. E., or 63O 1' 38" E. NoTE.-This example comprises the whole of the work of the method under any conditions. Where one or more altitudes are observed instrumentally these altitudes not having to be calculated, the reduction is so much shorter, the remainder of the operation being the same.

BPPENDIX 11. EXAMPLEOF MOON CULMINATINQ STA~ METHOD. On the 4th November, 1895, the transit of Moon I1 over the meridian was observed and afterwards that of B Tanri. The interval of mean time between the twotransits was lh. 3m. 1s. Requiredthe longitude of the place of observation. 0 , ,I Observed intervalbetween transits in meantime] 11 lh. 3m. 1s. = in sidereal time ...... Right ascension of B Tauriat transit (from Table) . . 5 19 43.91 Right ascension ofMoon I1 at transit on observed) 16 32.56 meridian ...... Siderealtime of moon's semi-diameter passage (from] 12.39 Table) ...... Right ascension ofmoon's centretransit at observed) 15 20.1, meridian ...... :. result A = 4h. 15m. 20.17s.

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330. 330 DOWNESLONGITUDEON IN LAND SURVEYING. [Selmted

0 I It Right ascension of Moon 11 at transit at Greenwich} 36 18.38 (TRble) ...... Siderealtime of moon's semi-diamcter meriditru pass} 12.39 (Tablc) ...... ~- Rightascensiontransit ruoou's centreGreenwich = side-} :sj real time of transit ...... 24 0 0

__-.- Greenwich mean time of meridian pnss moon's centre at Greenwich ...... :. :. result B = 1311.381x1. 55,976. The Ephemeris gives this time &8 13h. 39m.,but the Table is not reduced to seconds. secs. Moon's right ascension 4h. 14m. 25.33s. at Greenwich mean 23.674 time 4d. 5h...... variation in 1Om. = Moon's right aacensiou 4h. 14m. 25.33s. nt Greeuwich mean time 4d. tih...... variation at 10m. =l 23'730- 6)O 056 Averagechange of variationduc to 10 minutesbctrrecnj 5h. and Ch...... H. 31. S. D. H. 111. 8. Right ascension Of} 4 14 25.33 atGrceuwich mean time 4 5 0 0 moon ...

Right Of 4 14 37.167 at Greenwich meantime 4 55 0 moon ...l 23.674+0.009 = 0 0 23.683 changeinlom. ...0 0 10 0

Right ascension Of) 4 150.85 atGreenwich mean time 4 5 15 0 moon . . Result A . . k l5 20.17

Of accountedbeto for. ascension . .} 0019.32 (2s-683 +0.009): 19.32 = 10 : 8.155111. = .....0-- 0 8 9.3 Greenwichmean time when right asccnsion of moon S 23 = resultA ...... Greenwich mean time of meridian passage at Greenwich I =resultB...... --- Difference between meridians in mean time. ....0 8 l5 46.67 Acceleration ...... 0 0 l 21.44 -___- Difference between meridians in sidereal time ....0-- 8 17 8.11 = difference of longitude to east. = iu arc 12+O 15' 1* 65" E. Therefore longitude required 8h. 17m. 8s. E., or 12-1' 17' 2" E.

Minutes of the Proceedings of the Institution of Civil Engineers 1898.133:316-330.