259 ) on the Equations of Loci Traced Upon the Surface of The

259 ) On the Equations of Loci traced upon the Surface of the Sphere, as expressed by Spherical Co-ordinates. By THOMAS STE- PHENS DAVIES, Esq. F.R.S.ED. F.R.A.S. (Read 16th January 1832J 1 HE modern system of analytical geometry of three dimensions originated with CLAIRAULT, and received its final form from the hands of MONGE. DESCARTES, it is true, had remarked, that the orthogonal projections of a curve anyhow situated in space, upon two given rectangular planes, determined the magnitude, species, and position of that curve; but this is, in fact, only an appro- priation to scientific purposes of a principle which must have been employed from the earliest period of architectural delinea- tion—the orthography and ichnographyyor the ground-plan and section of the system of represented lines. Had DESCARTES, however, done more than make the suggestion—had he pointed out the particular aspect under which it could have been ren- dered available to geometrical research—had he furnished a suit- able notation and methods of investigation—and, finally, had he given a few examples, calculated to render his analytical pro- cesses intelligible to other mathematicians;—then, indeed, this branch of science would have owed him deeper obligations than it can now be said to do. Still I do not wish to be understood to undervalue the labours of that extraordinary man, indirectly at least, upon this subject; for it is certain, that, though he did succeed in placing the inquiry in its true position, yet it is ulti- mately to his method of treating plane loci, by means of indeter- minate algebraical equations, that we owe every thing we know in 260 Mr DAVIES on the Equations of Loci the Geometry of Three Dimensions, beyond its simplest elements, as well as the most interesting and important portions of the doc- trine of plane loci itself. I only wish to express, that the direct labours of DESCARTES on this subject are of very little value, if of any whatever; and this will be at once conceded, if we consult the Jesuit RABUEL'S Commentary upon his Geometria, published nearly a century after the original work, and containing the most amplified attempt to illustrate this particular part of the treatise that had then been made. The writings, too, of LEIBNITZ and the two BERNOULLIS may be searched in vain for any attempts to lay down new methods, or to develope and apply that of DES- CARTES for the general discussion of curve surfaces, or of curves any how situated in space. Indeed, we cannot but be struck with the very small number of investigations of this kind attempted by them ; and even those few might be said to have arisen out of the rivalry between the old geometry and the new. The singular facility possessed by these illustrious men, of de- vising methods of inquiry—of modifying those already in exist- ence, to accomplish the purpose they had in view—and their de- votion to every thing that could set the powers of the calculus in a more advantageous light,—these would certainly have led them to the consideration of the subject, had the method of DESCARTES been adequately stated, and its adaptation to any kind of calculus been even tolerably obvious. JAMES BERNOULLI did, indeed, throw out a casual observation in the Leipsic Acts, that an equation between three variables represented a curve surface, as an equation between two variables represented a plane curve ; but the remark seems to have suggested no further inquiries, and so far as JAMES BERNOUILLI was concerned, the subject dropped at this point. CLAIRAULT, by an independent train of inquiry, and whilst yet a minor, was conducted to the same result; but, pushing the investigation one step further, he was led to consider evert/ curve line in space as the intersection of two curve surfaces, or as a line traced upon the Surface of the Sphere. 261 traced upon them both at the same time. Every thing else fol- lowed, and it soon became the universal practice to investigate the properties of curves of double curvature by reference to three ca-ordinate planes, and determined by any two equations between the three variables. The method of DESCARTES at best led only to the discussion of curves of double curvature, by means of the equation sfzy and fzae, instead oif'xyz and f'ocyz, each equated to zero; whilst the latter process, which is that of CLAIRAULT, embraced the former as a particular case, granting, however, the resolution of equations of all orders: but that of DESCARTES could in fact only be applied in this particular case, whilst that of CLAIRAULT is independent of any such necessity, and applies with the same success, whether the elimination of any variable between the given equations be possible or not. The problem of VIVIANI, which was professedly proposed to bring the powers of the new Geometry to the test, must have awakened attention to this class of inquiries; and with respect to spherical loci, several problems of considerable difficulty were attempted during the next forty years, with varied degrees of success. The methods almost invariably employed in these inquiries was, to consider the character of the projection of the curve upon the equatorial plane, and, by means of these and the relations between the current vertical ordinates of the sphere, to express all the conditions, relations, and results. A few excep- tions, perhaps, might be made to this statement, the only one of which I can call to mind is that of JAMES BERNOULLI'S solution of VIVIANI'S Florentine Paradox, where he takes only account of arcs of the meridian of the current point, and the angles which those meridians form with a certain given meridian. He does not, however, seem to entertain the slightest notion of its applicability to any other class of problems than that one before him. When CLAIRAULT'S principle, however, became known, all those particular artifices were merged in that general method; and the sphe- VOL. XII. PART I. L 1 262 Mr DAVIES on the Equations of Loci rical curves, like all others, were expressed by two equations between three variables, one of which equations was that of the sphere itself. From that time forward, we find no traces of any attempt to devise special methods adapted to the discussion of curves traced upon specific surfaces, and to them only, amongst the continental mathematicians. No doubt, so far as the investigation of properties common to all classes and all orders of curves was concerned, this was an improvement of incalculable value, and one which for that purpose was altogether indispen- sable ; but it became at the same time a barrier to the inquiries respecting the peculiar characters of that class of curves which are traced on a spherical surface. The sphere, indeed, from the uniform character of all the regions about any point of its surface, would, we should at first sight expect, possess a great va- riety of interesting, and possibly, of important, properties belong- ing to itself alone, in the same manner as the circle does amongst plane curves ; and certainly, by methods whose essential character is their adaptation to the discovery of affections common to all curves, or to all curve-surfaces, we can scarcely hope to dis- cover those which are peculiar to one class amongst them. The importance of the sphere in astronomical inquiries (especially in that approximative astronomy which considers the earth a sphere, and its orbit a circle), as well as to the discussion of several cu- rious points connected with astronomical history,—these conside- rations render it an object of great importance that we should be able to investigate its loci by means of a system adapted to that purpose only, and therefore divested of every thing foreign to that immediate object. Some direct and simple methods of investigation are essential to our intelligibly expressing the varie- ties of the celestial motions, in a manner adapted to subsequent determination of the loci which result from their combinations. One instance of the advantages of such a method has been given in my paper on the " Antique Hour-Lines," which the Royal traced upon the Surface of the Sphere. 263 Society of Edinburgh has done me the honour to insert in its Transactions; and at a future time I hope to confer the same simplicity and completeness upon some other questions which have been repeated subjects of unsatisfactory discussion by every other process. My present purpose, however, is to lay down a sketch of the systematic principles of the method, to furnish so- lutions to the problems that naturally present themselves at the outset of the inquiry, and to apply it to the discussion of several problems, that, by their repeated agitation, have acquired a cele- brity which gives them a high mathematical interest, independent of any practical utility they might be supposed to possess in the affairs of common life. The method which I propose to employ is an equation between two spherical variables. Sometimes the latitude and longitude, sometimes the polar distance and the polar angle, will be the more convenient. These two cases have a much closer analogy on the sphere than the methods by polar and rectangular co-ordinates have in piano. The former is that generally employed in the hour-lines; the latter will be employed almost exclusively here, the class of problems here discussed seeming generally to admit of more ready treatment under this than the other form.

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