. , traced upon the Surface of the Sphere. 361 



Let N, taking the place of V in the former figure, be a point in the log. spiral, 

 whose equation is 



Draw NH, HG LL to CN, CA. Then 

 CN = v - k 1 = GH = sin AH 

 .-. sin? = A" .......... : .................... (2.) 



This is not the equation of the loxodrome ; and hence, also, this part of the sup- 

 position is erroneous also. The mistake probably arose from a momentary confusion 

 of the two methods of projection the stereographic and the orthographic. 



NOTE E. 







Ihejirst attempt, in all probability, that was made to assign a spherical curve, 

 by means of its equation between <p and 6, was by that very ingenious mathemati- 

 cian Mr JAMES SKENE of Aberdeen, and by the late Mr THOMAS WHITE of Dum- 

 fries. The latter gentleman proposed this question in the Gentleman's Diary for 

 1795, and it was answered by the former in the Diary for 1796. See DAVIS'S 

 Collection of those tracts, vol. iii. p. 258, under that date. I had completed the 

 researches contained in the preceding paper before I was aware of that circum- 

 stance, which was obligingly pointed out to me by Professor LOWRY of the Royal 

 Military College. The solution of Mr SKENE is very similar to the preceding 

 XXXVII, 13: and he shewed also that the orthographic projection of the curve 

 upon the equator (P being the pole) is an ellipse. 



In Professor LEYBOURN'S Mathematical Repository, O. S. Quest. 130, vol. ii. 

 p. 196, the spherical ellipse is proposed, and a solution by Mr LOWRY, inserted in 

 the following number, remarkable, like all the processes of that eminent geometer, 

 for the elegance of its methods, and the simplicity of the results. As a more ample 

 discussion of that curve will appear in the next number of the Repository, than I 

 could find room for here, I shall desist from further remark upon it at present. 



In HOWARD'S Spherical Geometry (1798), too, it is proposed, p. 115, to find the 

 locus of XXXVII ; which the author considers a new problem. Had it been so, 

 his investigation did not remove the necessity for a totally new consideration of it. 

 These, with the Spherical Parabola (Repos. v. p. 240), and the Hectemoria of my 

 former paper, are, so far as I know, all the attempts that have been made to treat 

 spherical loci by means of spherical co-ordinates. The curve of pursuit, treated by 

 SIMPSON and EMERSON in the Ladies' 1 Diary 1736, was considered as a plane curve; 

 and the same is true of Mr CUJ?LIFFE' I S note upon these, in LEYBOURN'S edition of 



