358 Mr DAVIES on the Equations of Loci 



these latter will be investigated in a paper which will be printed in the next number 

 (XXIII) of Professor LEYBOUHN'S Mathematical Repository. 



I have examined a great number of the loci thus produced, and in no case that 

 I have yet tried have I found the analogy broken. A few of the more curious, 

 elegant, and important ones will be inserted in the succeeding portion of . this in- 

 quiry, which I am preparing to send to the Royal Society of Edinburgh. In that 

 paper I shall examine those properties of spherical curves whose demonstrations are 

 dependent upon the differential co-efficients, or most readily investigated by means 

 of them, as their singular points, asymptotic circles, asymptotism in general, their 

 spherical involutes and evolutes, and other topics allied to these. I shall then enter 

 into a full examination of several paradoxical expressions that have occurred to me in 

 my researches, arising out of the application of general formulae to particular cases : 

 as, for instance, in the expression of the area of spherical curves taken between spe- 

 cified limits (see Art. XIX. of this paper), and other collateral subjects. Two other 

 topics will also claim a little consideration oblique spherical co-ordinates, and the 

 locus of the penetration of the sphere by any given surface, but especially by a con- 

 centric cone of the second degree. 



I may just remark, in addition to the little that is said in the article to which 

 this note alludea (XXIV), that, taking the focus for origin of <p and the line joining 

 the foci for origin of 6, the general equation of the conic sections on the sphere is 



tan <p 



cos 2 * cos 2 1 



sin 2 + sin 2 1 cos i 



where 2 E is the distance of the foci, and 2 at is the given sum or difference of the 

 focal radii vectores drawn to the points of the locus. This, for the sum, is, abating 

 the notation, the same as Professor LOWRY'S equation of the spherical ellipse, in the 

 old series of LEYBOUBN'S Repository, p. 196. See Note (E). 



In the case of the spherical parabola, 2 a. = - , and the expressions are for this 



case 



cos 2s 



tan <f c -. 



1 -+- cos 6 



The analogy between these and the focal polar equations of the plane conic sections, 

 is sufficiently remarkable to interest every inquiring mind. 



