78 



of any symmetrical intersection of a sphere with a cone of the 

 second order (i. e. when the centre of the sphere lies upon one 

 of the principal axes of the cone) may be expressed by elliptic 

 functions. This is easily deduced from the following formulae 

 for the rectification of the class of spheric curves represented 

 by the polar equation, 



sin^/3F(a>) -f. sin 2X cosp — 1 (a) 



where X is a constant. Let «i, So, denote the two arcs which 



correspond to the same value of the polar angle w, and we will 



have 



^ r. /S4F2 4-F'2-4F + sin22X>rfw 



«l + «2 = cos X \ V i 2^ i 



J ^ f F — COS'^X > F 



d(t) 



. ^ r, / MF2 4-F'2_4F + sin22X 

 ..-..zzsinX^V^ j, ^-_^ 



F 



f' denoting the derived function of F(fe)). 



Now in the case of the intersection of a cone and sphere, 

 such as we have described, f(w) is a linear function of cos 2 w, 

 and each of the foregoing integrals is reducible to an elliptic 

 function of the third kind. 



In the case of the spherical ellipse, the two arcs Si S2 are 

 equal; and when the vertex of the cone is upon the surface, the 

 second arc s.2 vanishes altogether. 



The same relation between the principal angles of the 

 cone, and the distance of its vertex from the centre of the 

 sphere, will cause the parameters in both the functions n to 

 vanish. And it is remarkable that, in this case, when the sum 

 and difference of the arcs Si, s^, are each expressible by a trans- 

 cendant of the first order, the curve will coincide with the 

 locus of the vertex of a spherical triangle, the base of which is 

 given, and of which the product of the sines of the semi-sides 

 is constant, and less than the square of the sine of the fourth 

 part of the base. 



This result is strikingly analogous to M. Serret's theorem 

 on the rectification of the Cassinian curve, published in M. 

 Liouville's Journal, vol. viii. p. 145. 



