Prof. Sylvester on Periodical Changes of Orbit, fyc. 295 

 2 = i/ttf? (c*-v*)du+(c*-u*)dv 



2 — fu—v (c^—v^du — (c 2 —u 2 )dv 

 V u + v */Jj^v*)du*+{(*--u*)dv % ' 



F being any form of function whatever ; the integral will always 

 be 



^ (tt W)4-(«)V« +L= o, 



where the relation between L and X depends upon, and may be 

 determined from, the nature of F*. 



As regards the expression for {p), the radius of curvature in 

 terms of vectorial coordinates, we may employ the well-known 

 formula 



1 = dp 



p V{d?a:)*+(d?y)*' 

 where 



/ 2 + c 2 — g 2 uv + c 2 



os = 



2c 2c 



v- 



, u + c u—c c + v 



2~ "2~""2 2~ _^V 



2c 



so that 



2< = 





{ll*-V*)\/(d*{uv)Y-(d*\/(c 2 -u2){c 2 -V*)f' 



which I have not thought it necessary to reduce further. As 

 regards the original question of determining the central force 

 towards a focus, say F, proper to make a body move in a Car- 



* By varying the curve to which ds refers, we may obtain innume- 

 rable classes of differential equations whose integrals can be determined. 

 Moreover, by taking ds the element of a curve in space referred to three 

 foci, ds can be expressed by aid of the theorem given in a previous foot- 

 note as a function of the three focal distances f, g, h and their differentials; 

 and consequently the lengths of the perpendiculars upon it from the three 

 foci can be expressed in like manner, and we may thus obtain integrable 

 forms of simultaneous binary systems of differential equations between/, <?, h, 



X2 



