Prof. Sylvester on Periodical Changes of Orbit, fyc. 291 



Mr. Crofton has also made a partial extension of the theorem 

 to the case of a plate of the form of either one of a conjugate pair 

 of Cartesian ovals, in a remarkable paper on the theory of these 

 curves, lately read before the London Mathematical Society. 

 In the " Prolusions" I raised the question of determining the 

 force at a focus required to make a body move in such oval. 

 This may easily be solved by aid of vectorial coordinates ; and 

 as it seems desirable to place on record the tangential affections 

 of a curve expressed in terms of such coordinates, which I am 

 not aware has hitherto been done, I subjoin the investigation for 

 the purpose. The results will be seen to be of great use in sim- 

 plifying the solution of the important problem of determining 

 the most general motion of a body attracted to two or more fixed 

 centres, a problem to which I purpose hereafter to return. 



If F, G be two foci, c their distance from one another, f, g 

 from any point in a curve, ds the element of the arc at the point 

 (/> 9)) 0> V the angles which ds makes with /and g, we have 



n df da 



coso=~, cos??=^-- 

 ds ds 



Call g -\-f~u, g—f—Vy and let co be the angle between / and g. 

 Then 



(cos d) 2 -f (cos ??) 2 + (cos co 2 ) — 2 cos (o . cos . cos 77 — 1 =0*. 



touch the radius, for the force becomes infinite in the direction of the radius, 

 and must tend towards the centre without becoming convex to it, on account 

 of the force being attractive. I do not see how these conditions can be re- 

 conciled, except by supposing the remainder of the motion to take place 

 along the radius itself, which involves the supposition of the transverse 

 velocity at immergence becoming instantaneously destroyed, and the same 

 at emergence when the force is repulsive. 



* The left-hand side of this equation, calling the directions of /, g, ds, 

 A, B, C, is 



cos AB 







cos BA 

 cosCA 



1 





 cosCB 



1 



cos AC 



cosBC 







1 



and in like manner for four lines in space A, B, C, D in spaces, the deter- 

 minant 







cos AB 



cosBA 







cos CA 



cosCB 



cos DA 



cosDB 



1 



1 



cos AC 

 cosBC 





 cos DC 



1 



cos AD 

 cosBD 

 cos CD 







1 



=0. 



This important equation is nowhere explicitly given in treatises on trigo- 

 nometry or determinants, but is virtually included in a theorem which is to 



