TO SEVERAL CENTRES. 199 



f T O \ 



also act on it, as 2 n' and 2' n' [ or and ) the same 



\ m m / 



circle will be described by the joint action of the forces. 

 This is even a more remarkable consequence than the 

 other ; because the forces acting to the centre would of 

 course give a uniform motion, and those acting to the 

 points in the circumference an accelerated motion, and the 

 forces combined will give an accelerated motion. At the 

 middle point II, the velocity will be, if only the forces 



mm V m r q , 



and r act, as -- -=- ; if the forces and also act, it 



r 5 q 5 2cr mm 



will be as * / + I* must, however, be added, 



V 4 a 4 m 



that Lagrange's solution does not contain this case of the 

 circle and two points in the circumference, and there is very 

 great difficulty in applying to it his analysis. Indeed, it 

 appears that if the problem be worked upon the datum 



* o 



of B = + 2 y r, and Q = - + 2 y q, there is no possi- 

 bility of obtaining an expression freed from the integral sign 

 in the same way as Lagrange does from his equation, 



founded upon the datum R = + 2 y r and Q = -f- 2 y q 



m = 2, and consequently m + 2 = seems necessary to 

 his process. 



There seems reason to suppose that the kind of reasoning 

 on which we have relied as to the identity of the trajectories 

 had influenced Legendre in confining his investigation to 

 the case of curves which have not infinite branches. He 

 expressly says (Ex. de Cole. Int. 11, 372), that he confines 

 himself to curves where the orbit is restricted to a definite 

 space. Certain it is, that the reasons applied to the identity 

 in the case of curves returning into themselves is wholly 

 inapplicable to curves having infinite branches. 



