APPENDIX. 433 



result. There is also given a very important generalisa 

 tion of Lagrange s solution, and of Legendre s theorem 

 already mentioned, by M. Ossian Bonnet. (Ibid, note iv.) 

 9. The same reason already given proves that if, instead 

 of two points not in the trajectory we take two in it, as 2$ 

 and 3 , and refer the forces to those two, and make the forces 



and -y in 2IT and 2 IT respectively, and the angle of 



projection and initial force the same, the same circle will 

 be described by the body; and that if two other forces 



also act on it, as SIT and 2TI (or and - ) the same 



V in 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 IT, the velocity will be, if only the forces 



m J m ^ m -c A i. r r j 9 i 



and , act, as - 5 if the forces and - also act, it 

 r 5 q 5 2 a 2 m m 



will be as \f 1+ - It 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 



n 



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

 bility of obtaining an expression freed from the integral sign 

 I I j in the same way as Lagrange does from his equa- 



/i 



tion, founded upon the datum R = -^ -f 2 y r and Q = ~ 2 



F F 



