198 ON FOKCES OF ATTBACTION 



the force being as any function of the distance, and of x, y, z, 

 being the co-ordinates. Pressed by the great difficulties of 

 the problem, and the impossibility of a general solution, he 

 first confines himself to the inverse square of the distance 

 (p. 97), and a general integration being still impossible, even 

 after obtaining a differential equation with the variables 

 separated, he makes a supposition which enables him to 

 obtain two particular integrals (p. 99), and this gives for the 

 orbit an ellipse in the one case and an hyperbola in the 

 other, with the foci in the two centres of force ; and it 

 follows, he observes, from the investigation, that the same 

 conic section which is described in virtue of a force to one 

 focus, acting inversely as the square of the distance, or to the 

 centre and acting in the direct ratio of the distance, may be 

 still described in virtue of three such forces (" trois forces 

 pareilles"*), tending to two foci and to the "centre." He 

 adds : " Ce qui est tres remarquable " (p. 101). It having 

 appeared to many persons that a portion of the demonstration 

 was not so rigorous as might be desired, M. Serret has very 

 ably and satisfactorily supplied the defect (Mec. An. torn. ii. 

 note iii. p. 329, ed. 1855), but he arrives at the same result. 

 There is also given a very important generalization of 

 Lagrange's solution, and of Legendre's theorem already men- 

 tioned, 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 I 

 and I', and refer the forces to those two, and make the forces 



and - in in' and I' n' respectively, and the angle of 

 r- <f 



projection and initial force the same, the same circle will 

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



* It is plain that " pareilles " does not mean of the same kind as r- 



and v ; for he resolves the force to the centre into two acting to the foci, 



a /8 



and calls the whole forces + 2 y r and 5- + 2 7 q. 



* 



