432 APPENDIX. 



sideration of the conditions stated, but without a full and 

 rigorous investigation, it was very satisfactory to find that 

 Lagrange had arrived at the same conclusion in one case 

 of his solution of the problem of two fixed centres {Mcc. 

 Anal. pt. ii. sect. 7. ch. 3.) That solution is marked 

 throughout with the stamp of his great. genius. Euler had, 

 in the Berlin Memoirs for 1760, treated the case of the 

 inverse square of the distance and the centres and orbit 

 being in the same plane. Lagrange s solution is general for 

 the force being as any function of the distance, and of 

 x, y, z, being the coordinates. Pressed by the great 

 difficulties of the problem, and the impossibility of a ge 

 neral 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 supposi 

 tion 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 offeree; 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 vir 

 tue of three such forces ( &quot; trois forces pareilles* &quot;), tending 

 to the two foci and to the &quot;centre.&quot; He adds: &quot;ce qui est 

 tres remarquable&quot; (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 



* It is plain that &quot; pareilles &quot; does not mean | of the same kind as 



q i 



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

 and calls the whole forces -^- +27?- and + 2 7 q. 



