TO SEVERAL CENTRES. 197 



one force -j or , or when described by the combined action 



r 



of both, or when described by the combined action of and 



TO 



q m m r q 



, or of -5-, ;, , and ; because in all those cases the 



m i* q* m m 



velocity will be different, and particularly the action of 



T- -I with - + - will occasion a different velocity in 



r* m q* m 



7l ITt 



each point from that occasioned by r H r. Thus to take 



i* q* 



the velocity at one point answering to C. If II a and II a' be 



taken as and . the diagonal He' is the force of and 

 i* (f r 



arn v* sy 



- combined, II C is the resultant of and combined 

 (f mm 



(supposing w = 1). Therefore the velocity in II will be as 

 II c' + II C, when all the forces act, and only as II c' when the 

 two former act alone, and as II C when the two latter act 

 alone.* But the curves appear to be the same in each case. 



8. These consequences seeming to follow from a con- 

 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 (Mec. 

 Anal, part ii. sect. 7, chap. 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 difference in velocity is easily obtained, in comparing the effect 

 of one force and of the combined forces, from the equation v? = 2 (/ X half 



2 t) T 

 chord osculating circle, the chord being = ' , p = perpendicular to 



the tangent, and E = radius of curvature. 



