170-174] SEVERAL CENTRES OF FORCE. 153 



174. Examples. 



1. A particle of mass m moves under the action of forces to two fixed 

 points A, A' of magnitudes mp/r 2 , mp/r 2 respectively, where r and / are the 

 distances of the particle from A and A', and /x and p are constants. The 

 equations of motion possess an integral of the form 



rV W = a (p. cos 6 - p! cos &} + const., 

 where a is the distance A A'. 



Fig. 43. 

 Resolving at right angles to the radius vector r, we have 



m ~ ~j (^G} = m ~ sin x, where x is the angle A PA', 

 so that r' 2 -, (r 2 0) = p'r sin x = /*' sin 0', 



similarly r 2 (r' 2 ff] = - pr' sin x = - M sin S. 



Multiplying by $', and $, adding, and integrating, we have an equation of 

 the given form. 



2. A particle of mass m moves under the action of forces to two fixed 

 points of magnitudes m^r, m//r f . Prove, with the notation of Example 1, 

 that there is an integral equation of the form 



/ir 2 ^+/iV 2 ^' = const. 



3. A given plane curve can be described by a particle under central forces 

 to each of n given points, when the forces act separately. Prove that it can 

 be described under the action of all the forces, provided the particle is properly 

 projected. 



Let f K be the acceleration produced in the particle by the force to the *th 

 centre K , v* the velocity of the particle at any point when the curve is 

 described under this force, r K the distance of the point from K , and p K the 

 perpendicular from K on the tangent to the curve at the point, p the radius 

 of curvature and ds the element of arc of the curve at the point. Then we 

 are given that 



