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 /z and // are constants. The 

 equations of motion possess an integral of the form 



rV 2 00 = a (/i cos 6 - p cos &} + const., 

 where a is the distance AA . 



A a A 



Fig. 43. 



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

 1 d 



so that 



similarly 



m ~ -J (r 2 0) = m ~,T 2 sin^, where x is the angle A PA , 

 r 2 -j (r 2 0) = fi r sin x p a sin &, 



r* d 



dt 



fir sin x= pa sin 6. 



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/j. r . Prove, with the notation of Example 1, 

 that there is an integral equation of the form 



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 #*, V K 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 



dv K . dr K v K 2 p K 



