168 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



of projection are at right angles to PQ. If b is the conjugate axis of the orbit 

 described by either P or Q, e its eccentricity, and &', e' those of the relative 

 orbit of P and S (in the absence of Q\ P being projected in the same manner 

 as before, then b = 26', and (1 - e)/(l +e) = (l- e')/(l + e'). 



42. If two bodies of masses E and M move under their mutual gravitation 

 and that of a fixed body of mass S so that the three are always in one plane, 

 prove that 



where h is the rate at which M describes area about E and H is the rate at 

 which the centre of inertia of M and E describes area about S. 



If the three bodies are free, prove that the equation must be altered to 



= const. 



43. If three bodies of masses m lt m 2 , m 3 , subject only to their mutual 

 attractions P 23 , P 31 , P 12 , remain at constant distances from one another, those 

 distances are in the ratios 



44. Three equal particles A, B, C, attracting each other with a force 

 proportional to the distance, and equal to /* per unit mass at unit distance, 

 are placed at the corners of an equilateral triangle of side 2a. The particle A 

 is projected towards the centre of the triangle with velocity c V/i, the other 

 particles being set free at the instant of projection. Prove that the three 

 particles will first be in a straight line after a time 



45. When a particle is at the nearer apse of an ellipse of eccentricity e 

 described about the focus, the force on unit mass at unit distance is increased 



by the small fraction - of itself : when the particle is at the further apse, the 



force becomes less than its original value by the same amount. Prove that 

 the time taken in this revolution is less than the original period by the fraction 



46. A particle describes an elliptic orbit about a focus and, when at the 

 end of the minor axis, it receives a small impulse towards the centre equal to 



-th of its momentum. Show that the eccentricity e is increased or diminished 



by -*/(!- e 2 } according to the direction of motion at the instant. 

 n 



