500 Mr. C. G. Darwin on the Collisions of 



of the astronomical process of reducing one of two bodies to 

 rest. We suppose them both incapable of rotation. Then if 

 points fixed in the bodies are chosen as their - centres,' and 

 if these centres are at x l9 y l5 z Y ; x 2 , y 2 , ^ tne mutual potential 

 W will be a function of x 1 — x 2> etc. 

 The equations of motion then are 



M t i ; ] = — ^ — etc., j 



° Xl > (5-1) 



mx 2 = — ^r — = -z- — etc. J 



Suppose that a is approaching along the direction of the 

 A'-axis from the positive side. The momentum integrals are 



M^ + mi- 2 =-MV, M 2 / 1 + ^2/ 2 = a, M£i + m£j=0. (5-2) 



Also if f = x x — x 2 etc. the relative motion is given by 



f=- snr etc (o-3) 



Suppose (5'3) to be solved for the relative orbit. Let the 

 angle between the initial and final asymptotes be 8, and 

 suppose that the y-axis is so chosen that the final motion is 

 in the plane x, y. From the conservation of energy the 

 final velocity is V, and so the final values of f etc. are 



£ = x x — x 2 = Y cos 8, i 



? = 2/i-2/2 = V sin 8 } > (5'4) 



£=*!-£, = <>. ) 



Solving for x 2 etc., we get 



i 2 =-MY(l+ cosS)/(M + m), 



y,= — MVsin5/M + m, 



k 2 =0, 

 and so the angle of projection is given by 



tan 6 — y 2 \x 2 = tan - . 



A 



8 * 



So we have 0— ~ an d we enunciate the result as follows. 



The angle of projection of H is half the angle between the 

 asymptotes calculated for the problem of relative motion, in 

 which one body is constrained to remain at rest at the 

 initial position of H, while the other, endowed with mass 



