196 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



Squaring this and using (9) we get 



(M - my'' + 2mc'-c - (M + m)c = (37) 



This is the equation of a sphere in velocity space which passes through 

 the point c' = c. For all other points c' is bigger than c (colHsion with 

 a stationary object always brings energy loss). The center of the sphere 

 lies on the line joining c to the origin; it lies on the side of the origin 

 from c when m < if, at infinity (making the sphere a plane) when 

 m = M and away from the origin when m > M. We make use of (37) 

 to express the polar angle k of c' with respect to c (which does not occur 

 as an integration variable in (34)) in terms of c'. We get 



(M + my -(M - m)c" ,„^^ 



COS K = ^r -. (,oo; 



2mcc 



The angle of scattering in the center of mass system also results from 

 squaring of (36) if the term my' is first taken to the right. We find 



°°^^ 2Mm c'2 2Mm ^ ' 



There is a more useful form of equation (34) which results if x is taken as 

 one of the integration variables rather than c'. Substitution is made 

 from the equation (39) above; it yields 



a ^ + -I- Ho) = 1 [' sin X d^ 5W («:)' r ,(c') d. (40) 

 dCz t{c) Ait Jq t{c') \c / Jo 



The magnitude of c' and its polar angle with respect to c are now auxiliary 

 parameters; the first is obtained from (39) 



c' = c ^ + ^ (41) 



V M2 + m2 + 2Mm cos x 



and the second from (38) and (41) 



m + M cosx .,^. 



cos K = (4^) 



VM2 4- m2 + 2Mm cos x 

 As previously, the azimuth co of c' about c is an independent variable. 



The simplifications of the equation (31) exhibited in (34) and (40) 

 still leave a double integral in the fundamental equation. The integration 

 over do) will now be eliminated by decomposition of h{c) in spherical 

 harmonics about the field direction. There is no loss of generality in 

 this step. 



h{c) = E K(c)P, (cos t?) (43) 



