108 KINEMATICS. [202. 



Transforming every term by the formula 2 sin a sin fi= cos (a /?) 

 cos (a+(3), we find 



. . . (26) 



Applying the trigonometrical formula 



( 2;? - I ) Ct / 2 \ 



sin a/2 y 



we obtain 



sin(2 i) (k m) sin(2 n i) (k + m) 

 2 = - - --- - - 



cos( w)7rsin(^ m) 



(k 4- ni\ cos( 4- m\it sin(/& + m\ v 



V ' 2n * x 71 " v ;^g 



If be different from tn, this reduces to 



and this is always = o, since k + m and k m are either both odd 

 or both even. 



If k=m, we find from (26) 



[7T . 27T , ( iWH 



COS 2m-+ COS 2*# -- 1 ---- + COS2*^ - ' 

 n n n J 



