Statistical mechanics 53 



Multiplying by d'\> and integrating, all odd powers of cos (tj; + vanish. 

 To obtain the value of the even powers we observe that 



If this series is multiplied by ^ and integrated, we find that all terms vanish 

 except the one for which 



2j) — 2,9 = 0 .-. s=p, 



so that 



0 



We thus obtain 



^ c^^ = sin 2-8- + cos ''èy + 



0 



+ (2) (1) ^2 + ^'^') COS ^0- sin ^Ô' {u, sin '^a + u, cos ^d)'-^ + 

 + 1^^ {v' + cos ^d- sin *9- (m^ sin + cos + 



+ (e) (3) ^6 («^' + '^')' cos sin (m, sin + u,, cos + 

 + -■ 



For «2'' a similar expression is obtained where only sin ^ and cos ^ change 

 places. In this manner we get 



[w'J = {u^ sin + u., cos '^d-y + [u^ cos ^O- + sin ^d)'' — u[ — u^^ -)- 



(112) + ^^j^(v^ + «t;^)cos2^sin2d((M,sin2d + M2COs2*X-2 + («^cos2{^ + M2sin2a)'--2^ + 



+ 



We infer from this formula that [u'''] is a homogeneous pohjuome in and 

 of the degree r. 



41. For r = 2 and r = 3 we get after some simple reductions 



[ »2] = cos sin 2* (— 3 + co^) , 



(113) [ = cos sin ^Ô' (— 3 + w^) , 



[w^] = cos ^0- sin 2^ (— ^w^ + 0)2), 



