64 C. V. L. Oharlier 



This is, indeed, obvious from symmetry, and can be directly shown in the 

 following manner. Put 



U = QcosBcosA, 

 F = ß cos B sin A, 

 W=i}sinB, 



so that 



dU dV dW = cos B dQ dB dA, 

 and the new variables have the following limits 



Q from 0 to oo , 



TT 7t 



B » -2 » 



A » 0 » 2t:. 



We now have 



= fi2 cos ^B cos 'A = \Çi\\ cos 25) (1 ^ cos 2^) 



and 



' 2 27t 



cosB = } 9Jf (3 cos + cos 3B) dB /(I + cos 2A) dA 



K 0 



2 



- g Ii 



so that 



^dUdVdW=^^^d9Q 'I{Q) ^ . 



But 



/// ß2 (Ï) COS B dB dA dü = ^njd^^ <ï> , 

 which gives «^qq = O, as we have 



— 2U' + + i72 = _3C/2 -I- Q2. 

 Let us now consider an integral of tlie form 



/ IJi VJ W (E> dlJd VdW 



where i j -{- h = odd number. 



Introducing polar coordinates we get 



/ i2'+7+A+2 JdQf (cos By+j+' sin '^i? cos '.4 sin JA dB dA = 



00 '2 2ir 



= / $ /c/fl X/ (cos sin ^B dB X / cos '.4 sin M dA . 



0 It 0 



~ 2 



