66 



C. V. L. Charlier 



^200 



(- 2Ü' + V + W')I^J~ — l]^dUdVdW 



-|- CO 



(— F2+ W')IU'^dUdVdW, 



as the remaining part of the integral vanishes. 



We may write this equality more conveniently in the form 



+ 00 



a.*a,,, = j| j" (— 3 r/^ + ß^) / U' ^dUdVdW, 



— 00 



or, if polar coordinates are introduced, 



(— 3 cos cos + 1) cos '^B cos ^ÄÜ^I^ dil dB dA . 



We liave here to evaluate integrals of the form 



But 



and 



so that 



y cos dB and j cos ^^A dA . 



0 



cos = I (1 + cos 2 A), 



cos == I (3 + 4 cos 2 A + cos 4^), 



cos ^B= i {3 cos B -\r cos 3B), 



cos ^5 = (10 cos 5 + 5 cos 35 + cos ÖB), 



2^ 



i 



0 



1 



COS ^A dA = 



cos *A dA 



3k 

 4 ' 



' 2 



It 

 2 



COS ^Zi^ = 



4 

 3 ' 



+ ; 



16 



