400 ON THE PRODUCT OF ANY THREE LAPLACE'S COEFFICIENTS. 

 Then r f Q* cos (AX + a) Q cos (X + /3) Q* cos (pX + r ) d/tdX 



Jo J -i 



will be - f*f J fir / (^) 2 ico 



Jo J - 1 



27T . , f 1 , ntv , , 



= 4 - cos (^-y- a) AM (Q^-dp. = Tr cos (fi --y- 



where A--(- IV (2?+l) - " r)1 (2g " 2r)! 



> -- 



-2r+ 1)! 



g)I 1 



m s) ! J ' 



v ' s! (q p s)\ (q+p 2r + s)\ (n 

 Another value determined above for 



- ( - m* (21+1) -_ 



l 'rl (?-)! (w-r)! (2/z + 2 g -2r+ 1)! 



X fs (-I)'- ( n + m + g ) ! (- m + 2 g- 2r - g ) ! 1 . 

 A ' ' ('/ ~P ~ s ) ( n + m ~ 2r + s) ! (TO m s) ! J ' 



Hence using the first of these values of K' we have 

 <$ cos (k\ + a) QH cos (mX + /3) 4); cos ( pX + y) c^cZX 



- cos (fl - y - a) ( - IV ( 



J r! 



x 2 ( IV w 



.s ! (^ p *) ! (<7 +p 2r + s) I (n m s) ! J ' 



