OF THE FORM JK^. 399 



it is evident that the product of the three Laplace's coefficients when 

 integrated with respect to X between X = and X = 27r will vanish except 

 when k = m+p or k = nip. 



First let k=-m+p and n+q lr l, and let K be the coefficient of 

 Qn+Z-ir ' n the above product. 



rf\ 



Then @* cos (k\ + a) Q cos (wX + /3) Qf 



J J -1 



will be = P" f 1 K (#*)' i cos (k\ + /3 + y) cos (k\ + a) 



cos X + 



o ./ -i 



where ^(- !)'(+!) " + ?-' -'^"' 



; rl (n-r)l (q-r)\ (2n + 2q-2r+l)\ 



X s ( - IV - - 

 ' 



- - 



s I (q - p - s) ! (q + p - 2r + s) I (n + m -s)\ 



Another value determined above for 



K - ( - i 



r\(q-r)!(n-r)\(n-m)l(2n + 2q- 



_ __ 



s ! (^ p s ) ! (w TO 2?' + s) ! ( + m s) ! 



Hence using this latter form for K we have 

 I'" P $ cos (JfeX. + a) ( cos (mX + /8) Q* cos (_pX + y) dpd\ 



Jo J -i 



w v (q+p)\(2n-2r}l (l + r)l (n + m)\ 



' r\ (q-r)l (n-r)\ (n-m)l (n + q + l+l)l 



x f* /_ IV (l + k + q-p-s)\(n-m + s)l _ 1 _ 



s ! (g p s) I (n m 2r + s)\ (n + m s) ! J ' 



Now let k = mp and n + q 2r = l, and let ^T' be the coefficient of 



<?n n +7-2r i n the above product. 



