398 ON THE PRODUCT OF ANY TWO LAPLACE'S COEFFICIENTS 



Hence we get 2Q x Q%cos m\cosp\ 

 _ , ., (2n + 2(/-4r+l)! (g+p)l (2n-2r)! (n + q-r)\ (n + m)l 



2, ( 1) - 



2g'-2rH-l)! r\ (q-r)l (n-r)l (n + q-2r)l (n-m)l (n + m + q+p-2r)\ 



T , v (n + m + 2q-2r-s)\(n-m + s)\ 1 -g;ff_ 2l . 



1 ' sl(q-p-s)\(n-m-2r + s)l~(n + m - s) ! J 2 n+ - + ^ A 



, S / 



- _ 



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



)x 

 p > A ' 



- 



s i (q-p-s)l ( 



where r has all values from to 2q, and s has all values from to q p. 



16. From Art. 14 it appears that the product 



2 /?;/<; or QHQl [cos (wi + p) X + cos (m -p) X] 



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

 hence 



J -1 



Also if R k , be another Laplacian coefficient of the same form, then 



except when k = m +p or when I; = m p. 



For -/?,", R f q let us take the value of the Laplacian coefficient in its 

 more general form, viz.: <X'cos(wX + /3) and Q p Q cos(p\ + y), 



where ( = ( 1 - p.JD m P n , 



and Q!=(l-p*)>IPP 9 , 



then Q* cos (k\ + a), Q cos (mK + /3), Q* cos (pX + y) 



represent any three such Laplace's coefficients. 

 Then since 2Q cos (mX + /3) x Q^cos (pX + y) 



= Qn x Q," {cos [(m +^>) X + ^ + y] + cos [(m -p) X + - y]}, 



