156 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



Thus from the first of equations (13) we have, on replacing $ by its value in 

 terms of ^ 5 W, 



On introducing the expansions for M^, 3 on the right we find 



. 4. fl. _ 

 . r P. | 



, ( + <7 - 1) ( - s) ( - <r + 2) (n + 8+1) 



~? n - l ~ 



which, with the help of (26), (27), reduces to 



//i 8\TT_?^VP f (n-s)(n + <r-l) p , 



v( L ~/*M ff / t ' r L (2w-l){w( + l)-(r} V h (2 



or from (49), 



.. . (58). 



This may also be written in the form 



y(i-^)u = ^s{ / ,a-D: ( }p: ! ...... (59). 



Again, from (13) 



~ /*$ (/.c; - D:) ? + sr n p n . . . (GO), 



whence, by means of (6), we may express \/([ ft 2 ) V by a series of surface- 

 harmonics. 



The corresponding formula? when the depth is variable may be obtained by 

 replacing h by K and C' t by 



so that we find 



where the quantities D' t are now defined by 



