146 MR. S. S. HOUGH ON THE APPLICATION OF HARMONid 



system inclusive of the ocean, by well-known properties of surface-harmonics it 



may be shown that 



,' _ v 

 - 2 



Thus from (15), on replacing the quantities involved by means of their expansions 

 in terms of associated functions, we obtain 



- IK P: M + V-P*. 00 ; 



whence, if we equate coefficients of P" 4 in the two members, we deduce 



r 4 = -0,d+y. ........ (22) 



where we have written for brevity 



From (16), (18), we Lave 



\l> -- (D + cr/i) v l'i + ' s M> a - -/* (A + <r) x ..... (24). 



13ut by means of (10) we iind, on replacing ^ by its expansion, 



M ) 4- - ,\ M/ v. r (rt ~ ff) ( " ~ " + ]) P< (/' + ^ + 1) (M + s) p. 1 



CT/i) ^- ~27TT" 1 " +1 " 2 + i 



while from (9), (6), we obtain 



p (A + ) V, - S< (cr - (u + 1)} p^rlV, + ^ Pi-.] - 

 Thus 



+ 2) + (r}(n + ) p.- n 



2 + l 



and the right-hand member of (24) is therefore equal to 



iv^.- g"- P , 



"*~ a "- ~" i " +1 " 



Hence, on comparing the coefficients of P; in the two members of (24), we obtain 





