392 Sir William Thomson on the General Integration 

 Then from (18) we have 

 [-^ 7 + 4m(l-^)]<£ 



« V[- ayW +*>(!-■■)] } (/•-•V** • ( 31 ) 



Going back to (15) we have 



7 #\ 



=V'(i-2^+^ 4 )(i+i)^-VW^ 



= 5>A*[t(l-tsr s ) 8 + l+2« 2 -3w 4 ]«r- , 7(tir)K i 



i« v P(i -** 2 ) + x + 3 ^ 2 ] (i -W*)""^ W K t 



= V[i(i -^) + 1 + 3 ^ 2 ] t 1 -^yHKw. (24) 



Lastly, using (24), (22), and (21) in (14), and equating to zero 

 the coefficient of [x iy we have 



{p(l--^)+l+(y + l) W 2 ](l-^) 7 H 



+ [-a7W+4w(l-«*)]7(/*-- f )« 2 }k j+1 =0. (25) 



By giving i successively in this formula all integral values from 

 ~oo to and +co , and attending to (19) and (20), we have a 

 succession of equations which successively determine K x , Kj, 

 K 3 , &c. in terms of the arbitraries C and K ; and using the 

 values found in (16) we have the complete solution sought. 

 6. Laplace takes fy=Z(l—^ 2 ), 



where / and q are constants ; so that the bottom and the undis- 

 turbed free surface of the water may be both elliptic spheroids 

 of revolution. With this or any other rational integral function 

 of fju for y, there is no difficulty in developing (§ 7 below) the 

 first member of (25), and working out a practical solution of the 

 problem. Laplace's most interesting and instructive results, 

 however, are confined to the case of an ocean of uniform depth 

 (for which in his notation ^ = 0, 01-7= constant). Taking this 

 case first, putting 4 m 



y= a > ( 26 ) 



