142 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



between the limits x=k and x=k + a : This gives 



/(a') = f cosm (a-*) (1 + e~ 2mfl ~) (f)' (m) dm by equation (6) ; 



Jo 



d(b 

 and y <W)=0, since -^7 = when t=Q. 



d) = ^ cosm yf (e^ + e-^ 1 ^ 2 ^ (p'(m) cos c m tdm 



Jo 



therefore in this case, writing a' for a k, and therefore x" for x k, we get 



e + e 

 and 7 () = "-2 m A cos 



/* 

 / 



*J o 



cos /*-*' cos c m / 



^ 



where F ()=/( -A). 



If/fa') be constant between the limits A and * + a, and equal to C, we have 



/ F( a r)?a-=Ca 



i/ - CO 



/CO Q 



F (w) cos /z (a-x) = - - (sin/xa r t k + sin px k) 

 CO // 



* = /" sin/Z-^^T^-f sinyu^^ ^y + ^ ^ ^ ^ 



^7 )U l + c- 2 '' A " 



From the circumstance that the present formula contains only cos c^t, where- 

 as the formula deduced for the preceding case contains only sin c^t, it follows that 

 if, at the same time, there be an elevation of the fluid and an impulsion at the sur- 

 face, the total effect will be the sum of the two effects estimated independently of 

 each other. This conclusion relieves us from the necessity of specifically con- 

 sidering the case in which both are united. 



With two or three conclusions from the above formula, we shall conclude 

 the present memoir. 



1. If *=0, we may for the surface omit y, which is very small, and we shall 

 obtain 



- . 



' l +<? -2*A P 



if 



, C r< sin u. (xk} sin tt(ir-k-d) 



and 0=-/ 



if , a ^ xk. 



11 f a s ' n 9 M , T 



But generally / f^ d /*= -- , 



J v r" * 



