510 PROFESSOK KELLAND OX THE THEORY OF WAVES. 



By the equations 



if-?(-^-(S))- 



we get 



dp dp 



-6'a(r-«.'' - e-"J)-sm6(ios.e\ dx 



-6^a(e'^''^-e-^''-'')cos^'0 j (^j/ 



.-. /)= —^(jii-vqCbac (~e''V + e~''Vcos 6 dx + e'^v — i~"-' iimOdy) 



= —.9 Q.V + Ql'C ((■"■ '■' + e~ " ■") sin 



+ p6^(cos20-|(.^"-"+e-2'-")) +P(4) 



This equation contains the value of jt?, and the use we purpose to make of it is 

 this. The quantity P is a function of t, and the depth of the fluid for the value 

 of 0.^ fixed on. If this depth be called z, we have 



jj — Qg{z—y)+gbcsm6{e'"J + e~'V) — gbcsin6{e''' + e~") 



Now the value of ^^ . or the expression for the force parallel to the axis of x, 



has been already formed, but another value of it may be obtained from this final 

 equation. If Ave equate the two, we get 



idp\ _dp dp dz 

 \dx] dx d:: ' dx 



where the quantity within brackets is the value of ^ from equation -(4). 



But if equation (4) be written ;j=^ {xy) + P, and (5) p-<^ (xy)-(f>(xz). 

 we get 



d (p (xy) _d (p (xy) d<p (xz) d z d<p{xz) 



dx dx dz dx dx 



(6) 



1 

 I 



Jen 



