510 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



By the equations 



dp _ / (du\ \ 

 'dx~ ^\ \dt) ) 



we get 



dp / /dv\ \ 



J dp dp , 



dp ^-i- d z + -f-dy 

 dx dy ' 



.oi 



^"(S)-^^"- ^iyy-^'^A 



- ^2^ (c«;/ _ ^- « y )2 sin ^ cos ^ 1 dx 



.-. p= —qy !J-\-qfbac ( — e'^y + 6—"'.' cos6 dx + e''-' — e~'".' sinOdy) 

 + d fV2 af { 2 sin 2 c^ a:- (e^ «?'_e-2 «2/) (^^ j 



= -^ P ;/ + P 6 (^ («■'''' •' + 6—" !') sin ^ 



+ pi^(cos20-l(e'"-"+e-'"-")) +P(4) 



This equation contains tlie value of p, 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 



Now the value of j- , 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 we equate the two, we get 



(dp) _<ip dp dz 

 \dx] dx dz' dx 



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



But if equation (4) be written p = (p {xy) + Y, and (5) p—(p (xi/) — <p (xz), 



we get 



d(p (xy) _d(p (xi/) d(p(xz) dz d(p(xz) . . 

 dx dx dz ' dx dx 



