PROFESSOR KELLAND ON THE THEORY OF WAVES. 539 



We are therefore reduced to precisely the old form, viz. : 



.u — ie^y + 6^"^^) {—gac sin a a: cos ac t + g ac GO^ax sin ac t+ij + c) 



= (e"2/ 4- e-«.y) (6 sin a . .1;- c ^4- C + c) 

 if b=-gac 



V— — (e'^y — e~ "^y) (J) cos ax cosac t+b sin a a; sinac #) 



= -b{e''y-e~''y) cosa.x-ct. 

 40. The corresponding value of z is 



z=h + {e''' ~-e~"'^) (p+g savac t cosax—g co$,act smax) 



TT 



Now D is independent of t (35) ; if, therefore, we put ax = --^,t=0, we get by 

 hypothesis z=h, - 



But we obtain ^=A + (e«* -e""^) {^-^^ 



hence — = D 



ac 



and 



^ \ac ac / 



^h^- — {f'-e-^') (1 + sin^). 

 ac 



41. Lastly, C?^p^(^'' (35) 



a p^ a? c^ sin^ act+g'^ a? c^ cos^ act 



(X p^ a? (^ ■.• p^=g^ 



.'. C is independent of t. 



TT 



And by making ax =-— and (=0, we obtain u=0, 



C + c = b 



and u=z{e'y + e-"y) b{l + 8m6). 



42. Having thus obtained values of u, v, and z, it remains that we substitute 

 them in the equations which determine the pressure. Now, in doing this, it must 

 be borne in mind that the values of u, % and z, are not correctly expressed by the 

 above formulae, and that consequently we must not expect accurately to satisfy 

 the conditions of integrability of the function which expresses the differential of 

 the pressure. Still whatever variation may be requisite in the above functions, 

 to enable them accurately to satisfy all the conditions, it cannot be doubted that 

 they hold true in the early part of the motion, as far as the large terms are con- 

 cerned. If, then, in the process of finding the pressure, we take no notice 

 of terms of the second order, our results will be a close approximation to the 

 truth. 



