PROFESSOR KELLAND ON THE THEORY OF WAVES. 53.9 



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



. M = (e"^ + e '■^) (—g ac sin a a; cos ac t+g ac cos ax sin a e/ + (J + c) 



= {e'y + 6^'") {b sma.x-ct + Q + c) 

 if b=—ga.c 



v= — (e"^ — e~''^) (b cos ax cosac /+ 6 sin a j- sin oc /) 



40. The corresponding value of z is 



^=/i + (e^' — e *") (D+^ sinac < cos ««— ^ cos ac< sin ax) 

 = h + {e'^ -e-"^ {l)+—amax~ce\. 



TT 



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

 hypothesis z=A. 



But we obtain ^= a + (.«*-.-"*) (d-^,) 



hence — = D 



ac 



and \ = k + {e"'-e-") (A + Asin0) 



^ \ac ac / 



ac 



41. Lastly, C^a/^H-^'' (35) 



a p^ a" c^ sin" a c t+g'^ a' c' cos" a c I 



a:p^«?(? ■.■ /=/ 

 .•. C is independent of t. 



TT 



And by making ax = — — and t=0, we obtain m=0, 



C + c=6 



and « = (f';' + e-"-'0 6(l+8in0). 



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

 them in the equations which determine the pressiu-e. Now, in doing this, it must 

 be borne in mind that the values of 11, v, 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. 



