PROFESSOR KELLAND ON THE THEORY OF WAVES. 507 



it is = J ~ {P-P')<iy- 1 ^"' pdy 



dz 1 2(/a; da. 



-'^'''T.^2^-Tr-dj 



^'''^ gQ{z-y)dy involves a^ as a factor. 

 Hence the accelerating force is 



da 

 d^x _ dz dx dt 

 IW ~ ~ ^ Jx'^ lit ' ~^ ' 



But it has been shewn (Art. 2), that 



1 da_ 1 dz 

 a dt z dt 



d^x _ dz dx dz 1 

 d¥ ~ ~^d^~Ji' dt ' z' 



27r 

 But " z=^h + a sm ^— - (ct—x') 



I A 



, dx , . 2'K , ^ -. 



and ^- .= sin — — (c t—x) 



dt A ^ ^ 



where h is supposed undetermined. By substituting these values, we get 



dx 

 2'Kb a ( dx\ 27r ^ dt 'lirac 



cos 



/ dx\ 27r a dt 



(c ) = ««.__ cos V 



\ dtl ^ \ h 



I c I = a a . -^— cos 6 — . — ^ — . cos 6 



A \ dt I A h \ 



2nrb /J , , /J, 2'7r /, 27roc6 . ^ a 



or — ;r — • cos V ic — b sin C7) = --— sa cos f . sin v cos v ; 



A A A/i 



and, equating coefficients, we get 



bc=ga; b^ = —-—; 

 h 



whence *=^- 



the values which we obtained before. 



8. Next, having these approximate formulae as our guide, let us proceed to 

 the general solution of the problem. 

 Retaining the same notation, 



