PROFESSOR KELLAND ON THE THEORY OF WAVES. 105 



Differentiating this with respect to t we obtain 

 dt \ 



' dt ' dx ' ' dz ' dt ' ' dz 



, , , 



+ 21' + . 22. M H --- V-tJ =0 



rf* d* 2 d-erf* ' dzdt ' 



/M. ~ rftv _ rfv. , \du d 2 n /ii x 



or ^ y + 2 M ,-' + 2, / _' + 2 M/ ,^ + (^-0^ + -=0 . 



The last equation is obtained by substituting for j-jy ^> their values 



du. dv. j , ... du. f dv, ^dv. f du. ta n -nA ff\ 



-j-', -i, and by writing r-' for -3-^, and ^- for -'... (6 and 8J. 



?/ dt dx dz dx dz 



It is remarkable that this equation, which is the condition that the pressure 

 at the surface shall be constant, does not contain any differential with respect to 

 z explicitly. 



By taking the small terms of an analogous equation, M. CAUCHY has obtained 

 results corresponding to the particular hypothesis, that the depth is infinite. 



50. Let us substitute in our equation (11) the values of u, and v t given by 



equations (1) and (2), and that of 0, given by (10). By denoting e" z +e~" ~ by S 

 and "*-* by D, we get 



-ffbDcos6-b 2 a . 4sin20 . c-26 3 aS . D 2 sin 2 6 cos 6 

 _63 ( C 2.*_ e 2.*) g acos2 COS Q + ba C 2 s cos + 2 p a S cos 0=0 ; 



or 



-6 2 (e 2a *" + e - 2a OScos20 + 26 2 S=^ 



a 



or c 2 S- 



or c"& 



a 



which equation is identical with that at the end of Art. 20. 



51. Thus we have three distinct but equivalent processes, by means of which 

 the same equation may be arrived at. It will not be worth while to follow the 

 different processes through the solution of the general problem. The result in 

 Art. 19 is in a form sufficiently simple. We propose rather to apply the formula 

 (11) to another case, that in which the velocity is expressed to two terms, of which 

 the second is not in the same phase as the first. Let us in fact assume 



VOL. XV. PART I. p f 



