PROFESSOR KELLAND ON THE THEORY OF WAVES. 117 



/<ZM\ _ M ?M du dw du dv 

 \dt) dx dy dx ds dx' 



All these equations are satisfied by the hypothesis that ^ ~J~ X ~' ~clz~ ~dx~ ^ 

 and 3^"^=; in which case udx + vdy+indz is a complete differential of 



some function with respect to x, y, and z. 



59. Let us proceed to apply the conditions just obtained to the case of wave 

 motion. We shall have u=f . sin a (xct), v=F . cos a (x ct), w=F . cos a (xet) 

 where f, F and F are functions of y and z. 



Then since P=0 ; d -+aF=Q '. '/. " . . (4) 



dy 



since N=0; ^+oF=0 . . . . (5) 



dz 



dF dF x-x 



since M = ; - -- -3- = .... (6) 



dz dy 



du dv dw . dF d F A ,. 



and since 3- + ^- + ^-=; a f + -j- + ^r =() ( 7 )' 



dx dy ds dy dz 



These form but three distinct equations; inasmuch as equation (6) is merely 

 a consequence of (4) and (5). 



By substituting in (7) the values of F and F deduced from (4) and (5), we 

 get 



d*f d 2 / d 2 f d 2 / , 

 a 2 /- -j-4 - -r4 = or -j-4 + ^4 = a / 



dy 2 dz 2 dy 2 dz* 



This equation admits of the following solution, 



and 2 embraces all possible values of n and >. 

 Equations (4) and (5) give us also, 



F=--S.e"( m y +n ^ .mb, F=--Ze*( m y +n -). nb. 



We must obtain the particular values of n and b, which satisfy our problem 

 from the restrictions which the problem itself imposes on us. 



Let us place the origin in such a point that when *=0, w=Q, and when y=o, 

 =0, whatever be the value of a? or t. 



If, as is commonly the case, the canal be symmetrical with respect to a ver- 

 tical plane running along its length, the origin will be situated at the bottom of 

 this plane, and the line in which this plane cuts the surface of the fluid, will be 

 the middle of the canal. Let the equation to a section of the surface of the ca- 

 nal made by a plane perpendicular to this line, or to the direction of motion, be 

 .'/=-H*); 



VOL. XV. PAET I. I 1 



