PROFESSOR KELLAND ON THE THEORY OF WAVES. 103 



/d\dvddvd 2 v d 2 v T.--U-I j.-u ru j.- 



(dl) Tx=Tt dl* + d7*- u + J^r y ' v > whlch 1S less than the <l uantl - 



ty which represents * ( d -?) \>yp+ or b ? ? fi? + ?) ' that is b ? ( 6 )- 

 dx\dt) ' dx dx dx dy J dx\dx dy) 



d (dv\ f d\dv 



rf - (jjj and (jj) j~ x are coincident. 



Hence equation (7) gives 



/ d\ du_ I d\ dv 

 \dt) ' 1y~ \d~t) ' dx' 



which, by integration, becomes 



dw dv 



,_, 



. . . (8) 



dy dx 



This is the condition which we obtain, therefore, when p is a complete dif- 

 ferential of x and y. The other condition (6) is general, and must hold in all 

 cases of fluid motion. 



47. We do not purpose to solve these equations, having already done so in 

 Art. 8. We proceed, then, to the discussion of the equations (4) and (5). 



If C=0, as is the case when the solution assumes the form given it in Art. 8, 

 or when the motion is oscillatory, we perceive that udx + vdyisa, complete dif- 

 ferential of x and y, t being considered constant. Call i 



d <t> d(b 



.: u=--!-, v= 



dx dy 



, dp fdu du du\ . /dv dv dv\ 



and -f- = ~ffdy [ + + v ]dz I +u + v ) 



p \d( dx dy) \dt dx dy) 



Sd 2 (b d(b d 2 (b d(b d 2 (b'\ , 

 = gdy I - J- H 2- . . -\ 2- . y ) dx 

 \dtdx dx dx 2 dy dxdyj 



. _ _ 



\dtdy dx ' dxdy dy ' 



d 2 d> '. d(h d 2 

 -L- -. _ 



. -- -- -. _ _ -- 



L dx dx 2 dx dydx dy dy 2 dy dxdy J 



/ d 2 d) , d 2 d) , \ 

 ( - dx H -- dy I 

 \dxdt dydt y ) 



"-{<$)'+(%)'}-<% ' .;;. 



all the differentials except that of y being performed as though t were constant. 

 By integration, then, we obtain 



where f(t) is an arbitrary function of the time. 



