ON WAVES IN A VARIABLE CANAL 



To the equations (1) and (2) it is requisite to add the con- 

 ditions relative to the exterior surfaces of the fluid, and if 

 A = be the equation of one of these surfaces, the corre- 

 sponding condition is [Lagrange, Mec. Anal Tom. u. p. 303. 



(i.)]- 



_ dA dA dA dA 



= 77 +-J-W+ -= V+-j- W. 



dt dx dy dz 



Hence 



. dA , dA d6 dA d6 dA d6 , , A _ N /AX 

 v = 77 + -7 ' -^ f T~ . V- + 7- . ~r (wnen^L = UJ...(A). 

 dt dx dx dy dy dz dz ^ 



The equations (1) and (2) with the condition (A) applied to 

 each of the exterior surfaces of the fluid will suffice to determine 

 in every case the small oscillations of a non-elastic fluid, or at 

 least in those where 



udx 4- vdy + wdz 

 is an exact differential. 



In what follows, however, we shall confine ourselves to the 

 consideration of the motion of a non-elastic fluid, when two of 

 the dimensions, viz. those parallel to y and 2, are so small that 

 $ may be expanded in a rapidly convergent series in powers of 

 y and 0, so that 



Then if we take the surface of the fluid in equilibrium as the 

 plane of (a?, #), and suppose the sides of the rectangular canal 

 symmetrical with respect to the plane (x, z), </> will evidently 

 contain none but even powers of y, and we shall have 



Now if 



represent the equation of the two sides of the canal, we need 

 only satisfy one of them as 



