458 Mr green, ON WAVES IN A VARIABLE CANAL 



supposing the disturbance so small that the squares and higher 

 powers of the velocities &:c. may be neglected. In the above formulee 

 j9 = pressure, /> = density, and (p is such a function of .r, y, s and f, that 

 the velocities of the fluid particles parallel to the three axes are 



"-S)-=©. -CD- 



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

 relative to the exterior surfaces of the fluid, and if ^ = be the equa- 

 tion of one of these surfaces, the corresponding condition is [Lagrange, 

 Mec. Anal. Tom. II, p. 303. (I.)], 



_dA cLA clA dA 

 ~ dt dx dy dz 



Hence 



, . , „ dA dA deb dA d(p dA dxb . , . ^, 



<^) « = rfT ^ 7?^ v/l + ^-.^ + d^-£ ("'J^«^^ = o)- 



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 + vdy + wdz 

 is an exact differential. 



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

 ation of the motion of a non-elastic fluid, when two of the dimensions, 

 viz. those parallel to y and x, are so small that (p may be expanded in a 

 rapidly convergent series in powers of y and », so that 



(p = fi>o+ <p'l + cp,\ + <p"~+<p: y^ + <t>,.Y:^ + &c. 



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

 of {x, y,) and suppose the sides of the rectangular canal symmetrical with 

 respect to the plane {x, k,) <p will evidently contain none but even powers 

 of y, and we shall have 



(3). <p = (po+<p,& + <p"^ + <p„^ + &^c. 



