252 Mr. J. McCowan on the 



before the liquid was disturbed, bad the common abscissa x. 

 Let z denote the height of the free surface of the liquid above 

 the axis of x, where it is cut by the plane £, and let h be the 

 undisturbed height. Since the form of the cross section of 

 the channel is supposed given, the area of the section of the 

 liquid made by the plane f is a function of z which can be 

 determined and may therefore be supposed known, say <£(/), 

 and the area of the undisturbed section being the same 

 function of h must therefore be (f>(h). 



Consider now two planes, perpendicular to the length of 

 the channel, which move with the fluid and so contain always 

 the same particles, and which, before the liquid was disturbed, 

 were at the small distance Sx apart : at time t their distance 

 apart will have become o%=&x .d^/dx, but the quantity of 

 liquid between them will be unaltered and therefore 

 tjy(z)8^=<f>(h)Bxj or 



dj_m m 



dx~ $(z) [L) 



This is the " equation of continuity/' and we have also, if 

 u denotes the component of the velocity parallel to the axis 

 of x, 



§ =• w 



Next, let p be the density of the liquid and let p be the 

 pressure at any point on the plane f : the motion, parallel to 

 the axis of x, of the particles near this point will therefore be 

 given by the dynamical equation 



<P£__(dp\ 



p dt*~ W ; 



or, by (1), . 



n d?Z__<ttz)_dp r ~, 



p dt? " $(h)dx w 



Now, from the initial conditions, it is clear that the vertical 

 component of the acceleration will be negligible, and therefore 

 that the variation of the pressure from point to point will be 

 due entirely to the variation in the depth below the surface : 

 thus, for points on the same horizontal plane, Sp=gpSz, and 

 therefore (3) gives 



d^_ <Hz)dz 



dt 2 ~~ y <p{h)dx w 



Thus the component of the acceleration parallel to the axis 

 of x is the same for all the particles in any plane f perpen- 

 dicular to the length of the channel, and therefore all particles 



