﻿372 On the Uniform Motion of an Imperfect Fluid. 



Differentiating successively in respect to x and y, and observing 

 that fz 2 x dx is not a function of y, nor J z \dy of x } 



If the fluid descends by its weight only, , , =0 ; 

 (dz . dz\ , 



T . iv sin i 



Let m— —= > 



2{a 



dz dz 

 dx dv 



The general primitive of this partial differential equation is 

 readily found by the method of Lagrange to be 



x=Ce mx+ ^- x ) ; 



or, since x and y enter symmetrically, 



z = Q £ my +$(*-?/) 



where cj> is an arbitrary function of (#—«/). A complete primi- 

 tive is found by the method of Charpit to be 



mz = Ae mx + Bs m y. 



The whole volume Q of the fluid discharged in a unit of time 

 through a rectangular portion of the transverse section repre- 

 sented by xy is 



C x py 1 n x Cv 



Q= I 4 zdxdy=—\ \ (Ae mx + 'Be m y)dxdy ) 



... Q=^ 2 {Ay(€ OT *-l)+Ba?(€^-l)}. 

 Or, substituting the value of m, 



9#. w sin i , 10 sin i 



z=~^ \kfW'+Be-*r*\ 9 ...... (1) 



wsnu ( 3 



(O ... v 2 w; sin i w sin i 



STiW |Ay(e^*-l)+B,(eV»-l)}. . (2) 



In equation (1) z represents the velocity of the stream at the 

 point xy. This equation only applies to one-half the section — 

 from one side to the centre. The whole discharge is therefore 

 to be determined by substituting a for x and b for y in equation 

 (2) and taking twice the result. 



