Caaon Moseley on the stead// Flow of a Liquid, 35 



Substituting for a and /3 their values, 



2^( - + Zsinz ) 

 t;^*= ^^ ^ (48) 



1 — s 1— 2^( — \- Isim) — >6i^\yi ^ '■ -' 



The conditions of equation (46) are satisfied by the above 

 equation, whatever may be the form or value of the function 

 (f){y—x). But these conditions are not the only ones to which 

 the flow of the liquid is subjected. They do not include the 

 form of the channel, or the degree of roughness or smoothness 

 of its sides. The term U3 in equation (46) represented these ; 

 but it disappeared in the double differential, and has no place in 

 equation (47), from which equation (48) is deduced. The inde- 

 terminateness oi(j)(y—x) results from the neglect of this condi- 

 tion ; and the function is to be determined by taking it into 

 account. Let us suppose it to be so determined. 



Liquid Films, 

 If in any cross section a curve be taken whose equation is 



x + (f>{ij-x)-^(0)=p, 



where p is constant, that curv^e will represent the intersection of 

 it by a film. For, by equation (48), the velocity v of the liquid 

 at every point in that curve will be the same. By varying the 

 values of p, all the films of a given stream flowing uniformly may 

 thus be determined. In closed channels of symmetrical forms 

 and uniform roughness the velocities of such particles as flow 

 nearly in contact with the sides approach probably to equality. 

 A film nearly in contact with the sides has therefore nearly the 

 form of the channel itself; and as the films are geometrically 

 similar, it follows that approximately all the films, from the fila- 

 ment of maximum velocity to the internal surface of the channel, 

 take approximately the geometrical form of that surface. The 

 degree of that approximation can only be determined by com- 

 paring theoretical results founded upon it with the results of 

 experiment. That is the object of what remains of this paper. 



* To compare this equation with equation (10) (Phil. Mag. September 

 1871), we must make sini=0, y=\, 2ffh=v'^, j=i. We shall thus ob- 

 tain 



Assuming x-\-(P{y — x) — (j){0)=r, this equation becomes identical with 

 equation (10). 



D2 



