XXIII. On the Steady Motion of Incompressible Fluids. By G. G. 

 Stokes, B. A. Fellow of Pembroke College. 



[Read April 25, 1842.] 



In this paper I shall consider chiefly the steady motion of fluids 

 in two dimensions. As however in the more general case of motion 

 in three dimensions, as well as in this, the calculation is simplified when 

 udx + vdy 4- wd% is an exact differential, I shall first consider a class 

 of cases where this is true. I need not explain the notation, except 

 where it may be new, or liable to be mistaken. 



To prove that udx + vdy + wd% is an exact differential, in the case 

 of steady motion, when the lines of motion are open curves, and when 

 the fluid in motion has come from an expanse of fluid of indefinite 

 extent, and where, at an indefinite distance, the velocity is indefinitely 

 small, and the pressure indefinitely near to what it would be if there 

 were no motion. 



By integrating along a line of motion, it is well known that we get 

 the equation 



£ = V- \{u- + v* + w*) + C.... (1), 



where d V = Xdx + Ydy + Zd%, which I suppose an exact differential. 

 Now from the way in which this equation is obtained, it appears that 

 C need only be constant for the same line of motion, and therefore in 

 general will be a function of the parameter of a line of motion. I shall 

 first shew that in the case considered C is absolutely constant, and then 

 that whenever it is, udx + vdy + wdz is an exact differential. 



To determine the value of C for any particular line of motion, it 

 is sufficient to know the values of p, and of the whole velocity, at 



3D2 



