440 Mr STOKES, ON THE STEADY MOTION 



any point along that line. Now if there were no motion we should 

 have 



£ = v+ c (2), 



p 



p x being the pressure in that case. But considering a point in this 

 line at an indefinite distance in the expanse, the value of p at that point 

 will be indefinitely nearly equal to p u and the velocity will be indefinitely 

 small. Consequently C is more nearly equal to d than any assignable 

 quantity : therefore C is equal to C t ; and this whatever be the line 

 of motion considered ; therefore C is constant. 



In ordinary cases of steady motion, when the fluid flows in open 

 curves, it does come from such an expanse of fluid. It is conceivable 

 that there should be only a canal of fluid in this expanse in motion, 

 the rest being at rest, in which case the velocity at an indefinite distance 

 might not be indefinitely small. But experiment shews that this is 

 not the case, but that the fluid flows in from all sides. Consequently 

 at an indefinite distance the velocity is indefinitely small, and it seems 

 evident that in that case the pressure must be indefinitely near to what 

 it would be if there were no motion. 



Differentiating therefore (1) with respect to x, we get 



1 dp v du dv dw 



p dx dx ax dx 



. ■ 1 dp v du du du 



but - -/- = A — u-j v -j » T ; 



p dx dx dy dz 



/dv du\ (dw du\ 



whence v -3 ti + w \ j t) — °- 



\dx dyi \dx d%l 



a- -i 1 (dw dv\ , (du dv\ 



Similarly, w (^ - jgj + u (^ - ^ j = 0, 



/du dw\ (dv dw\ 



u (di-dx-) +v \d- % -«£•]-*' 



dv du dw dv du _ dw 

 whence Tx = Ty , jj - ft, di'dx' 



and therefore udx + vdy + wdz is an exact differential. 



