448 Mr. STOKES, ON THE STEADY MOTION 



orifice, in the first vessel, the pressure will be approximately p u and the 

 velocity nothing. At a distance in the second vessel, the pressure will 



be approximately^, and therefore the velocity = V ^' - ^ , nearly. 



r 



The result is the same if forces act on the fluid. Hence the velocity 

 must be approximately constant; and therefore, the fluid which came 

 from the first vessel, instead of spreading out, must keep to a canal 

 of its own of uniform breadth. This is found to agree with experiment. 

 Hence we might expect that in the case of the hyperbolas, if the end 

 at which the fluid entered were narrow, the entering fluid would have 

 a tendency to keep to a canal of its own, instead of spreading out. 



In ordinary cases of steady motion, when the lines of motion are 

 open curves, the fluid is supplied from an expanse of fluid, and conse- 

 quently udx + vdy + wd% is an exact differential. Consequently, cases 

 of open curves for which it is not an exact differential do not ordinarily 

 occur. We may, however, conceive such cases to occur; for we may 

 suppose the velocity and direction of motion, at each point of a section 

 of the entering, and also of the issuing stream, to be such as any case 

 requires, by supposing the fluid sent in and drawn out with the 

 requisite velocity and in the requisite direction through an infinite 

 number of infinitely small tubes. 



In the case of closed curves however, in whatever manner the fluid 

 may have been put in motion, it seems probable that, if we neglect 

 the friction against the sides of the vessel, the fluid will have a tendency 

 to settle down into some steady mode of motion. Consequently, taking 

 account of the friction against the sides of the vessel, it seems probable 

 that the motion may in some cases become approximately steady, before 

 the friction has caused it to cease altogether. 



Motion symmetrical about an axis, the lines of motion being in planes 

 passing through the axis. 



Before considering this case, it may be well to prove a principle 

 which will a little simplify our equations. 



