OF INCOMPRESSIBLE FLUIDS. 447 



vector, and the value of p has the form pV+A + B(a?+ y 2 ). When 

 however the ellipse becomes a circle, P and Q vanish in the equation 

 P<p"(Ui) + Q(t>'{U^) — 0. Consequently the form of <p may be any 

 whatever. The value of J7, being a? + y 1 , we have 



u = 2cj>'(U 1 )y, v = -2(p'(U l )a;; 



whence, « 2 + * = 4 \<f>' (U,)}* (& + f) = 4 U, {<f> (t/,)} 2 . 



Hence, the velocity may be any function of the distance from the centre. 

 It is evident that we may conceive cylindrical shells of fluid, having a 

 common axis, to be revolving about that axis with any velocities what- 

 ever, if we do not consider friction, or whether such a mode of motion 

 would be stable. The result is the same if we enquire in what way 

 fluid can move in a system of parallel lines. 



In any case where the motion in a certain system of lines is possible, 

 if we suppose two of these lines to be the bases of bounding cylindrical 

 surfaces, and if we suppose the velocity and direction of motion, at 

 each point of a section of the entering, and also of the issuing fluid, to 

 be what that case requires, I have not proved that the fluid must move 

 in that system of lines. When the above conditions are given there 

 may still perhaps be different modes of steady motion ; and of these 

 some may be stable, and others unstable. There may even be no stable 

 steady mode of motion possible, in which case the fluid would continue 

 perpetually eddying. 



In the case of rectangular hyperbolas, the fluid appeared, on making 

 the experiment, to move in hyperbolas when the end at which the 

 fluid entered was broad and the other end narrow, but not when the 

 end by which the fluid entered was narrow. This may, I think, in 

 some measure be accounted for. Suppose fluid to flow out of a vessel 

 where the pressure is p t into one where it is p 2 , through a small orifice. 

 Then, the motion being steady, we have, along the same line of motion, 



^ = C — £ v *, where v is the whole velocity. At a distance from the 



3E 2 



