OF INCOMPRESSIBLE FLUIDS. 453 



As before, the ratio of the coefficients of <p" (£/,) and <£'(t/i) must be a 

 function of U t alone, when as, r and £7", are connected by the equation 

 F(r, z) — Ui. If the motion be possible, it will in general be deter- 

 minate, U being of the form Af(r, z) + B. If U = r however, the 

 form of (p remains arbitrary. In this case the fluid may be conceived 

 to move in cylindrical shells parallel to the axis, the velocity being any 

 function of the distance from the axis. 



Particular cases are, where the lines of motion are right lines directed 

 to a point in the axis, and where they are equal parabolas having the 

 axis of z for a common axi^. In these cases udx + vdy + wdz is an 

 exact differential. 



We may employ equations (20) and (21) to determine whether the 

 hypothesis of parallel sections can be strictly true in any case. In this 

 case, the sections being perpendicular to the axis of ss, we must have 



1 dU 



w = -j- = /'(*); 



r dr v / 



dU sv^ 



U= -\r*F(z) +/(»). 



Substituting this value in (21), we find, by equating to zero coefficients 

 of different powers of r, that the most general case corresponds to 



U= (a + bz + cs s ) r 2 + ez +f. 



If udx + vdy + wdz be an exact differential, the most general case 

 corresponds to 



U = (a + bz) r* + c + ez. 



G. G. STOKES. 



Pembroke College, Cambridge, 

 April, 1842. 



Vol. VII. Part III. 3F 



