L 363 ] 
XVI. On the Motion of Fluid, part of which is moving Rotationally and part 
Irrotationally. 
By M. J. M. Hill, M.A., Professor of Mathematics at the Mason Science College, 
Birmingham. 
Communicated by Professor Stokes, D.C.L., Sec.R.S. 
Received December 29, 1883,—Read February 7, 1884. 
Introduction. 
Clebsch has shown that the components of the velocity of a fluid u, v, w, parallel to 
rectangular axes x, y, z, may always be expressed thus 
u— 
d X_ L X 
dx dx’ 
v= d f+\ 
dy 
dp 
dy’ 
dp 
~dz ’ 
where \, \p are systems of surfaces whose intersections determine the vortex lines; and 
the pressure satisfies an equation which is* equivalent to the following 
H _L y_ _ c hd_ l 
p dt 2 
dx y 
dx) 
+ 
dy 
+ 
dx 
dz 
+JX 3 
Idpf /dp 
\dx 
+ 
dy 
+ 
dp 
dz 
where p is the pressure, p the density, and V the potential of the forces acting on the 
liquid. 
It is shown in this paper that an equation of a complicated nature in X only can be 
obtained in the following cases (that is to say, as in cases of irrotational motion, the 
determination of the motion depends on the solution of a single equation only):— 
(1.) Plane motion, referred to rectangular coordinates x, y. 
The equation is somewhat simpler when the vortex surfaces are of invariable form, 
and move parallel to one of the axes of coordinates with arbitrary velocity. 
(2.) Plane motion, referred to polar coordinates r, 9. 
The equation is somewhat simpler when the vortex surfaces are of invariable form, 
and rotate about the origin with arbitrary angular velocity. 
(3.) Motion symmetrical with regard to the axis of z in planes passing through it, 
referred to cylindric coordinates r, z. 
The equation is somewhat simpler, when the vortex surfaces are of invariable form, 
and move parallel to the axis of z with arbitrary velocity. 
* British Association Report for 1881, p. 62. 
3 A 2 
