366 
PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
Example III. treats of the revolution of an elliptic vortex cylinder round its axis, 
where the angular velocity is not restricted as in the last case. The irrotational 
motion outside may be supposed to be limited by a smooth rigid confocal elliptic 
cylindric surface, rotating with the same angular velocity. The last example is the 
particular case of this, obtained by supposing the elliptic section of the external 
confocal cylinder to become infinite. 
Example IV. treats of the motion of the fluid in a fixed circular cylindric surface, 
where the vortex strength is any function of the distance from the axis, the irrotational 
motion continuous therewith being supposed to extend to an infinite distance. 
Example V. treats of a possible case of rotational motion inside a certain hollow 
smooth rigid surface of annular form, which moves parallel to its straight axis with 
arbitrary velocity. 
1. Clebsch’s * forms for the components of the velocity of a liquid u, v, iv parallel to 
fixed rectangular axes x, y , z in space are :— 
dx ax ay ay 
w=^+\ d i 
dz dz 
where the surfaces \= const., \[>= const, determine by their intersections the vortex 
lines, and always contain the same particles of liquid. 
If F(X, x\i) be an arbitrary function of X, xjj 
=W,yW 
dx dx 
tF(A,, -xfr) ( ^ ^ d,y\r 
b\]r 
+ x| &' t 
&F(A,, xjr) d\ 
clo: 
bX 
and similar expressions for v, w. 
Thus these expressions for the components of the velocity are still in Clebsch’s 
form. 
X', i jj' are each functions of X, xfj. 
X satisfies the same equation as y. 
# Taking as independent variables three families of surfaces, always containing the same particles, and 
the time, the writer obtained independently Clebsch’s forms in an article published in the Quarterly 
Journal of Pure a?id Applied Mathematics, February, 1880, vol. xvii., entitled “ On Some Properties of 
the Equations of Hydrodynamics.” 
A demonstration of the same forms for any fluid in which the density is any function of the pressure 
is contained as a particular case in a paper entitled “ On Some General Equations which include the 
Equations of Hydrodynamics,” which is published in the Transactions of the Cambridge Philosophical 
Society, vol. xiv., part i., the writer having previously seen Clebsch’s paper, “ Ueber Die Integration der 
hydrodynamischen Gleichungen,” ‘ Crelle,’ Bd. lvi., p. 1. 
