of Edinburgh, Session 1885-86. 
377 
These equations give explicitly the position of any chosen particle 
at any time, and of course it would be easy to find from them what 
the path is ; but it is easier to do this from the unintegrated 
equations (8). Multiplying the first of these by a/a 2 , the second by 
/3/b 2 , and the third by y/c 2 and adding, we find 
B dj y_d A=() 
d 2 dt b 2 dt c 2 dt 
which proves that the orbit lies in the plane 
(U); 
P 
a? * + J2 9 + c 
S=H 
(18), 
where H denotes a constant. 
Again multiplying the first of equations (8) by %/a 2 , the second 
by S)/b 2 , and the third by $/c 2 and adding, we find 
b dfi j_ dl = 0 
d 2 dt b 2 dt e 2 dt 
( 19 ), 
and integrating this we have 
( 20 ), 
where K denotes a constant. 
This proves that the orbit lies on the ellipsoid (20) ; and we con- 
clude that the orbit is the ellipse in which this ellipsoid is cut by 
the plane (18). 
Going back now to the explicit fully integrated solution (15) and 
(16), we see that a particle of the fluid describes, relatively to the 
moving solid in which the fluid is contained, the ellipse specified 
by (18) and (20), according to the law of a single particle describing 
an ellipse under the influence of a force towards a fixed centre 
varying in simple proportion to distance from the centre. 
Now the period of revolution of the containing shell round its 
axis of rotation (w, p, a) is 27r/e 
where € = x /(*r 2 + p 2 + o- 2 ) 
which is easily seen to be less than 2ttIw [the value of w being 
given by (15) above]. Hence considering the shell and contained 
liquid at any instant, and again at the later instant when the shell 
is again in the same position after a single complete revolution 
