of Edinburgh , Session 1885-86. 
375 
which is the explicit solution of the problem, so far as concerns 
merely the absolute velocity at any point of the fluid, which is 
generally considered far enough in the solution of a hydrodynamieal 
problem. But it would be interesting in every case, and it is easy 
in this case, to complete it up to the determination of the posi- 
tion of every particle of the liquid at any time, and we may 
therefore go on to do so. Relatively to the axes of the ellipsoid 
let (£, ty, $) be the coordinates at time t, of any particular particle 
of the liquid. The component velocities ( dic/dt , dty/dt, d^/dt) 
of the particle relatively to the ellipsoid are equal to the 
differences between the components ( u , v , w), of the absolute 
velocity of and the corresponding components of the absolute 
velocity of an ideal point ( x , y, z ) rigidly connected with the 
ellipsoid, and coincident with (£, ty, §) at the time t. These last 
components are 
P$-crty, i wfy-pXi • • ( 7 )- 
Hence, and from (6), at the instant ($, ty, j) coincident with 
(; x , y, z) we have 
where 
di _ 
dt 
-« 2 (r9-^5) 
II 
h$ 
- H a 5 - 7£) 
di = 
dt 
- c 2 (/?£ - a\j) 
2-nr 
R- 2 P 
2 0 - 
5 2 + c 2 ’ 
H 0 . 0 » 
c 2 + cr 
^ a 2 + b 2 
K8). 
These are linear differential equations of the first order for determin- 
ing (£, ty, j) in terms of t Denoting d/dt by S, we may write 
them as follows — 
4* £+ 79 -/ 35 =° 
-rS + p-9 + “3=° 
/?£ — at) + .A J = 0 
(°)- 
