324 MR. S. S. HOUGH OH THE ROTATION OP AH ELASTIC SPHEROID. 
In the case where n is zero, it will be noticed that the last equation reduces to 
Poincare’s differential equation for the oscillations of a rotating mass of liquid.* 
It may easily be shown by retaining any one of the four quantities u, v, w, iff, and 
eliminating the other three, that each one of these quantities is a solution of the 
equation (6). 
If the motion relatively to the moving axes consists of a simple harmonic vibration 
in period 27 r/A, we may suppose u, v, w, \}j each proportional to e lXt , and the equations 
(4), (5), (6) will then become 
(n V 2 -J- A 2 ) u -f- 2 (di'kv = — 0i///0te,'"j 
(n V 2 -f A 2 ) v — 2coi\u = — dxjj/dy, j 
(n V 2 + A 2 ) w — - dxfj/dz, \ . 
du/dx + du/dy-\-dw/dz — 0, 
[V 2 (n V 2 + A 2 ) 2 - 4w 2 A 2 0 2 /0z 2 J iff = 0.(8). 
§ 2. Boundary Conditions. 
The conditions to be satisfied at the boundary are that the components of surface- 
traction should vanish at all points of the displaced surface. We proceed to replace 
these conditions by certain analytical conditions at the mean surface. 
Take a point P on the mean surface and let the normal at P meet the surface of 
the distorted body in PA 
Let cos a, cos (3, cos y be the direction-cosines of the normal PP 7 and cos a + l lt 
cos /3 -j- m y , cos y -f n-y the direction-cosines of the normal at P' to the displaced 
surface ; also let PP' = £. Then l u m u %, £ will be small quantities of the order of 
the displacements u, v, iv. 
Let P, Q, P, S, T. U denote the components of stress at P and the same letters 
* ‘ Acta Mathematical vol. 7, p. 356. 
