MR. S. S. HOUGH OH THE ROTATION OF AH ELASTIC SPHEROID. 321 
t of the particle, which, when the body is unstrained, is at the point x Q , y 0 , z 0 . The 
velocity components of this particle will be 
u — oo (y 0 + v), 
V + CO (x 0 + u), 
IV. 
The components of acceleration will be 
• •• • • 
U — Vft) = U — VOO — {v + CO (x Q -f- u)\ 00 = U — 2 COV 
• ‘ • • • ••• 
A -f- L co == V -f- W<u -{- — co fy^ -j- r?) | oo — V —j- 2 coU 
W = VK 
If P, Q, R, S, T, U denote the six components of stress at the point x, y, z (where 
OC - OCq + u, &c.), X, Y, Z the components of bodily force at this point, and p the 
density of the material, the equations of motion may be written down in the 
same manner as if the axes were fixed, provided that we replace the accelerations 
cV'vjdfi, dhvjdfi by the values we have found above. Thus we have* 
— oo~U —- co' 2 X 0 , 
— oo 2 v — oo z y 0 , 
U — ^ (x Q + u) — oo (y 0 + v) — 
1 = 1ft + v ) + w ( x o + u ) = 
W =|fe+D 
ap 
+ 
au 
aT 
dx 
ff/ 
+ 
a z 
a u 
H" 
3Q 
+ 
as 
ox 
3T 
+ 
cfc 
+ 
3E 
dx 
Sy 
a z 
p (u — 2 cov — a fu — co'X Q ), 
p (v -f 2aou — oo z v — oryf), 
pw. 
We have here employed two different sets of independent variables. The quantities 
involved on the left have been supposed to be expressed as functions of the variables 
x, y, z, t, that is to say, the coordinates of a definite point on which we fix our attention 
and the time ; these variables are analogous to the Eulerian system in Hydrodynamics. 
On the other hand, the quantities u, v, w on the right have been regarded as functions 
of x 0 , y 0 , z Q , t, which correspond to the Lagrangian system of independent variables 
in Hydrodynamics. It is desirable for us to retain only one set of variables ; we propose 
to select the former set and proceed to examine the modified form of the equations of 
small motion. 
If the symbol djdt be used as above to denote partial differentiation with respect 
* Loye, ‘Elasticity,’ vol. 1, p. 60. 
2 T 
MDCCCXCYI.—A. 
