MR. S. S. HOUGH OH THE ROTATION OF AH ELASTIC SPHEROID. 323 
If further, the bodily forces be derivable from a potential function Y, so that 
x = av/ax, Y = av,%, z = av/a 2 .( 3 ), 
on substituting the values (2). (3), in (1), we obtain 
7 v3 “ + S { v + ^ G* + 2O - 7} = “ ~ 2 “’ 
J Vh + D | V + !</ (*» + f) - ^-} = v + 2<«, 
7 v5w + 4{ v + + s*) - 7} = “■ 
Putting n/p = n, Y + |<y 3 (x 2 + y~) — p/p = rp, these equations take the form 
nV 2 u + m'dx = u — 2 ojv, j 
nV 2 v + dxjj/dy = v + 2cou, r*.(4)* 
nVhv + cxp/dz = w. J 
We must in addition express the fact that the material is incompressible; this 
is done by the equation 
du/dx + dvjdy -f- div/dz = 0.(5). 
The equations (4), (5), which are the rigorous equations for the vibrations of a 
rotating, incompressible, elastic solid, are theoretically sufficient to determine u, v x w, xp, 
subject to certain boundary conditions. Eliminating v, u in turn from the first two 
of equations (4), we obtain 
nV 2 
nV 3 
S 3 \ 3 , , a 3 2 ‘ 
dfi J + 4(0 ' ¥ 
a t* 
r)° 
+ 4 “ 2 a? 
U = 
V 
a.r 
+ 2 “ St 
J 
Applying the operators d/dx, 0/8y, and making use of (5), 
nV 2 — 
*\* o3:' 
■se) +4lo s t\ 
(ho ( 02 \ 
0* ~ r V ' “ 8*0(aa? + 
Hence, on eliminating 
equation for xp :— 
w by means of the third of (4), we obtain the following 
v2 ( )iV3 -avJ + 4 -SUsb = 0 .W- 
2 T 2 
