MR. S. S. HOUG-H ON THE ROTATION OF AN ELASTIC SPHEROID. 
325 
accented the components of stress at P', ancl let clri denote an element of the normal 
to the mean surface. Then to our order of approximation we have 
F = P + £ dP/dn', Q' = Q -f £ 3 Q/dn, &c. 
Thus the component of surface-traction at P' parallel to the axis of x is 
P (cos ci -T /|) -p U (cos /3 -p wq) *p 1 (cos y -p ?q), 
= f P + ^ f5) ( cos “ + + ( U + £ U) ( cos ^ + m i) + (^T p £ (cos y + n x ), 
0P 
dii' 
-p ^P + m x U + ?qT, 
if we neglect small quantities of the second order. 
Now in the small terms we may replace P, Q, F, &c., by their values in the 
steady motion. Since we have supposed the body when undisturbed to be free 
from tangential stress in its interior, we have in this case 
P = Q = It = - p, S = T = U = o 
throughout, while at the surface also = 0. 
Therefore /jP -p m^J -p n 1 T, dPJ/dn', 0T/0 n may be put equal to zero, and 
|p = — = — p p7 {Y + ^co 2 ( x 2 + y 2 )} = + pg say, where g denotes the value 
of gravity (inclusive of centrifugal force) at the surface. Thus the ^-component 
of surface-traction at P' is 
P cos a -p U cos j8 p T cos y + P9t cos a. 
= P cos a -p U cos f3 -p T cos y -p £ 
, SU , 0T 
cos a + 0 ^/ cos P -p yy cos. y 
dn’ 
Introducing the values of P, U, T in terms of the displacements and equating this 
expression to zero, we obtain 
du , du „ . \ , du 
— p COS a 4- tt X— COS a + COS p 4- — COS y 4- n — cos a 
\ox dy oz / \ dr 
dv 
dx 
cos /3 -p ~ cos y 
= — pgC cos a ; 
and, in like manner, by considering the components of surface-traction in the 
directions of the axes of y and z. 
