WITH THE TIDES OF A VISCOUS SPHEROID. 
547 
three by x, y, z respectively and add them; then bearing in mind the fourth equation, 
we have v 2 p'= 0, of which yf=0 is a solution. 
Thus the equations to be satisfied become 
o , w id w / A 
V-a=--- 2/ , V -/3=- V-y=0. 
Solutions of these are obviously 
M s 
O' l w M 3 
P =10 - ~^ r x > 
y 
■=o 
1 
i w 3 • a • i i ■ /) , 
-To ~ sm 0 sm <p =tu ~ r ,' r sm “ cos r 
V V L. 
( 10 ) 
These values satisfy the last of (8), viz. : the equation of continuity, and therefore 
together with p — 0, they form the required values of ct, ft', y, p. 
We have next to compute the surface stresses corresponding to these values. 
Let P, Q, II, S, T, U be the normal and tangential stresses (estimated as is usual 
in the theory of elastic solids) across three planes at right angles at the point x, y, z. 
Then 
Q, Pi, T, U being found by cyclical changes of symbols. 
Let F, G, H be the component stresses across a plane perpendicular to the radius 
vector r at the point x, y, z ; then 
Fr = Px -\-\Jy-\-Tz ~ 
Gr = U x -f- Qy -f- Sz > 
Hr=Tx +S y + Rz_ 
( 12 ) 
Substitute in 
d d , d 
x T~ + y~T J r z T- 
dx J dy dz 
(12) for P, Q, &c., from (11), and put C— + fi'y + y'z, and r~ for 
Then 
Fr= — p'x- 
r d~ l ) a ' + % 
, Gr=&c., Hi — &c. (13) 
These formulas give the stresses across any of the concentric spherical surfaces. 
In the particular case in hand p— 0, y — 0, £'=Q, and a, /3' are homogeneous functions 
of the third degree, hence 
