334 MR. S. S. HOUGH OH THE ROTATION OF AH ELASTIC SPHEROID. 
The remaining boundary equations can be written down by replacing x by y, z 
respectively. Hence, all three boundary equations will be satisfied provided A, B 
are subject to the relations 
— Art 3 -|- lOnB + -^npci 2 ^2B — \ ^ ) = cr, 
A« 2 - -A-Aa a + -hrrpo? (A - + ) = a*. 
\ H j 
Solving these equations, we obtain 
A = 
21 
I9 + -| 
irpci 
o, B n = 
4a 2 
n 
19 + 4 
. irpa~ 
n 
2 B - \ — = 
5 a 2 !n 
n 
19 + 1^ 
n 
Now, let e = 5&ra 3 /38n, so that e will denote the ellipticity which would be 
induced in a sphere of radius a by centrifugal force when distortion is resisted by 
elasticity alone.* Then the above values may be written 
._21_ 3 _ 8e / /5 
A ~ 19 {1 + e'/e} ! hC °* ~ ! + e'/e ’ 
2 B - 
i A " 2 \ . 
' OT 
n 
2e' 
1 -I- e'/e 
Finally from (24) 
. a , T> ~ n r. Aft-tB- r~ . „ Aft-o)- x ( 6c>xz — 0,uz) 
u — u 0 -b u, = zOo -F Barzfch — -A- — + A-—— 5 —— 
ncc~ n a~ 
a I , , f>' - + e 'l . 4 x(6pz-0fj£) 
1 -r-'•+ = rT7Te-?- 
1 + e'/e 
s e > _ 
V ~ V Q + Vi —' — zd 1 { 1 + 1_ 
1 + e'/e 
« a 6 !> + i 
_ 
5 1 + e'/e ft 2 
10 = w Q + tiq = (— £cd 3 -b y0\) <] l — ft" 6 
1 + e'/e 
4 d s (9,xz - 6 x ys ) 
5 1 + e'/e ft 2 
* Of. Thomson and Tait, “Natural Philosophy,” Part IT., § 837. 
