SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 
7 
A. — ry—i + 1__ / A A. __ L^ r i+Z^L( r -i-ia \ 
^- 1- 2i + l dx { z+1 ~ 2i+l dx> 
Similarly - S*, - S; may be expressed as B i _ 1 +B /+1 and Q_ 1 + Ci +1 , where the B’s 
and C’s only differ from the As in having y, z written for x. 
We have now to form the auxiliary functions ' v ly_ 1 , <ff corresponding to A;_ 1} B;_ l5 
C;_ x and Mt,, <3>; +a corresponding to A z+1 , B, +1 , C ; +1 . 
Then by the formulae (6) 
(2t+l)V i _ s _^+^+^pS,—0 
2»+l_ d 
<h;= 
r 2i+i * dx 
-3f+l 
d_ 
dx 
+ 
„2i+l 
-(2i+i)^=|A +3 A- M s.) 
da;'' 
+* 
] i,j 
+5 
— — (7+ l)(2i+3)r'Si 
2t + l , _/'d 2 d 2 . d 2 , 
r 2i+5 *^ +2 [dx?^dy^dz^ 1 J 
Thus 
¥,- a =0. 
2i+l 
(i + l)(2i+3) 
US,:, y;= K —drrz --US„ 4> i+a = 0 
2i+.l 
Then by (5) we form a corresponding to A,_ l5 Bi_ 1? C;_ l5 and also to A,- +1 , B z - +1 , 
C,- +1 , and add them together. The final result is that a normal traction S, gives, 
vcv 
i{i + 2) 
XV 
(i+l)(2i+3) 
L 12(d — l)[2(d + 1) 2 + 1] 2(27 + l)[2(t +1) 2 +1] 
d 
(27 -h 1)[2(7 -h l) 2 + 1] 
„2i+3JA ( r -;-lg.) 
dx 
.(7) 
and symmetrical expressions for /S' and y. 
d, ft, y are here written for a, /3, y to show that this is only a partial solution, 
and v is written for n to show that it corresponds to the viscous problem. If we 
now put S i=—gw(n, we get the state of flow of the fluid due to the transmitted 
pressure of the deficiencies and excesses of matter below and above the true spherical 
surface. This constitutes the solution as far as it depends on (iii). 
There remain the parts dependent on (i) and (ii), which may for the present be 
classified together; and for this part Sir W. Thomson’s solution is directly applicable. 
The state of internal strain of an elastic sphere, subject to no surface action, but 
under the influence of a bodily force of which the potential is ¥7may be at once 
adapted to give the state of flow of a viscous sphere under like conditions. The 
solution is— 
