548 MR. G. H. DARWIN ON PROBLEMS CONNECTED 
F = sin 6 sin <f>, G sin 6 cos </>, H = 0 . . . . (14) 
and at the surface of the sphere r—a. 
Then according to the principles above explained, we have to find a, /3 /5 y / so that 
they may satisfy 
— ~ 1 -\-vV~a = 0, &c., &c., 
ax ' 
throughout a sphere, which is subject to surface stresses given by subtracting from (6) 
the surface values of F, G, H in (14). Hence the surface stresses to be satisfied by 
a,, (i,, y / , have components 
-A-3— "G ft3 (t — sin 0 d) sin 6 sin <f>, B 3 = — ^ sin 2 9) sin 6 cos (f>, C 3 = 0. 
These are surface harmonics of the third order as they stand. 
Now the solution of Sir W. Thomson’s problem of the state of strain of an incom¬ 
pressible elastic sphere, subject only to surface stress, is applicable to an incompressible 
viscous sphere, mutatis mutandis. His solution" shows that a surface stress, of which 
the components are A,-, B„ C; (surface harmonics of the i ih order), gives rise to a state 
of flow expressed by 
a — 
i r (« 2 - a) 
ddh-1 . 
1 
+ 
cr 
+ 
va l ~ l ]_2(2r + 1) 
dx 4 
i -1 
_(2i* + l)(2i + l) 
dV" *' +1 )4- 
i fl>hi 
2?(2/ +1) dx 
+ A/A 
(15) 
and symmetrical expressions for /3, y. 
Where W and <J> are auxiliary functions defined by 
v +1 =+ f (C,- ; - 
‘) 
(16) 
In our case i — 3, and it is easily shown that the auxiliary functions are both zero, 
so that the required solution is 
w 
a = 
8a C 
4^ . w 
(■f — s i n ° @) r?J sin 6 sin <j>, = — sin 2 6) sin 6 cos (/>, y / — 0. 
It we add to these the values of a, ft', y' from (10), we have as the complete 
solution of the problem, 
* Thomson ancl Tait’s ‘Nat. Pliil.,’ § 737. 
