SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS. 
D 
true sphere, and of the positive and negative matter filling the space Scr„ but (in) 
subject to certain surface forces. 
Since (i) and (ii) together constitute a bodily force, the problem only differs from 
that of Sir W. Thomson in the fact that there are forces acting on the surface of the 
sphere. 
Now as we are only going to consider small deviations from sphericity, these surface 
actions will be of small amount, and an approximation will be permissible. 
It is clear that rigorously there is tangential action'" between the layer of matter 
2 <t ; and the true sphere, but by far the larger part of the action is normal, and is 
simply the weight (either positive or negative) of the matter which lies above or 
below any point on the surface of the true sphere. 
Thus, in order to reduce the earth to sphericity, the appropriate surface action is 
a normal traction equal to —qyuAo-,, where g is gravity at the surface, and iv is the 
mass per unit volume of the matter constituting the earth. 
In order to show what alteration this normal surface traction will make in 
Sir W. Thomson’s solution, I must now give a short account of his method of 
attacking the problem. 
He first shows that, where there is a potential function, the solution of the problem 
may be subdivided, and that the complete values of a, /3, y consist of the sums of two 
parts which are to be found in different ways. The first part consists of any values of 
a, (3, y, which satisfy the equations throughout the sphere, without reference to surface 
conditions. As far as regards the second part, the bodily force is deemed to be non¬ 
existent and is replaced by certain surface actions, so calculated as to counteract the 
surface actions which correspond to the values of a, /3, y found in the first part of the 
solution. Thus the first part satisfies the condition that there is a bodily force, and 
the second adds the condition that the surface forces are zero. The first part of the 
solution is easily found, and for the second part Sir W. Thomson discusses the case of 
an elastic sphere under the action of any surface tractions, but without any bodily 
force acting on it. The component surface tractions parallel to the three axes, in this 
problem, are supposed to be expanded in a series of surface harmonics ; and the 
harmonic terms of any order are shown to have an effect on the displacements inde¬ 
pendent of those of every other order. Thus it is only necessary to consider the 
typical component surface tractions A/, ifi, C, of the order i. 
He proves that (for an incompressible elastic solid for which m is infinite) this one 
surface traction A/, If, Cf produces a displacement throughout the sphere given by 
1 f a»-r» 
. 1 
nct i -'\2(2i 2 + 1) 
dx 
‘ t-1 
,3e'+l 
_(2i a + l)(2i+l) dx 
;('F;— 1 r-* i+l )+; 
1 cMfi- 
+1 
2i(2i +1) dx 
-A r 
(5)t 
* I shall consider some of the effects of this tangential action in a futare paper, viz. : “ Problems con¬ 
nected with the Tides of a Viscous Spheroid,” read before the Royal Society qn I) ecem b ei ' 19th, 1878. 
t Thomson and Tait’s ‘Nat. Phil.,’ 1867, § 737, equation (52). 
