8 
MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC 
1 
f i(i + 2) o 
(i +1) (2i + 3) o | dWi 
V 
Ll2(t-l)[2(t + l)»+l] 
2 (2i +1)[2 (i + 1) 2 + 1]^ J dx 
d 
(2? + l)[2(t + l) 3 + l] dx 
r2i + 3 
0 
• (sr 
with symmetrical expressions for /3" and y". 
I will first consider (ii); i.e., the matter of the earth is now supposed to possess the 
power of gravitation. 
The gravitation potential of the spheroid r=afl-cr, (taking only a typical term 
of cr) at a point in the interior, estimated per unit volume, is 
gw 
(3a 2 - r 2 )- 
6gw 
2 i+ l\a 
C T; 
according to the usual formula in the theory of the potential. 
Now the first term, being symmetrical round the centre of the sphere, can clearly 
cause no flow in the incompressible viscous sphere. We are therefore left with 
3gw ffV 
2 i+ 1\« 
Now if 
3gw fry 
2 i+ I\« 
cr, be substituted for W; in (8), and if the resulting expression be 
O 
compared with (7) when —gwcri is written for S„ it will be seen that —a"= ——-a'. 
Thus 
a -pa = a 
2t + l\t 2,. \ // 
= ■ 
3 
And if v ( =^ Q’ov, 
' i //_ 
a -j- a = 
i(i + 2) 
2(t-l)[2(t + l)» + l] 
i 
a~ — 
(t+l)(2*+3) ,1 d /2 
2(2t+l)[2(t+l) 2 + l] / J dx\o 
(i— 1)Y 
(2i + l)[2(i+ 1) 2 +1] dx 
■ (9) 
with symmetrical expressions for and y'-j-y". 
Equation (9) then embodies the solution as far as it depends on (ii) and (iii). And 
2 
since (9) is the same as (8) when—-(?’—l)V 2 - is written for W*, we may include all 
O 
the effects of mutual gravitation in producing a state of flow in the viscous sphere, by 
adopting Thomson’s solution (8), and taking instead of the true potential of the layer 
* ‘Nat. Phil.’, § 834, equation (8) when m is infinite compared with n, and i — 1 written for i, and v 
replaces n. 
t The case of § 815 in Thomson and Tait’s ‘Nat. Phil.’ is a special case of this. 
