ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 
219 
where the density p is expressed in gravitational units, 
density of the earth as a whole, including the ocean, 
degree of approximation, 
g = iTTcra, 
and therefore 
But if cr denote the mean 
we have, with the same 
V 
71=00 
= S 
op 
n=l (2/i + 1) cr 
gO,^n {p)- 
As this only involves the ratio pjcr, it is independent of the unit of mass employed. 
Next suppose that the surface-value of the disturbing potential can be expressed 
by means of the series 
ty,^n {p)- 
Then equation (9) of § 2 gives 
^ = — gi-\-v 
gU - 
= - s 
or, if we write for brevity, 
op 
(2n + 1) cr 
op 
0 
11 Yn 
yn 
(a^). 
we 
have 
Now, from the equation 
d 
dp. 
— yn ■“ gif^n 
4, = trn'Pn{p) . . 
( 19 ). 
( 20 ), 
which defines the zonal harmonics, we find 
Hence, if we integrate (18) with respect to p, we obtain 
^ ^ ^ = 4aVSC„rP„ dp 
i' Oa J 
P-p^ dp 
= A — (l — p^) 'Z 
Cn dP„ 
n (n 4- 1) dp 
where A is an arbitrary constant, which may be seen to be zero by putting p. = zb 1. 
Therefore 
2 F 2 
