582 MR. G. H. DARWIN ON PROBLEMS CONNECTED 
Hence 
p—w{ 3cr— r~) ~ + solid harmonics + a constant. 
Now when r—a, at the mean surface of the sphere, p=gwcr l , therefore 
p=w(ar—r^+gwa 
Then substituting this value of p in the equations of motion (73;, 
dec d \ , 0 
2i + l \a 
da_ d 
dt dx 
and two similar equations 
(74) 
The expression within brackets [ ] on the right is the effective disturbing potential, 
inclusive of the effects of mutual gravitation, and thus this process is exactly parallel 
to that adopted above in order to include the effects of mutual gravitation in the dis¬ 
turbing potential in the case of the viscous spheroid. 
OQ 'll Z 
Now p, the radial velocity of flow, is equal to 
QrJ 'll £ 
Therefore multiplying the equations (74) by-, -, - and adding them, we have, by 
the properties of homogeneous functions, 
But when r—a, p—~. 
CtO 
Therefore 
dp 
dt 
w-1 
S; 
2d —1) d- 1 
ia 
27(7-1) 
2i+l 
(75) 
Now suppose S;=Q; cos vt, and that the tidal motion is steady, so that a t must be of 
the form XQ* cos vt ; then substituting in (75) this form of a n we find 
Whence 
2i(i — l) 
2i + l 
— ia} 1 . 
m= 
ia‘ 1 
2i(i—l) g 
2i +1 a 
Q; COS Vt 
(76) 
This gives the equation to the tidal spheroid. 
