720 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
The function W only dilfers from the ordinary disturbing function by a constant 
factor, and so W will be referred to as the disturbing function. 
I will now explain why it has been convenient to depart from ordinary usage, and 
will show how the same disturbing function W may be used for giving the perturba¬ 
tions of the rotation of the planet. 
In the present problem all the perturbations, both of satellites and planet, arise 
from tides raised in the planet. 
The only case treated will be where the tidal wave is expressible as a surface 
spherical harmonic of the second order. 
Suppose then that p=ct-\-o- is the equation to the wave surface, superposed on the 
sphere of mean radius a. 
Then the potential V of the wave a, at an external point p, must be given by 
A = J f npica 
(! 5 ) 
Here w is the density of the matter forming the wave ; in our case of a homogeneous 
earth, distorted by bodily tides, iv is the mean density of the earth. (If we con¬ 
template oceanic tides, the subsequent results for the disturbing function must be 
reduced by the factor yi, this being the ratio of the density of water to the mean 
density of the earth.) 
Now suppose the external point p to be at a satellite whose mass, radius vector, and 
mean distance are r.i, r, c. Then if we put r= fpm/c 3 , and observe that C= -^nwa 5 , we 
have 
(16) 
Where cr is the height of tide, where the wave surface is pierced by the satellite’s 
radius vector. 
But the ordinary disturbing function B for this satellite is this potential V 
augmented by the factor because the planet must be reduced to rest. 
Hence our disturbing function 
w =i) 8 ;. 
where cr is the height of tide at the place where the wave surface is pierced by r. 
Now let us turn to the case of the planet as perturbed by the attraction of the 
same satellite on the same wave surface. The whole force function of the action of 
the satellite on the planet is, by ( 16 ), clearly equal to 
