492 
MR. G. H. DARWIN ON A PLANET 
The last paragraph contains a discussion of the evidence adduced in this part ot the 
paper, and a short recapitulation of the observed facts in the solar system which bear 
on the subject. This is probably the only portion which will have any interest for 
others than mathematicians. 
I. 
THE THEORY OF THE TIDAL FRICTION OF A PLANET ATTENDED BY ANY 
NUMBER OF SATELLITES. 
§ 1. Statement and limitation of the problem. 
Suppose there be a planet attended by any number of satellites, all moving in 
circular orbits, the planes of which coincide with the equator of the planet; and 
suppose that the satellites all raise tides in the planet. Then the problem proposed 
for solution is to investigate the gradual changes in the configuration of the system 
under the influence of tidal friction. 
This problem is only here treated under certain restrictions as to the nature of the 
tidal friction and in other respects. These limitations however will afford sufficient 
insight into the more general problem. The planet is supposed to be a homogeneous 
spheroid formed of viscous fluid, and the only case considered in detail is that where 
the viscosity is small; moreover, in the tidal theory adopted the effects of inertia are 
neglected. I have however shown elsewhere that this neglect is not such as to 
materially vitiate the theory.* The satellites are treated as attractive particles which 
have the power of attracting and being attracted by the planet, but have no influence 
upon one another. A consequence of this is that each satellite only raises a single 
tide in the planet, and that it is not necessary to take into consideration the actual 
distribution of the satellites at any instant of time. We are thus only concerned 
in determining the changes in the distances of the satellites and in the rotation of 
the planet. 
If the mutual perturbation of the satellites were taken into account the problem 
would become one of the extremest complication. We should have all the difficulties 
of the planetary theory in determining the various inequalities, and, besides this, it 
would be then necessary to investigate an indefinitely long series of tidal disturbances 
induced by these inequalities of motion, and afterwards to find the secular disturbances 
due to the friction of these tides. 
It is however tolerably certain that in general these inequality-tides will exercise a 
very small influence compared with that of the primary tide. Supposing a relationship 
between the mean motions of two, three, or more satellites, like that which holds good 
in the Jovian system, to exist at any epoch, then it is not credible but that such 
relationship should be broken down in time by tidal friction. General considerations 
* “ Problems Connected with the Tides of a Yiscons Spheroid—No. III.,” Phil. Trans., Part. II., 1879. 
