ATTENDED BY SEVERAL SATELLITES. 
493 
would lead one to believe that the first effect of tidal friction would be to set up 
amongst the satellites in question an oscillation of mean motions about the average 
values which satisfy the supposed definite relationship; afterwards this oscillation 
would go on increasing indefinitely until a critical state was reached in which the 
average mean motions would break loose from the relationship, and the oscillation 
would subsequently die away. It seems probable therefore that in the history of such 
a system there would be a series of periods during which the mutual perturbations of 
the satellites would exercise a considerable but temporary effect, but that on the whole 
the system would change nearly as though the satellites exercised no mutually per¬ 
turbing power. 
There is however one case in which mutual perturbation would probably exercise a 
lasting effect on the system. Suppose that in the course of the changes two satellites 
came to have nearly the same mean distance, then these two bodies might either come 
ultimately into collision or might coalesce so as to form a double system like that of 
the earth and moon, which revolve round the sun in the same period. In this paper I 
do not make any attempt to trace such a case, and it is supposed that any satellite may 
pass freely through a configuration in which its distance is equal to that of any other 
satellite. 
§ 2. Formation and transformation of the differential equations. 
In this paper I shall have occasion to make frequent use of the idea of moment of 
momentum. This phrase is so cumbrous that I shall abridge it and speak generally of 
angular momentum, and in particular of rotational momentum and orbital momentum 
when meaning moment of momentum of a planet’s rotation and moment of momentum 
of the orbital motion of a satellite. I shall also refer to the principle of conservation 
of moment of momentum as that of conservation of momentum. 
The notation here adopted is almost identical with that of previous papers on the 
case of the single satellite and planet; it is as follows :— 
For the planet let : 
M= mass ; a =mean radius; g— mean pure gravity; w=mass per unit volume; 
1 ;=viscosity ; £=:§ g/a\ angular velocity of rotation; C= moment of inertia about 
the axis of rotation, and therefore, neglecting the ellipticity of figure, equal to \Ma % . 
For any particular one of the system of satellites, let : 
m = mass; c=distance from planet’s centre ; fl = orbital angular velocity. 
Also fj. being the attraction between unit masses at unit distance, let r=|lp,m/c 3 ; 
and let v—M/m. 
These same symbols will be used with suffixes 1, 2, 3, &c., when it is desired to 
refer to the 1st, 2nd, 3rd, &c., satellite, but when (as will be usually the case) it is 
desired simply to refer to any satellite, no suffixes will be used. 
Where it is necessary to express a summation of similar terms, each corresponding 
to one satellite, the symbol % will be used ; e.g., Skc i will mean /cyp-b/cyfy-b &c. 
3 s 2 
