ATTENDED BY SEVERAL SATELLITES. 
499 
The dissipativity, converted into heat by Joule’s equivalent, expresses the amount 
of heat generated per unit time within the planetary spheroid. This result has been 
already obtained in a different manner for the case of a single satellite in a previous 
paper (“ Problems, &c.,” Phil. Trans., Part II., 1879, p. 557). 
§ 3. Sketch of method for solution of the equations by series. 
It does not seem easy to obtain a rigorous analytical solution of the system (19) of 
differential equations. I have however solved the equations by series, so as to obtain 
analytical expressions for the As, as far as the fourth power of the time. This solu¬ 
tion is not well adapted for the purposes of the present paper, because the series are 
not rapidly convergent, and therefore cannot express those large changes in the con¬ 
figuration of the system which it is the object of the present paper to trace. 
As no subsequent use is made of this solution, and as the analysis is rather long, I 
will only sketch the method pursued. 
(IE 
If fgd be taken as the unit of time — = — 
Cll 
Differentiating again and again with regard to the time, and making continued use 
of this equation, we find d-EJdt 3 , d^E/dE, See., in terms of bE/b£. 
It is then necessary to develop these expressions by performing the differentiations 
with regard to £ 
An abridged notation was used in which 
k 
. With this notation the whole operation may be shown to depend on 
the performance of b/b£ on expressions of the form 
a,., bfi j k > 
. P r _ 
a l> 
ki 
a 2 , b 0 
n 
A 
_ i 
i 
i_ 
«;(a/2— lori) 
cc? 
a, b 
. P . 
k , ( aXar 8 — bn/c \ k 
represented (-—-) or 
bEf 
where y is independent of but may be a function of the mass of each satellite. 
Having evaluated the successive differentials of E we have 
h — E {l +1 
fdE\ f~ lcPE\ $ lEE 
UWo 1 - 2 \ clE ) 0 + 1.2.3 \ dt 3 
) + &c. 
/ 0 
Where the suffix 0 indicates that the value, corresponding to t— 0, is to be taken. 
It is also necessary to evaluate the successive differentials of bE/b£ with regard to 
the time, and then we have 
* U/o 1.2 dt Jo 1.2.3 dt* ) Q 
The coefficient of t 4 was found to be very long even with the abridged notation, and 
involved squares and products of 2fs. 
• MDCCCLXXXI. 3 T 
