498 
MR. G. H. DARWIN ON A PLANET 
where x 7 =£; A is a certain time to be computed as above shown in (10) ; k an 
angular velocity to be computed as above shown in (17) and (18); and X the square of 
an angular velocity to be computed as above in (17) and (IS). 
Also 
If v be large compared with unity, £ is very approximately proportional to the seventh 
power of the square root of the satellite’s distance. 
The solution of this system of simultaneous differential equations would give each 
of the £’s in terms of the time ; afterwards we might obtain n and E in terms of the 
time from the last two of (19). 
These differential equations possess a remarkable analogy with those which repre¬ 
sent Hamilton’s principle of varying action (Thomson and Tait’s ‘ Nat. Phil.,’ 1879, 
§ 330 (14)). 
The rate of loss of energy of the system may be put into a very simple form. This 
function has been called by Lord Rayleigh (‘Theory of Sound,’ vol. i., § 81 ) the 
Dissipation Function,* and the name is useful, because this function plays an 
important part in non-conservative systems. 
dE 
Tn the present problem the Dissipation Function or Dissipativity is — C —. 
Now 
dE ^ bE d£ 
dt ' 6% dt 
From (19) the dissipativity is therefore either 
bE\* _ fdf^ 
dt 
CA AA % 
This quantity is of course essentially positive. 
bE 
It is easy to show that ^ = —* - ( n —42) 
Then on substituting for the various symbols in the expression for the dissipativity 
their values in terms of the original notation, we have 
dE t " , . 0 
~-yr=2— tn-nf 
dt gp v ' 
Or if N be the tidal frictional couple corresponding to the satellite m, 
dE 
— C = %N(ii — fl) 
This last result would be equally true whatever were the viscosity of the planetary 
spheroid. 
* Sii’ W. Thomson prefers to modify tlie name by calling it Dissipativity. 
