THE ELEMENTS OF THE ORBIT OE A SATELLITE. 
851 
Therefore by putting y—~ in equation (284) we have 
— ^ log 7)=^cf>( i) + 2(1 — lo^^ii) — A 2 a (l — Y^)</>(iii) — 5 78i70(iv) 
—1(1 + 3^)#)—^^#(ii) 
and by putting y=0 in (284) 
t 2 ^ — + 2 ( 1 “ 1 0r 7+^V)^(ii) + i f ll 7( 1 — + 1 156^(iv) 
+l^?(l + 4i7)#)-{-^ 3 ^(ii) 
The equations may be also arranged in the following form :— 
lo g’ V ■= WW + #(ii) - 49^>(iii)-9i//(i)] 
+ 17 [-20^(ii) + 301^(iii)-578^(iv)—W(i)“¥V<ii)] • ( 289 ) 
+^[W) — 20<£(ii) + 1 |-^(ii i ) + 
+T[— 2 < M i ) + 73( M ii )—^ J ^^(iii) + 115 6 ^(iv) + 1 8V»(i)+-2 1 V , (ii)] • • ( 29 °) 
The former of these apparently stops with the first power of rj, but it will be 
observed that we have cl log n/dt on the left-hand side so that clrj/dt is developed as far 
as rf. 
These equations give the required solutions of the problem. 
§ 25. Application to the case ivliere the planet is viscous. 
If the planet or earth be viscous, we have, as in § 7, F x =cos2f x , G x =cosg x , 
H x =cos(xh x ), G x =cosg x , F x =cos 2f x . 
When these values are substituted in (289) we have the equation giving the rate ol 
change of ellipticity in the case of viscosity. The equation is however so long and 
complex that it does not present to the mind any physical meaning, and I shall 
therefore illustrate it graphically. 
The case taken is the same as that in § 7, where the planet rotates 15 times as fast 
as the satellite revolves. 
The eccentricity or ellipticity is supposed to be small, so that only the first line of 
(289) is taken. 
I took as five several standards of viscosity of the planet, such viscosities as would 
