THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
853 
If the viscosity be very small, the equations (289-90) admit of reduction to very 
simple forms. 
In this case the sines of twice the angles of lagging are proportional to the speeds 
of the several tides, and we have (as in previous cases)— 
sin4f x , sin2g x sin2xh x . 
——77=1— ixA, ———=+ —Ax A, ———=ixA, 
sm4f ^ sm4f 4 2 sm4f 2 
sin 2g x 
sin 4f v 
— I—|_Lv-\ 
— 2 r 
sin 4f x 
sin 4f 
= l+£x\. 
Therefore 
<£(x)=i sin 4f [P 8 +2P 6 ^ 2 -2P 2 g 6 -Q 8 -lxX(P 8 +4P 6 g 3 +4P 2 Q° + Q 6 + 6P 4 Q 4 )] 
sin 4f (cos i —IrxX) 
V»(x)=i sin 4f [—• 2P 4 $ 4 xX— 2P 2 Q 2 (P 2 — $ 3 ) 2 xX—+rX(P 4 —4P 2 $ 2 +Q 4 ) 2 ] 
= ~i sin 4f (i xX )(I) 
And 
</>(i) + 4<£(ii)— 49<^>(iii) — 9 i//(i)= — sin 4f(1 1 cosi—-18X) 
— 2 0(f) (ii) +301 4>(ni) — 5 7 8 <f>( iv) - (i) — 4+//(ii) = — | sin 4f(297 cos i - 75 6 X) 
Whence from (289) 
— u f ^ l°g v ) = “"§■ 84n 4f{ 11 cosz‘(1+- 2 - l > 7) — 18X(1 + 21>7)} 
or 
| \ t lo g Y y^ T -( i: +or^ 7 )V sin 4f { cos ^~xff (1 +¥^?)} • • • (291) 
From this we see that, in the case of small viscosity, tidal reaction is in general 
competent to cause the eccentricity of the orbit of a satellite to increase. But if 18 
sidereal days of the planet be greater than 11 sidereal months of the satellite the 
eccentricity will decrease. Wherefore a circular orbit for the satellite is only 
dynamically stable provided 18 such days is greater than 11 such months. 
• 
Now if we treat the equation (290) for ~ in the same way, we find—- 
The first line sin 4f( cos i —X). 
The second = -^77 sin 4f(27 cos i— 46X). 
The third —brf sin 4f (273 cos i—6 97X) 
MDCCCLXXX. 5 n 
