854 
MR, G. H. DARWIN ON THE SECULAR CHANGES IN 
Therefore 
- - -f=£ sin 4f[(l + 2777 + 273^) cos i-X(l +46^ + 697^)] ~1 
or 
r 2 k dt 
i§=F(l+27, + 273,>in4f 
n 
cos i — —(1 —(- 19-17 — 8 9rj~) 
. (292) 
From this it follows that the rate of tidal reaction is greater if the orbit be eccentric 
than if it be circular. Also for zero obliquity the tidal reaction vanishes when 
— = 1 — 1977-|-45 Gy 
Hence if a satellite were to separate from a planet 111 such a way that, at the 
moment after separation, its mean motion were equal to the angular velocity of the 
planet, then if its orbit were eccentric it must fall back into the planet; but if its 
orbit were circular an infinitesimal disturbance would decide whether it should 
approach or recede from the planet.* 
Now suppose that the viscosity is very large, and that the obliquity is zero. 
Then 
— ~ ~ y log 7 ?=-g-(sin 4f 1 +4 sin 4f"— 49 sin 4f lu +6 sin 2lT) 
T" - ' ro CtZ 
and the sines are reciprocally proportional to the speeds of the tides, from which they 
take their origin. As to the term in sin 2 h l , which takes its origin from the elliptic 
monthly tide, the viscosity must make a close approach to absolute rigidity for this 
term to be reciprocally proportional to the speed of that tide ; for the present, there¬ 
fore, sin 2 lT will be left as it is. 
Then the equation becomes, on this hypothesis, 
9 f d 
t 2 k dt 
log 77 = -g- sin 4f 11 
!-A, A 49(1—N)‘ 
l-X ' 
1 — 
l-%\ 
+f sin 2 If 
gf d 
t 3 k dt 
log y) 
1 sin 4f u 
44-63A + 20X 2 fi 
sin 2 h‘. 
(293) 
The numerator of the first term on the right is always positive for values of X less 
than unity, and the denominator is always positive if X be less than §-. Hence if the 
viscosity be not so great but that the last term is small, the eccentricity always 
increases if X lies between zero and 
* See Appendix (p. 886) for further considerations on this subject. 
