AND ON THE REMOTE HISTORY OF THE EARTH. 
499 
Dividing the first and second equations by the third, and observing that 
2di 
we 
have 
sm l 
=■ cl log tan 2 
• 1 + (“T 1 see i 
d , „ i V r / \ t / 11 
—r log tan~ -=- 
dN 
N[ 1 — — sec i 
n 
sin 2 i 
cos % 
+v- 
/2 
• • 
sm ^ tan i - 
t/ n 
, 42 
1 -sec ^ 
n 
(84) 
If there be only one disturbing body, which is an interesting case from a theoretical 
point of view, the equations may be found by putting r=0, and may then be written 
. 2/2 n 
7 . mos«— — 
d , , 2 i 1 ft 
10 P' tan- — , T 
adP 6 2 N . fl 
cos l - 
n 
dN 
j • o • 42 
1 — 7 sirr i — — cos i 
* n 
~ /2 
COS l — - 
n 
d% i • A T [ • 42\ 
n—= A sin 4e.— cos i — - 
r dt 4 c*ft 0 \ «/ 
(85) 
From these equations we see that so long as fl is less than n cos i, the satellite 
recedes from the planet as the time increases, and the planet’s rotation diminishes, 
42 
because the numerator of the second equation may be written cos iy cos \ sin 3 i, 
which is essentially positive so long as fl is less than n cos i. But the tidal friction 
1 + cos 3 i 
vanishes whenever fi¬ 
ll- 
_!_ (>Qg2 
The fraction —-— is however necessarily greater 
2 cos i ’ 2 cos i 
than unity, and therefore the tidal friction cannot vanish, unless the month be as 
short or shorter than the day. The obliquity increases if fl be less than \n cos i, 
but diminishes if it be greater than \n cos i. Hence the equation fl — \n cos i gives 
the relationship which determines the position and configuration of the system for 
instantaneous dynamical stability with regard to the obliquity (compare the figures 
2 , 3, 4, Plate 36). From this it follows that the position of zero obliquity is one of 
