AND ON THE REMOTE HISTORY OF THE EARTH. 
479 
Therefore our equation may be written 
/ gl+lA* 
\9 cr ~ 
Now let 
n, 
dt 
2 o/t , \ j • , • di! 
= fcr(l -f-v) i — cos i-f-n sin i 
dt 
_2 
^ 2 (l+c) 
f 1 72/ 
, and let sw 0 /2 0 *=-, and let N- — 
[a no 
And we have 
B' -, AT • M f 
Urr=~ COS id-A Sill 1'- 
r <* C?? 0 dt 
(51) 
(52) 
It is not hard to show that the moment of momentum of the.orbital motion of the 
two bodies is C-Fs/2% and that of the earth’s rotation is obviously C n. Hence 
snffi is the ratio of the two momenta, and /x is the ratio of the two momenta at 
the fixed moment of time, which is the epoch. 
In the similar equation expressive of the rate of change in the earth’s orbital motion 
round the sun, it is obvious that the orbital moment of momentum is so very large 
compared with the earth’s moment of momentum of rotation, that /x is very large and 
the earth’s mean distance from the sun remains sensibly constant (see Section 19). 
Then by (16) and (29), remembering that 
we have 
i . i di )u i 
P= cos-, q= sin-, — = 
dx /u a i j n 
— . and JS — , 
n n<; 
Nsiiii~~=^-2pq[E 1 p 7 q sin 2e 1 — E2 p 3 q 3 (p 3 — q~) sin 2e — E 2 p<p sin 2e 3 
- \-E\p 5 q(p~-\-3q 3 ) sin e l — E'pq(p 2 —q 2 ) 3 sin e—E'. 2 pq 5 (3p 2 -\-q 2 ) sin e'. 2 
— E"2>p^(f sin 2e"J .(53) 
And by (21) 
cos i = Jp(y> 3 —q 3 )[Ai p s sin 2e Y -\- E ip A '(f sin 2 e-\-E. z q s sin 2e 3 
-\-E\2p 6 q 2 sin ^ l -\-E'2p 2 q 2 {p 2 —(fY sin e -f E\2p 2 cf sin e' 3 ] . (54) 
By (33) and (34), and remembering to take the halves of and and that 
sin i=Q, cos i—P 
N sin — T1lq^lePQ z sin 2e-^-^E'P s Q sin e'] .... (55) 
\ (It J flllg 
cos sin 2 e+±E'P 2 Q 2 sin e'].(56) 
(j7l 0 gft 0 
3 Q 
MDCCCLXXIX. 
