AND ON THE REMOTE HISTORY OF THE EARTH 
501 
Then our equations are 
cl 
1 + 
log, tan 2 - = 
> 
dN 
i-l) 
( 86 ) 
The experience of the preceding integration shows that i varies very slowly com¬ 
pared with the other variables N and hence in integrating these equations an 
average value will be attributed to i, as it occurs in small terms on the right-hand 
sides of these equations. 
The second equation will be considered first. 
T ( - • /*j* t 
We have r=—, so that if we put /3=yA — I , y=-T 4 -— sin i tan i, and omit the last 
S W T o 
term, we get by integrating from 1 to N and from 1 to £ 
l+/*{ 1 -£+/3(l -P)+y(l - f)} .(87) 
as a first approximation. This is the form which was used in the previous solution, 
for, by classifying the tides in three groups as regards retardation of phase, we virtually 
neglected fl compared with n. 
This equation will be sufficiently accurate so long as — is a moderately small frac- 
71 / 
tion; but we may obtain a second approximation by taking account of the last term. 
Now 
— (sec i — 1 )=\ sin 3 i — • rr~ very nearly 
— 1 
— 2 
S1U“ l 
*^0 
1 
1 + /i ^ 
. d 
by substituting an approximate value for N. 
A more correct form for the equation of conservation of moment of momentum will 
be given by adding to the right-hand side of equation (87) the integral of this last 
expression from 1 to ^ and multiplying it by /x. And in effecting this integration i 
may be regarded as constant. 
Let &=—— M ’. Then since 
