AND ON THE REMOTE HISTORY OF THE EARTH. 
491 
Combine these seven values by Weddle’s rule, viz. : 
r6& 
u 3 dx=j^h[u Q +u i + A+ 5 K + M .3 + %)] 
J o 
and so find the time corresponding to £=’ 88 . It must be noted that the time is 
negative because d£ is negative. 
In the course of the work the values of — corresponding to £=1, ‘96, '92, '88 
have been obtained. Multiply them by 2W; these values, together with the four 
values of sin 4e, sin 2e', sin 4e" and the four of W, enable us to compute four of 
—l°g 10 tan {(!—-§- sin 3 i), as given in (75). 
Combine these four values by the rule 
rz/t 3 /, 
| ujx=— O 0 + R 3 + 3K + ^)] 
- o 
37/ 
and we get 
, tan 7(1 — §• sin 3 i) 
° 10 tan i 0 ( 1 — §• sin 2 i 0 ) 
from which the value of i corresponding to £='88 may easily be found. It is here 
useless to calculate more than four values, because the function to be integrated does 
not vary rapidly. 
We have now got final values of i, N, t corresponding to £=' 88 . 
Since the earth is supposed to be viscous throughout the changes, therefore its 
figure must always be one of equilibrium, and its ellipticity of figure c=W 3 e 0 . 
Also since £=( ^'j = /y/~, where c is the moon’s distance from the earth, therefore 
^=£-^ "j, which gives the moon’s distance in earth’s mean radii. 
The fifth and sixth column of Table IV. were calculated from these formulas. 
The seventh column of Table IV. shows the distribution of moment of momentum in 
the system; it gives /r the ratio of the moment of momentum of the moon’s and earth’s 
motion round their common centre of inertia to that of the earth’s rotation round its 
axis, at the beginning of each period of integration. 
Table I. shows the values of e, e, e" the angles of lagging of the semi-diurnal, 
diurnal, and fortnightly tides at the beginning of each period. 
Tables II. and III. show the relative importance of the contributions of each term 
to the values of — and — log 10 tan 7(1 — sin 3 1 ) at the beginning of each period. 
The several lines of the Tables II. and III. are not comparable with one another, 
because they are referred to different initial values of fl and n in each line. 
I will now give some details of the numerical results of each integration. The 
