AND ON THE REMOTE HISTORY OE THE E 4RTH. 
487 
This is the third of the simultaneous differential equations which have to be treated. 
The four variables involved are i, N, t, which give the obliquity, the earth’s rotation, 
the square root of the moon’s distance and the time. Besides where they are involved 
explicitly, they enter implicitly in Q, II, U, Y, W, X, Z, sin 4e, sin 2 d, sin 4e". 
Q, B, &c., are functions of the obliquity i only, but P is a constant. Also 
sin 4e=-r4-^w;= 7 —wy—), sin 2 e'= sin 4:e"= . I made several 
4n a + p- 4 n a N* + p v n a N a +p v 4fi 0 H 
attempts to solve these equations by retaining the time as independent variable, and 
substituting for ^ and N approximate values, but they were all unsatisfactory, because 
of the high powers of d which occur, and no security could be felt that after a con¬ 
siderable time the solutions obtained did not differ a good deal from the true one. 
The results, however, were confirmatory of those given hereafter. 
The method finally adopted was to change the independent variable from t to 
A new equation was thus formed between -ZV and £ which involved the obliquity i 
only in a subordinate degree, and which admitted of approximate integration. This 
equation is in fact that of conservation of moment of momentum, modified by the 
effects of the solar tidal friction. Afterwards the time and the obliquity were found 
by the method of quadratures. As, however, it was not safe to push this solution 
beyond a certain point, it was carried as far as seemed safe, and then a new set of 
equations were formed, in which the final values of the variables, as found from the 
previous integration, were used as the initial values. A similar operation was carried 
out a third and fourth time. The operations were thus divided into a series of periods, 
which will be referred to as periods of integration. As the error in the final values in 
any one period is carried on to the next period, the error tends to accumulate ; on this 
account the integration in the first and second periods was carried out with greater 
accuracy than would in general be necessary for a speculative inquiry like the present 
one. The first step is to form the approximate equation of conservation of moment of 
momentum above referred to. 
Let A=W sin 4e+X sin 2e', B = Z sin 2e'. 
Then the equations of friction (67) and reaction ( 68 ) may be written, 
■' ! ° 5 T-(p+v) A +T' B 
(69) 
d £ r 0 2 . 
(70) 
We now have to consider the proposed change of variable from t to £ 
The full expression for ~ contains a number of periodic terms; also contains 
dt 
dN 
terms which are co-periodic with those in —. Now the object which is here in view 
dt' 
3 R 
MDCCCLXXIX. 
