770 
ME. G. H. DARWIN ON THE SECULAR CHANGES IN 
§ 14. On the small terms in the equations of motion clue directly to tided friction. 
The first step is the formation of the disturbing function. 
As we shall want to apply the function both to the case of the earth and to that of 
the moon, it will be necessary to measure longitudes from a fixed point in the ecliptic; 
also we must distinguish between the longitude of the equinox and the angle y, as they 
enter in the two capacities (viz. : in the X'Y' and functions) ; thus the N and A' 
of previous developments must become N — ip, N' — xp' ; e, e must become e — xp, e' — xp' ; 
and 2(y-^y') mus f be introduced in the arguments of the trigonometrical terms in the 
semi-diurnal terms, and y^y' the diurnal ones. 
The disturbing function must be developed so that it may be applicable to the cases 
either where Diana, the tide-raiser, is or is not identical with the moon ; but as we are 
only going to consider secular inequalities, all those terms which depend on the 
longitudes of Diana or the moon may be dropped. 
In the previous development of Part II. we had terms whose arguments involved 
e—e'; in the present case this ought to be written ( nt-\-e — xp) — (fl't- f-e'— xp'), for 
which it is, in fact, only an abbreviation. 
Now a term involving this expression can only give rise to secular inequalities, in the 
case where Diana is identical with the moon ; and as we shall never want to differentiate 
the disturbing function with regard to fl' , we may in the present development drop 
the fit and fl't. 
Having made these preliminary explanations, we shall be able to use previous results 
for the development of the disturbing function. The work will be much abridged by 
the treatment of i,j, i',f as small. 
Unaccented symbols refer to the elements of the orbit of the tide-raiser Diana, or 
(in the case of i, y, xp) to the earth as a tidally distorted body; accented symbols refer 
to the elements of the orbit of the perturbed satellite, or to the earth as a body whose 
rotation is perturbed. 
Then since i, i and j, f are to be treated as small, (22) becomes 
(128) 
The same quantities when accented are equal to the same quantities when i,j, N, xp are 
accented. 
Then referring to the development in § 5 of the disturbing function, we see that, for 
the same reasons as before, we need only consider products of terms of the same kind 
in the sets of products of the type X'Y' X Hence the disturbing function W is 
the sum of the three expressions (37-0) multiplied by rr'/g. Now since we only wish 
5 } =Pp-Qqe i <*-*>= l-tf-if-ive"-*' 
