488 MR, G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
is to determine the increase in the average value of W per unit increase of the average 
value of £ The proposed new independent variable is therefore not hut it is the 
average value of £; but as no occasion will arise for the use of £ as involving periodic 
terms, I shall retain the same symbol. 
In order to justify the procedure to be adopted, it is necessary to show that, if f(t) 
be a function of t, then the rate of increase of its average value estimated over a 
period T, of which the beginning is variable, is equal to the average rate of its increase 
estimated over the same period. Now the average value of f(t) estimated over the 
period T, beginning at the time t is 
1 G+ T 
-j f\f)dt, and therefore the 
rate of the increase 
d 1 [ t+T . . 1P +T 
of the average value is — — J f(t)dt, which is equal to -J f(t)dt; and this last 
expression is the average rate of increase of f(t) estimated over the same period. This 
therefore proves the proposition in question. 
Now suppose we have -y-=— M+ periodic terms, where M varies very slowly; 
ctz 
then — M is the average value of the rate of increase of N estimated over a period 
which is the least common multiple of the periods of the several periodic terms. Hence 
by the above proposition —M is also the rate of increase of the average value of N 
estimated over the like period. 
( lp 
Similarly if -j = X + periodic terms, X is the rate of increase of the average value 
of t estimated over a period, which will be the same as in the former case. 
But the average value of N is the proposed new dependent variable, and the average 
value of £ the new independent variable. TIence, from the present point of view, 
= ~ w- This argument is, however, only strictly applicable, supposing there are not 
dg N 
periodic terms in ~ or ^ of incommensurable periods, and supposing the periodic terms 
dt 
are rigorously circular functions, so that their amplitudes and frequencies are not func¬ 
tions of the time. 
It is obvious, however, that if the incommensurable terms do not represent long 
inequalities, and if M and X vary slowly, then the theorem remains very nearly true. 
With respect to the variability of amplitude and frequency, it is only necessary to pos¬ 
tulate that the so-called periodic terms are so nearly true circular functions that the 
integrals of them over any moderate multiple of their period is sensibly zero, to apply 
the argument. 
Suppose, for example, i fj(t) cos (v£+x(0) were one °f the periodic terms, then we have 
only to suppose that i p(t) and y(7) vary so slowly that they remain sensibly constant 
2jt 
during a period — or any moderately small multiple of it, in order to be safe in 
2tt 
assuming cos as sensibly zero. 
Jo 
Now in all the inequalities in Wand £ 
