70 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
2. When (as in the present instance) the inequality is so small that we may 
be satisfied with the principal part of it, we may in the expressions omit the 
powers of e' . Thus we have 
da! 2 n' a" 2 d R 
dt ju/ d s' 
dv! 3 n n a! d R 
~dt = "1 ~d7 
de n' a' d R 
dt ' fjJ e' dvd 
d ■ut' n ' a 1 d R 
dt fjJ d de' 
de' 3 n'° a! d R 2 n' a' 5 <iR 1 n' a! d d R 
dt fd d s' ‘ t "r ^ da ' 2 ’ pf ' dd 
3. Hitherto this method has been actually used (I believe) only for the cal- 
culation of secular variations. But it can be applied with great advantage in 
almost every case : and in the instance before us it is particularly convenient, 
as it requires only the development of a single term. For if in the development 
of R we take the terms depending on cos {13 n't — 8 n t + A}, whose coefficient 
is of the 5th order, it will be found that 
da' dn' 
dt 5 dt ’ 
and jj, are of the 5th order. 
jj of the 4th order, and -jj of the 3rd order. Integrating these expressions, 
and substituting them in the formula for v\ there will be produced terms of the 
forms ( j 3 ? y/,_ 8 „y 2 sin {13 rlt— 8w/+B} and 13w , _ 8w shi { 12 n't — 8/U + C}, 
where p is of the 5th, and q of the 4th order. And a little examination will 
show that no other argument will produce terms of the same or of a lower 
order, which are divided by the small quantity 13 ri — 8n : inasmuch as this 
do! 
divisor is introduced only by integration of the expressions for &c. Our 
object then at present is to select in the development of R all the terms of the 
form A cos {13 ?/ 1 — 8 nt And as the inequality which we are seeking 
will probably be small, we may confine ourselves to those terms in which the 
order of the coefficient is the lowest possible : that is, to terms of the 5th order. 
