92 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
It will be seen hereafter, that for any one of the terms whose union com¬ 
poses L*' \ &c., a is greater than — A, and that it may, on the mean of 
values, be said to differ little from — 12 A. This reduces the ratio of the terms 
to 400 : 1. Now though we cannot assert that the sum of one set of terms 
will have to the sum of the other set of terms a ratio at all similar to this, yet 
the great disproportion of the terms related to each other seems sufficiently to 
d R 
justify us in the a priori assertion that the terms depending on are not 
d. R 
worth calculating. It will readily he seen that the terms depending on are 
still more insignificant than those depending on 
32. We stated in (1) that the variations of the elements would be sufficiently 
taken into account in the expression for R if we put E + F t for e, &c.; which 
amounts to taking only the secular variations. There will be no difficulty in 
doing this for ee, , ■&,/, and 0 : but if such terms existed in the approximate 
^R ^ 
expressions for a' and a, they would require the use of the differentials 
But a' and a have no secular variations : and therefore these differentials are 
not wanted. We may therefore proceed at once with the numerical Calcula¬ 
te) (9) 
tion of the terms L , L , &c. 
Section 12. 
(O) (1) (2) (A) (A) (A-) 
Numerical calculation of C r , C i , C , 8$c., C, , C 5 , fyc. to C,,. 
■Z Z Z 
33. If we put k — '2 co for v' — v , we have 
l 
( 0 ) ( 1 ) ( 2 ) 
= iC: — C x . COS 2 co + C z . COS 4 co — &c. 
^ { a n + 2 a! a. cos2 a> + cr} 2 s s 
Integrating both sides with respect to a, from u = 0 to u = f-, and putting 
for the symbol of integration with respect to co between these limits. 
S • 
= 4 - c! 0) 
a * /{a' s + 2a'a. cos 2 w -f a~ } 4 § 
