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 — 1 2 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 be seen that the terms depending on are 
d R 
still more insignificant than those depending on — - f . 
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 e', e , rs , rz,f, and 6 : but if such terms existed in the approximate 
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- 
(8) (9) 
tion of the terms L , L , &c. 
Section 12. 
(0) (1) (2) (k) (A) (A) 
Numerical calculation of C r , C r , C i , 8§c., C 3 , C 5 , 8$c. to C M . 
7S V TS 
33. If we put t — 2 co for v' — v, we have 
l . JP) ^(0 
w v - # (2) 
, t n,ri i a — ; — = h Ci — C T . cos 2 a + C z . cos 4 a — &c. 
V {«” + 2a'a. cos2ts + a 2 } 2 s i 1 f 
Integrating both sides with respect to a, from a — 0 to w — and putting 
for the symbol of integration with respect to u between these limits, 
S . 
= 4- c! 0) 
" * y' { a' 2 + 2 a' a . cos 2 w + a 2 ] 4 § 
