106 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
+ L ( \ c 5 . sin (5 zff) -f- M ( \ . sin (3 sr ? + 2 6) 
( 10 ) . , (ii) 
+ M . e 2 ef 2 . sin (2 m + nr -f- 2 &) + M . e e 2 f 2 . sin (&' + 2 zs + 2 6) 
-f- M ( \ c 3 / 2 . sin (3 vs + 2 6) + N ( \ e'f 4 . sin (V -f- 4 &) 
+ \ ef* . sin (nr + 4 $) j- . sin { 13 (w'£ -j- s') — 8 (n t + e) } 
The elements e', c, &c. are all subject to small permanent variation ; and 
(considering the great length of period of the inequality which we are calcu- 
lating,) those variations may have a sensible influence upon it. It is prudent 
therefore, as well as interesting, to take into account these variations. 
5 1 . Let P and Q be the values of the coefficients of cos {13 (n't + z’) — 8 (nt + s ) } 
and sin {13 (n't + s') — 8 (n t + s)} in the expression above, giving to the ele- 
ments the values which they had in 1750. Then, as all the permanent varia- 
tions are small, the powers of t above the first may be rejected, and the coeffi- 
cients at the time t after 1750 may be represented by P -f pt and Q + q t. 
Thus the term of R becomes 
(P ~\~ p t') cos {13 (n't-\- s') —8 (nt-\- g)} + (Q + <jrf) sin {13 (w7 + e f ) — 8 (nt- j- g) } ; 
and by (2), omitting the terms depending on and for the reasons in (31), 
d J} = — 39 y a ( p + P 0 sin { 13 (n’t + s') — 8 (n t + s) } 
QQ 
+ — — } — (Q + q t) cos { 13 {rl t + s') — 8 (n t + s) } 
Tt~ + (P t + pt 2 ) sin { 13 (rl t + 1) — 8 (nt + s)} 
: — 7 — (Q t -J- q t 2 ) cos { 13 (n r t -f- s') — 8 (n t + s) } 
Integrating these, (considering n 1 , s', n, and s, on the right-hand side, as 
constants,) and substituting in the expression n 1 1 + s', it becomes 
N'< + E' 
+ ~T^ { (13n'-‘«ly + ( - 13 8 nf } sin { 13 (rl t + l) - 8 (« t + s ) } 
