106 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
(13) (9) 
+ L . e 5 . sin (5 vs) -f- M . e' 3 f 2 . sin (3 vs' -f- 2 0) 
-j- M ( . e' 2 e f 2 . sin (2 w' -f- vs + 2 6) -f- \ e e 2 f 2 . sin (vs' -f- 2 vs -f- 2 6) 
+ M (12) . e 5 /2 . sin (3 sr + 2 0) + N (10) . e'f . s j n _p 4 6 ) 
+ N ( \ef*. sin (vs -f- 4 6) . sin {13 (n't -f- s') — 8 (n t + g)} 
The elements e', e, &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. 
51. Let P and Q be the values of the coefficients of cos { 13 (n't + s') — 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 + pt and Q -f q t. 
Thus the term of R becomes 
(P+j9 1) cos {13 (n’t- \-4) — 8 (nt-\-s)} -j- (Q + </ 1) sin { 13 (w7 + s') — 8 (nt-\-e )} ; 
and by ( 2 ), omitting the terms depending on andfor the reasons in (31), 
dv! 
d t 
ds' 
d t 
__ _ 39 a /p _j_ p g * n { 13 ( n 't -j- 4) — 8 (n t -f- g)} 
+ (Q + 9 t) cos {13 (n't + 4) — 8 (n t -}- z )} 
r 
— -j- — —(P t + y t 2 ) sin {13 (n! t + 4) — 8 (n t -f- g)} 
V“ 
SQ y 
--— 7 — (Q t -f- q t 2 ) cos {13 (?i't + 4) — 8 (n t + g)} 
r 1 
Integrating these, (considering n', 4, n, and g, on the right-hand side, as 
constants,) and substituting in the expression n't s', it becomes 
N'* + E' 
39 n'- a' f P + p t , 2 q 
+ 
1 n n a! 
(13 n' — (13tt'-8w) 
j> sin { 13 (n't + 4) — 8 (n t -f- g)} 
