88 PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
The arguments of the cosines multiplied by these coefficients are not similar ; 
their forms may be found by the reasoning in (10). 
25. The next term of R to be developed, by (14), is 
— m.~ — 7 / 4 . cos (2 v' + 2 v — 4 0). 
{V 3 — 2 r' r . cos (d — v) + r 2 } ' 
(k) . 
We shall put T s for the general term in the expansion 
* 2 " 
( 0 ) (!) . ( 2 ) 
r=$IY+r, . cos (v'- *;) + r s . cos(2v-2v) + &c. 
{pt-Qdr. cos (d-v)+r 2 }* T 
(^) 
and for the general term in the expansion 
{a n — 2 a' a. cos(v'— z^ + a 2 }^ 
= i ^ \ cos (v'— w) + C ( 5 \ cos (2v'— 2i>) + &c. 
Section 9. 
Development of cos (kv' — k v) . f 4 . cos (2 v' + 2 v — 4 0), to the fifth order. 
26. As the multiplier /* 4 is of the fourth order, we need to develope 
cos (2 v' + 2 v — 4 0) only to the first order. Now by (13), 2 v' + 2v — 4 0 — 
(2+2) — 4 0 
+ 4 e 1 . sin (1 +0) + 4 e . sin (0 + 1) 
and consequently cos (2 v' + 2 v — 40) = 
cos (2 + 2 — 4 0) — sin (2 + 2 — 4 0) . {4 e' . sin (1 + 0) + 4 e . sin (0 + 1) } 
= cos (2 + 2 — 4 0) 
+ 2 e' cos (3 + 2 — 4 0) + 2 e . cos (2 + 3 — 4 0) 
Multiplying this by / 4 it will be seen, as in (22), that we may omit 4 0 in the 
argument. Thus we have for the development of / 4 . cos (2 v' + 2 v — 40), 
Term of the fourth order , 
/' 4 . cos (2 +2). 
