84 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
21. The next term of R to be developed, by (14), is 
. f 2 . cos (v* + v — 2 6) 
— m 
{r 12 — Qr 1 r . cos ( v 1 — v) + 
(Jc) 
We shall put T 3 for the general term in the expansion 
r r 
— 2 r 3 r . cos (t/ — v) + r 2 } 
.(*) 
( 0 ) ( 1 ) ( 2 ) 
=^T 3 + F 3 . cos(v'— «;) + T 3 ,cos(2«/— 2 v) + &c. 
And C 3 for the general term in the expansion 
(°) . JXi 
a ’ a 
S?) 
— = £C, + C 3 cos (t/ — v) + C 3 cos (2t/— 2«>) + &c. 
{a! 2 -9.a! a.cosid -v) + T 
Section 6. 
Development of f 2 . cos (v + v — 2 &), to the fifth order. 
22. As the multiplier f 2 is of the second order, we want cos (v 1 ' + v — 2 0) 
only to the third order. Now, by (13), v' + v — 2 6 = 
(1 + 1 ) — 2 0 
+ 2 e' sin (1 + 0) + 2 e . sin (0 + 1) (A) 
+ ^ e' 2 . sin (2 -f 0) + ~ e 2 . sin (0 + 2) (B) 
is is 
+ -e ,3 sin (3 + °) H-^^.sin (° + 3) (C) 
Its cosine, as in (15), is 
cos(l + l — 2 0).{l - }_ sin ( 1 + i_20).{ A + B + C- A 8 J 
Following the rule of (8) in the expansion, we find for the value of cos 
(v + v — 2 6). 
cos ( 1 + 1 — 2 6) 
Principal Term, 
