84 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
21. The next term of R to be developed, by (14), is 
dr 
— m 
{r 12 _ 2r' r. cos(t/ — v ) + r-}' 
. f 2 . cos (v* + v — 2 6) 
(Jc) 
We shall put T 3 for the general term in the expansion 
dr 
{r'- — 2 dr . cos (d — v) + r 2 } - 
( 0 ) ( 1 ) ( 2 ) 
-py + I\ . cos(v'—v) + T 3 .cos(2 1 >' — 2 v)+&c.; 
.(*) 
And C 3 for the general term in the expansion 
a' a 
AD 
( 2 ) 
{a! 2 — 2 a' a . cos (i/ — v) + a 2 1" 3 " 
= 4C 3 + C 3 cos (v 1 — v) + C, cos(2z/—2v) + &c. 
Section 6. 
Development off 2 . cos {v' + v — 2 6), to the fifth order. 
22. As the multiplier f 2 is of the second order, we want cos (v 1 + v — 2 6) 
only to the third order. Now, by (13), v' -{• v — 2 6 = 
(1 + 1 ) -2 6 
+ 2 e' sin (1 + 0) + 2 e . sin (0 + 1) . (A) 
+ -|- e' 2 . sin (2 + 0) + ~ e 2 . sin (0 + 2) . (B) 
is is 
+ — e 3 sin (3 + 0) + ^ e 3 • sin (0 + 3) .(C) 
Its cosine, as in (15), is 
cos (1 + 1- 2 d). { 1 - Ai + g g A D } -sin (1 + 1- 20 ).(a + B + C-^ S | 
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 , 
