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 
. —-—- 7/ 4 . cos (2 v' + 2 v — 4 6). 
{r 12 — 2 r’ r . cos {d — v) + r 2 } r 
(k) 
We shall put I\ for the general term in the expansion 
T 
r n r 2 
{? J2 —2 r’r. cos (d — f) + r+ 
( o) (1) (2) 
= ^T 5 + r s . cos (v 1 — v) +T, . cos( 2 t/— 2 y) + &c. 
A*) 
and C 5 for the general term in the expansion 
{a!°--Qa'a. cos(d —v) + a 2 } ; 
^ + C ( ; \ cos (v 1 — ij)-f C ( 5 \ cos (2 v'~ 2 y) + &c. 
Section 9. 
Development of cos {k v 1 — k v) .f x . cos (2 v' + 2 v — 46), to the fifth order. 
26. As the multiplier / 4 is of the fourth order, we need to develope 
cos (2 v' + 2 v — 4 6) only to the first order. Now by (13), 2 v' + 2 v — 4 6 — 
(2+2) — 4 6 
+ 4 e 1 . sin (1 + 0) + 4 e . sin (0 + 1) 
and consequently cos (2 v' + 2 v — 4 6) = 
cos (2 + 2 — 4 6) — sin (2 + 2 — 4 6) . {4 e'. sin (1 + 0) + 4 e . sin (0 + 1)} 
= cos (2 + 2 — 4 6) 
+ 2 e' cos (3 + 2 — 4 6) + 2e. cos (2 + 3 — 4 6) 
Multiplying this by / 4 it will be seen, as in ( 22 ), that we may omit 4 6 in the 
argument. Thus we have for the development of / 4 . cos (2 v’ + 2 v — 46), 
Term of the fourth order , 
/ 4 . cos (2 + 2 ). 
