86 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
Section 7. 
Development of cos {kv' — kv) .f 2 . cos (v 1 + v — 2 &), to the fifth order. 
23. We must multiply the expression in (16), (of which only the terms to 
the third order will be wanted), by the expression just formed, according to 
the rule of (8). Thus we obtain the following expression : 
Term of the second order, 
~ f 2 . cos {k + 1 — k — 1). 
Terms of the third order, 
O' k + 4 ~) e 'f 2 • cos + 2 “ k ~ X ) + (— \ k + o -) e/ 2 cos(I+l — k — 2). 
Terms of the fourth order, 
(t k2 + il k + Tb ) e ' 2 / 2 • cos (^+ 3 - 
+ ( ~ ¥ k2 + 4") e ' e f 2 ' cos ^ + 2 ~ k ~ 2 ) 
+ (t^ 2 ~ 7b A + lb) e2 / 2 ' cos ( A + 1 ~ k — 3) 
Terms of the fifth order, 
( T5 + f~6 kl + 1 * + 1 ) e ' 3 / 2 • cos (A + 4 - ) 
+ (— -J^-TB^ + T^ + Ts) e ' 2 «A cos (* + 3 ~ r~~2) 
+ (-J- A 3 - TB A2 - ^ A + ^) e' e 2 / 2 . cos (A + 2 — A — 3) 
+ (~ i^A> + -|A 2 - 1 lc + |) e>/Kcos (T+ I - F^4) 
