78 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
-f- (77- /&— ^ k 2 ) e' e 2 . cos (k + 1 — k — 2^ 
+ (~¥ A 3 +4^ 2- §I^) e3 * C0S (k- k - 3 ) 
Terms of the fourth order, 
(4 T6 m k *+ T^l k ) ^ • cos (A + 4 - A;) 
+ (- ^A4-^-A; 3 -^A; 2 )e' 3 <?.cos(A; + 3-A;— l) 
+ (-j Ar 4 — ^ A; 2 )e' 2 <? 2 . cos (A; + 2 — A: — 2^ 
-f (-4^ + 4 ^-4> 2 ) ^'^.cos (a + 1 — A:- 3 ) 
+ (5 ■ **-ra ■ H * 2 - w *) «* • cos (* - ^ ) 
Terms of the fifth order, 
(4o ^ 5 + 48 ki+ 4 A2 + ^ cos ( /<; + 5 ~ k ) 
+ (- k' k >~ H V- ^ kt ) ^ * • cos (^R - A~l) 
+(4^+J^-i1^ 3 -t4 A2 ) *' 3<?2 - cos (^+ 3 -^ 2 ) 
+ ( ~ 4 ^ 5 + 4 kiJr Wi k2> ~ ^ 2 ) e3 ' C0S (^' + 2 -A:— 3 ) 
+ (4 4 Hi^ 3- 41 /<;2 ) ^ • cos ( A +! - ^- 4 ) 
+ ( - 4o ^ 5 + 4 /<;4 “ A3 + 4 ^ 2 - 1 ^ 5 A ) e5 • cos ( k - k ~ E ) 
This development includes every argument whose coefficient is of an order 
not exceeding the fifth. The coefficients however here exhibited are only the 
first terms of the series which represent the complete coefficients. 
