72 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
— TT Sill 
•Jr cos (v' — v) — cos (v' + V — 20) j> • 
m r r 
cos ( 1 / — v) + 
+ &C. 
5. The first line of this may be expanded in the form 
r i (o) . (i) , ( 2 ) 
— 7w <1 — Ti + Fj cos (v 1 — v) +Ti cos (2 v' — 2 v) + &c. > 
( 0 ) ( 1 ) 
where r x , r x , &c 
are functions of r' 
and 
We must then express r' and 
, ( 0 ) ( 1 ) ( 2 ) 
r in terms of nt and nt, and must substitute these values in T t , T t , , &c. 
"E- 72 ~<2 
and must express v' and v in terms of n't and n t ; and on multiplying the re¬ 
spective expressions we shall have the development necessary for our method. 
6. Now upon expressing r in terms of n't, the following remarkable law 
always holds : The index of the term of lowest order in the coefficient of such 
an argument as cos ( pn't -f- A), is p. The same is true with regard to the de¬ 
velopment of r, v', and v. 
7. Now such a term as A cos [13 n't — 8 wf + B} can be produced only by 
the multiplication of ^ (hrit — k nt -f- k — k (from the first term in the 
development of cos ( k v' — kv)^, with (13 cv> k) ( n't + e' — ■&') and 
sin ^ k) (n t + s — zj) ^occurring in the development of kv 1 — kv, or of T| ^ . 
The largest term in the coefficient, according to the rule just explained, will 
be of the order whose index is the sum of 13 cv> k and 8 cv> k. Now if k be < 8, 
as for instance if k be 7, the index of the order is 6 + 1 = 7? or the term is of 
the 7th order, and therefore is to be rejected. And if k be >13, as for instance 
if k — 14, the index of the order is 1 + 6 = 7, and the term is to be rejected. 
But if A'be 8, or 13, or any number between them, as for instance 10, then 
the order of the term is 3 + 2 = 5, and the term is to be kept. It appears 
therefore that the only terms which we shall have occasion to develope, are 
( 8 ) ( 9 ) ( 13 ) 
Tx . cos (8 v’ — 8 v), Ti . cos (9 v’ — 9 v), &c. as far as T, . cos (13 v 1 — 13 v) 
inclusively. 
