72 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
o /" "j 2 fyi y* 
— 77 - sin 4 7 T < cos (v 1 — v) — cos (v'-\-v — 20 ) > r 
2 2 1 v ’ V ’) {r'*-<lr'r . cos + r 2 }^ 
+ &C. 
5. The first line of this may be expanded in the form 
r i (o) (i) (2) -) 
— in < — Tr + Fj cos {v 1 — v) +T, cos (2 v' — 2 v) + &c. > 
( 0 ) ( 1 ) 
where T* , T, , &c., are functions of r' and r. We must then express r' and 
r in terms of lit and n t, and must substitute these values in T* \ ri \ T* \ &c. 
7Z cf 
and must express v' and v in terms of rit 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 {frit + 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 rit— 8nt-\-V>} can be produced only by 
the multiplication of (krit — knt + k i — kz j, (from the first term in the 
development of cos {k v' — kv)^j, with (13 cv> k) {n't + e f — w) and 
sin ( 8 0X3 ( n t + s “ ^occurring in the development of k v' — kv, or of T ( 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 -f- 1 = 7 , or the term is of 
the 7 th 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 -f- 6 = 7, and the term is to be rejected. 
But if k 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 ) 
T, . cos (8 v — 8 v), Tj . cos (9 v — 9v), & c. as far as T, . cos (13 v' — 13 v) 
inclusively. 
