74 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
have arisen from the product of e' 2 ^ ^2w7+2s'— 2 ro-'^and 3^; 
its coefficient therefore can only be e’ 2 eh And conversely, from seeing this 
coefficient, we should be certain that the argument would be 2 ( tit + s' — v;') 
-J- 3 (n t + s — w). Instead therefore of writing 
e' 2 e 3 . cos (2 rit + 3 n t + 2 i -j- 3 s — 2 v?'— 3 w) 
we might simply write 
e' 2 e 3 . cos 
omitting the argument entirely. But it will be found more convenient to re- 
tain the figures in the argument, writing it thus, 
e' 2 e 3 . cos (2 + 3) 
the first figure being always appropriated to the accented argument. And 
when this term is multiplied by cos {\\n!t—\\nt -\-\ \ t' — 11 s) or cos (1 1 — 11 ), 
we may write down the result 
A cos (13 -8) 
without any fear of mistake. For we know that the argument must have been 
produced by adding 2 {n't -(- s' — ■&'), 3 {n t + s — rs), and 1 1 {n't — nt-\-s! — s), 
and thus when a result is obtained the term can be filled up. 
10 . If we examine the second line in the last expression of (4), it is easily seen 
that sin 2 - 77 , a quantity of the second order (considering sin -f- as of the same 
order with e' and e) enters as multiplier into two terms : of which the first, or 
sin 2 77 . cos {v’ — v), when developed will have in every term one part of the 
argument produced by a subtraction ; and therefore, when combined with the 
expansion of the term multiplying it, will produce terms cos (13 — 8 ) of the 
7 th order at lowest ; the first term therefore is useless. But the second, or 
— sin 2 ~ . cos {v' + v — 2 &), is exactly analogous to e 2 cos {v' v — '2 ■&) , which 
would arise from the product of e 2 cos {2v — 2 w) and cos {v' — v), and to which 
all the preceding remarks would apply; and examination would show that in 
the development of this term, in which products of sin 2 with powers of e' and e 
