74 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
have arisen from the product of e' 2 ^ ^2 n't +2s'—2and ^sin 
its coefficient therefore can only be e' 2 e 3 . And conversely, from seeing this 
coefficient, we should be certain that the argument would be 2 ( n't + s' — nr') 
-f 3 (ra t -f- s — ts). Instead therefore of writing 
ra' 2 e 3 . cos (2 nt + 3 ra t + 2 s' + 3 £ — 2 s/— 3 w) 
we might simply write 
e 2 ra 3 . cos 
omitting the argument entirely. But it will be found more convenient to re¬ 
tain the figures in the argument, writing it thus, 
ra' 2 ra 3 . cos (2 + 3) 
the first figure being always appropriated to the accented argument. And 
when this term is multiplied by cos (lira'/ — llra/-J-lls' — 11s) or cos (11 — 11), 
we may write down the result 
ra' 2 ra 3 . cos (13 — 8) 
without any fear of mistake. For we know that the argument must have been 
produced by adding 2 (nt + s' — w'), 3 (ra / -f- e — nr), and 11 (ra'/ — ra/ -f- 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 77 as of the same 
order with e' and e) enters as multiplier into two terms: of which the first, or 
sin 2 77 . cos (ra' — ra), 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 
7th order at lowest; the first term therefore is useless. But the second, or 
— sin 2 . cos (ra' + ra — 2 &), is exactly analogous to e 2 cos (ra' + ra — 2 w), which 
A 
would arise from the product of e 2 cos (2 ra — 2 ■&) and cos (ra' — ra), 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 
