76 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
-\-(—e 2 — — + sin (2 nt + 2 s — 2 sr) 
+ ^jf^+Scc.) sin (3rc£ + 3s — 3w) 
+ (-^ e 4 — &c.^ sin (4 m t + 4 s — 4 sr) 
+ c 5 -&c.) sin (5 n f + 5 s — 5 sr) 
+ &c. 
but for our purposes it will be sufficient to take v = (0 + 1) + 2 e . sin (0 + 1) 
+ ^ e 2 . sin (0 + 2)+ ^c 3 . sin (0 + 3)+ — e * . sin(0+4) + .sin(0+5). 
For none of the terms can be of any use to us till they are multiplied, so that 
the largest term of the coefficient is of the 5th order ; and then all the other 
parts will be of a higher order. 
14. Putting/ for sin it will be seen that (in conformity with the remarks 
in this section), the terms of R to be developed are 
m 
\/ {V s — 2 r' r . cos (f — v) + r 2 } 
j. 0 mr ] r . cos (t/ + v — 2 $) 
{r 2 — 2 / r . cos [d — v) + r 2 f 
„ m r' 2 r 2 . cos (2 v' + 2 u — 4 0) 
_ _ T s - - 
{V 2 — 2r’ r . cos (t/ — d) + r 2 ] T 
Section 3. 
Expansion of cos {kv 1 — kv), to the fifth order. 
15. By (13) the value of kv' — kv is 
{k — k) 
+ 2 Ae' . sin (1 +0) — 2 ke . sin (0+ 1) (A) 
+ ~ k e' 2 . sin (2 + 0) — k e 2 . sin (0 + 2) (B) 
T T? 
