76 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
e 1 — ~ i^ + Scc.^ sin (2 nt + 2 s — 2 sr) 
V 4 24 J 
+ ^ ^ ^+&c.) sin (3 n t + 3 g — 3 ■or) 
+ (^ e 4 —&c.^ sin (4ra£ + 4g — 4x?) 
+ (12?Z e 5 —&c.^ sin (5 rc t + 5 g — 5 sr) 
+ &c. 
but for our purposes it will be sufficient to take v = (0 +1) + 2 e . sin (0 + 1) 
+ |-c 2 .sin(0 + 2) + ^.sin (0 + 3)+ .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 w r ill be seen that (in conformity with the remarks 
in this section), the terms of R to be developed are 
m 
-P 
y' { r 12 — 2i J r . cos (xf — v ) + r 2 } 
r 
mr 1 r . cos (d + v — 26) 
-I/ 4 - 
{r' 2 — 2 dr. cos [d — v) + r 9 } T 
m r n r 9 . cos (Q d + 2 v — 4 0) 
{ r’~ — 2 dr. cos [d — d) + r 9 ] T 
Section 3. 
Expansion of cos {Jc v' — k v), to the fifth order. 
15. By (13) the value of k v' — k v is 
(k — k) 
+ 2 ke '. sin (1 +0) — 2 ke . sin (0+ 1) . . . . 
+ ~ k e ' 2 . sin (2 + 0) — f k e 0 -. sin (0 + 2) 
! rr t? 
(A) 
(B) 
