82 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
Every argument is included whose coefficient is of an order not superior to 
the fifth: but only the lowest order of each coefficient is taken. 
Section 5. 
Selection of the coefficients of cos (13 — 8) in the development of 
m 
aJ[ r n — ( 2r' r . cos if — v) + r 1 } 
19. For this purpose, as the general term in the expansion of 
m 
is -mfi . cos (kv' — kv ), we ought to mul- 
y' {r' 3 — 2 dr. cos ( v 1 — v) + r 2 } 
tiply together the expressions of (16) and (18), to multiply the product by m, 
and then giving different values to k to select those terms which have for argu¬ 
ment (13 — 8). But without going through this labour we may, when a value 
is assumed for k, select by the eye the terms required. As we have explained 
in (7), the values which it is proper to give to k are 8, 9, 10, 11, 12, 13. 
20 . Thus we obtain the following coefficients of cos (13 —8): 
k — 8. 
m 
x{- 
- w 0* 1 ) - W ( 3 ’°) + 7S5 ( 2 ’ ] ) + We ( 4 >°) - <a ( 3 >‘) 
* By (0,0) C r is meant the same as C, . 
