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 
a/ [ d 2 — 2 r' r . cos (u' — y) + r 2 } 
19. For this purpose, as the general term in the expansion of 
m 
„(*) 
- _ 2 r . cos - v) + r*} is “ m F i • cos (* v ' “ k v )> we ou £ ht to mul “ 
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) : 
m X 
{- 
k — 8. 
* , 178109 4217 407 ( . 
240 (0^) + 708 (1;0) iQ2 * 384 (^,0) 
192 3840 j" ' 
n (8) ' 5 \ 
(L . e 5 ) 
k = 9. 
« X pfee (0,0) - ^ (1,0) + ^ (0,1) + (2,0) 
- W (M) - W + iS & 1 ) + its ( 4 >°) - WA ( 3 >D 
+ ^(4,l)}cf.e' 4 e (L (9) . e' 4 e) 
( 8 ) ( 8 ) 
* By (0,0) C is meant the same as C, . 
2 £ 
