98 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
ct cf + a A cf = - cf , or (1,0) cf + ( 0 , 1 ) cf = - cf . Again (as 
* 
^4 (^) 
another instance) ^ ^ C, is a function of a' and a of — 5 dimensions; con- 
, , d 5 ^(*) . rf 5 r d * A k) u- , 
sequently a'^ir^ c s + C, =-5jjKTa C * = or, multiplying 
fA -1 (£) (&) 
both sides by a ' 3 a, (4,1) C s + (3,2)0, = -5(3,l)C, . It is indifferent 
which coefficient of each order we calculate first; and for the algebraical pro¬ 
cess it is rather most convenient to differentiate successively with regard to 
the same quantity (as a'). 
43. Now -r-i 
da J */a! a 
1 1 1 
* /a, a n V 
(a + 7 - 200S X ) 
+S- 20 « x y 
+ K-v + ^-^ + ._; eosx y+- 
or, taking the coefficient of cos&x in the expansion on both sides, 
1 1 ^(*) / 1 . a \ A k ) 
_ f 1 — _ 
d i 
d (*) _ i i r « , / 2 , JL\ o 
Differentiating this formula with respect to a, and using the same formula to 
d- (*) 
simplify the differential coefficient, we get C s . In the same manner 
<f, &c - are foimd; multi P l y in & them (beginning with c[ ^ by a, a 2 , 
a 3 , &c., we obtain the following expressions: 
l A*) / l . \ AW 
(M)cf , = -i<f+(-^ + .)*.C 1 
( 2 , 0 ) cf = +1 cf + + - 3.). • c®, + ( - i + + . *. 7Ti ■ C , 
(3,0)cf=-f 5 cf + 4(-i + 5K ) s .c«, 
