98 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
, a (*) d (*) w /A lN a) . . , 
^ C * + a Ta C * = “ C s > or 0,°) c , + (°A) c s = — c , • Again (as 
d* (*) 
another instance) da C s is a function of a' and a of — 5 dimensions; con- 
.1 , d 5 ^(*) , d 5 J!<) „ <Z* ^(*) .... 
sequently a c s + « do ». C, =-5^0, : or, multiplying 
(i) (*) (4) 
both sides by a* a, (4,1) C s + (3,2) C s = — 5(3,1) C 4 . 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 7 — i I —j=f- . —7 
da J a/ a! a ( a 
(a ' a n y 
(~a +7?- 2c0S %) 
1 1 1 
2 ‘a 1 ' */ a' a' /a' 
/ a' , a _ \ s 
U + a'- 2008 *) 
+ S {-'7 + ^)l7Ta- (a< a “ \ 
fv + 7- 2 cos x) 
or, taking the coefficient of cos k x in the expansion on both sides, 
A «n w -11 r (4) . / 1 JL\ 0 r (t) 
da' 2 ‘ a' s 4" ( a ‘ a 73 / A * + 1 
Differentiating this formula with respect to a, and using the same formula to 
d 2 _(*) 
CL \ ,c ) 
simplify the differential coefficient, we get C s . In the same manner 
(TaJ* \ &c. are found ; multiplying them (beginning with C s ( ^ by a, a 2 , 
a 3 , &c., we obtain the following expressions : 
„ _(*) 1 „(*) , / 1 . \ ^(*) 
0>°) C, = - -2 C s + (~ “I + “) S - C s+1 
(2.0) C s = + — C S + (— - 3 a) 5 . C 5 + x + ( - — + a) . s . s -f 1 . C s 
(k) 1 5 a) q / 1 \ (k) 
( 3 . 0 ) c, =- -g- c; +|(--+ 5«),.c; +1 
■'s + 2 
