94 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
(0 4 „ cos 2 w 
i. 7r a! " * y'-f 1 + 2 a cos 2 co + « 3 } 
Making the same substitution as in (33), there will be produced three terms ; 
of which one vanishes in the definite integral, the second is similar to the ex- 
pression of this article, and the third similar to that of (33). Making a simi- 
lar substitution in the second term, new terms are produced. Pursuing this 
method, it will be found that the 6nly terms whose values are ultimately sen- 
sible are those which are similar to the expression of (33) : and at last we get 
(f = cf 
{- 
sin /3 sin (3 sin /3' sin (3 sin /3' sin /3" 
+ 
2 
2 
&C 
•}=-sx°, 
9424137 
35. Putting % for v —v, and differentiating with respect to % the logarithms 
of both sides of the equation 
1 1 (0) (1) (2) 
*/{a' 3 -2a'«.cos % + a 3 } = "2 C i + C i • COS X + C, COS 2 % + &C. 
multiplying out the denominators, and comparing the coefficients of cos 4%, 
(*+ 0 _ 2 1c , \ p (*) _ S*-l p (*- 0 
5 ~ 2 k + 1 \ a W 2 k + 1 
where + a = 2,1058226. Making A; successively 1, 2, 3, 4, &c., we get the 
following values : 
c ( . 0) = 1 
(oj J (6) 1 
i = X 2,3863750* C, = X 0,0903724 
(12) j 
Cx =—, x 0,0093812 
ri) i (7) i ( 13 ) i 
C, =-j X 0,9424137 Cx = -7 X 0,0609432 Cx = - 7 - X 0,0065274 
2 it 2 U 2 (4 
c ( . 2) = 1 
I'-ej 1 (8) l 
=^r X 0,5275791 C 4 = 7 X 0,0414571 
( 3 ) 1 
C 4 = ^ X 0,3233422 C 
,C 4 ) l 
,( 9 ) 1 
) 4 = -j X 0,0283925 
X 10 ) l 
(14) I 
Cx = ^ x 0,0045503 
(15) 1 
Cx =-j X 0,0031744 
(16) 1 
Cx = ^ X 0,0022123 
(4) ] ( 10 ) | 
c 4 = -J X 0,2067875 Cx = ^ X 0,0195495 
( 5 ) 1 ( 11 ) 1 ( 17 ) 1 
C. =-7 X 0,1355852 Cx = -j X 0,0135189 Cx = -r X 0,0015356 
1 14 * U ^ tv 
( 18 ) 1 
c 5 = X 0,0010554 
Laplace’s numbers, which are somewhat different from these, are computed by the less accurate 
method of summing a slowly converging series. 
