94 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
c ( : } 
2 
4 ^ cos 2 w 
it a' " ‘ y' { 1 + 2 « cos 2 m + a 2 } 
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 6 nly terms whose values are ultimately sen¬ 
sible are those which are similar to the expression of (33): and at last we get 
sin (3 sin S' , sin 6 sin 3' sin/3" , „ 1 1 ~ « , „ ,, ^ „ 
• 2 + 2' 2 • -W L + &C.|=-,X 0,9424137 
35. Putting ^ for v — v, and differentiating with respect to % the logarithms 
of both sides of the equation 
1 1 ( 0 ) ( 1 ) ( 2 ) 
\/ {a 12 - 2 a' a. cos ^ + a 2 } = ~2 ' C0S ^ C0S 2 ^ + &c - 
multiplying out the denominators, and comparing the coefficients of cos k%, 
2 k ( 1 . \ _(*) 
c? +,, = 
■2 
2 k — 1 1 ) 
Ci 
/I \ (k) 2 
2 k + I \~a. °7 Q ~ ~ok + 1 -s 
where + a — 2,1058226. Making h successively 1 , 2 , 3, 4, &c., we get the 
following values : 
( 0 ) ] 
C 4 = -^ X 2,3863750* C 
(i) l 
c, =-r X 0,9424137 C 
2 a 1 
( 6 ) 1 
* = X 0,0903724 
,(V) l 
# = -j X 0,0609432 
(12) J 
Cr =—, X 0,0093812 
(13) 1 
Cx = 7 X 0,0065274 
( 2 ) 1 ( 8 ) 1 
Cx = 7 X 0,5275791 Cx =yX 0,0414571 
2 U ^ Cl 
(14) 1 
Cx =TX 0,0045503 
2 w 
(3) 1 (9) 1 (15) 1 
Cx = -r X 0,3233422 C r =— X 0,0283925 Cx = — X 0,0031744 
2 a 7Z CU ^ ^ Cl 
(4) ] (10) 1 (16) 1 
Cx = X 0,2067875 Cx =-^ X 0,0195495 Cx = -j X 0,0022123 
(17) 1 
(5) 1 (11) 1 (17) 1 
Cx =— X 0,1355852 Cx =— X 0,0135189 Cx = — X 0,0015356 
2 cv J 2 a ? a 1 
(is) 1 
Cx = - X 0,0010554 
2 a 3 
* Laplace’s numbers, which are somewhat different from these, are computed by the less accurate 
method of summing a slowly converging series. 
