20 
ON ROOTS OF FLUCTUATING FUNCTIONS. 
[560 
561] 
we have as follows: 
a" 1 = 2 2 . n + 1, 
b~ l = 2 5 . n + 1 . n 4- 2, 
c -1 = 2 7 . 3 . n + 1 ... n 4- 3, 
d~ x = 2 n . 3 . n + 1 ... n + 4, 
e~ l = 2 13 .3.5 . ?i + 1 ... ft + 5, 
/ _1 = 2 1(i . 3 2 .5 . n + 1 ... /1 + 6, 
g~' = 2 18 .3 2 .5.7 . n + 1 ... n + 7, 
// _1 = 2 23 .3 2 .5.7 . ?/ + 1 ...?/ + 8, 
a! -1 = 2 4 . (?i + l) 2 . n + 2, 
b{~ 1 = 2 9 . (n + 1. n + 2) 2 . n + 3 .+ 4, 
Cj -1 = 2 13 .3 . (n + 1 ... n + 3) 2 . n + 4 ... n + 6, 
d~ l = 2 19 .3 . (n + 1 ... n + 4) 2 . n + 5 ... n + 8, 
5n + 11 
a * ~ 2 8 . (n + l) 4 {n + 2) 2 n + 3 . n + 4 ’ 
^ 25 v? + 231?i + 542 
2 2 17 . (n + 1 . n + 2) 4 (n + 3. n + 4) 2 n + 5 ... n + 8 ’ 
_ 429?/ 5 + 7640n 4 + 53752n 3 + 185430n 2 + 311387?/ + 202738 
aj 2 16 (n + l) 8 (n + 2) 4 (n + 3. n + 4) 2 n + 5 . n + 6 . n + 7 . n + 8 
If n = 0, 
^ 16 101369 
Vp-i«. = a * = 227.33.5 7 =Pi > suppose ; 
whence 
p x = 2*404825. 
[The quantities p x , p. 2 ,\.. are the roots of the function J n (x) in increasing order 
of magnitude, so that, as these roots are all real, it folloAvs that for J 0 (x), 
a = %>r 2 , «i = Sj?r 4 , a, = 2pr a , a 3 = Ipr 16 , •••] 
ON T] 
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