On account not only of its historical interest, but of its direct-
ness and simplicity, we reproduce here Bessel's original proof of
the theorem that the equation J^ (ic) = has an infinite number of
real roots*.
It has been shown (see 46 above) that

Ja (x)— - 1 cos {x sin A) dd>
ttJo

dt



11:



- I cos (xt) -



Suppose that « = ■ ■ tt where m is a positive integer, and

m' a positive proper fraction ; then

(2 m + VI ) irt dt



h{^)-l\y



ir], 2 ^{{'i.m + mj-

" Berlin Abliaudliineen (1891), Art. 14



y Google



v.] THE ZEllOES OF THE

and hence, by taking « = /(-!, 6 = A + 1, and putting h succes-
sively equal to 1, 3, 5,... (2m - 1), we obtain, on writing /* for
2m -I- to',

J, (=.) = y_" ™ f ^. f ( J ^ ,,,, + v;™+^. - ■ • ■

V^^-(2to-1+m)'J

2 - -,, T"' ■'!"« c^w
+ - (- I)™ cos -s" , -. ■ --- ■ - — ■

The integral

j_, ^''^ T v';^A:+^;i^

and is therefore positive ; moreover

I cos -^ ,

U 2 \/^3_(2to + m)'

is evidently positive. Therefore J^ {x) has been reduced to the
form

J«{x)^-Ui+V,^ — tis+ ... +{--)'^1lw,,

where m,, u^, ... u^ are a series of positive quantities and

J(l< %<-;%... <Mm.
