( 102 ) 
Mathematics. — “A definite integral containing Bessel’s functions” 
by Prof. W. KAPTEYN. 
If Z(t) and J, (¢) are two Bessel’s functions of the first kind 
and of orders m and n, then 
OG dn lL die m2 
+(t _5) ti) 
dt? tat 
and 
9 
dt? t. di t? 
If we multiply the first equation by J, and 
find by subtraction 
ad? Je i aa. (as 
a ae — 7) n= 0. 
the last by Jm, we 
d? Ln d? I, 1 ad, dln m2 — n? 
nu? AE os i ( age aes a ani. ig fn tn 
By putting 
dn at. u 
” lt m dt —— 
we obtain 
a! 1 m2 — n° 
ie + ia == In it 
or 
d m2 — n° 
oA (U t) = „pn i 5 
and after integration between 0 and oo 
eo PI Te 
(Ui ne f ned. 
0 
0 
Now for t=oo we have 
