( 494 ) 
Mathematics. — “On a series of Bessel functions.” By Prof. W. 
KAPTEYN. 
(Communicated in the meeting of December 24, 1904). 
In the following we shall try to determine the sum of the series 
T, (@) I, (©) +3 1, (@) A, (@) +57,@1,(@)+-..= = n In (@) In («). 
To this end we begin to determine the sum of tie simpler series 
Ss = = Lj (w) cos ng. 
If we introduce, being an odd number, for the Bessel function 
the form 
then 
x Le 
ka) iz) 
S= de ST (t cos p Ht cos 3p -+ ..-), 
and 
t (1—+t?’) cos p 
tcosgttcs8y+...= TE (mod t <1), 
hence 
x 1 
Gas) ea 
g ij e (1—#’) cos p 
faa fe Ae Ea yas EE 
If we put 
ae cos p En 
1—2 t? cos 2p-+t* 
then 
x 1 
mins 
S= — bal an — t*) R, 
or 
: A 
= 1, cosng = — En e (i — t°) R. 
1.3 
Differentiating this equation, we get 
x 1 
aa IR 
wD » r 
= nl,(x) sin ng= re) e 
1.3 0 
