1249 
where the latter integral, which may be written 
fe ge (0) Gn (a) da 
0 
vanishes according to (2) if n > 0. 
Therefore three successive coefficients of this expansion are related 
in the following way 
(n + 1) anp = 2 (n+ 1) an — nay) (n > 0) 
so that all the coefficients may be expressed in a, and a,. 
Now 
p‚ (a) =1l—a 
6 fF oe =(= trade da — 2a 1 
l Ha En lia eae 
0 
which proves that all the coefficients are dependent on the first 
“e—4de Wai 
TA — — eli CC) = 0,596347 ... 
e 
0 
hence 
These coeffieients may also be obtained in another way. 
From ABEL’s expansion 
e — > pn (a) v" 
1 ae 0 
which holds where 
mod v <1 
we see, by putting 
u 
br 
1—-v 
that 
ed 1 t t? 
a ae Ey TG Lt Ae (149) P, (#) + … 
if 
t 
d—— <1 
mo “Er 
Multiplying this equation by e-tdt and integrating between the 
limits O and oo, we obtain 
i 
Te ay =F a, P, (.c) + 4, P, (w) Tine *s 
where 
