12538 
pn (@) and p41 (2) 
tend to the same limit. 
If, therefore n is very large, the second integral, tends to 
1 it 
femternntede= [egaloypal(a) = 
0 0 
l 
epa («) |} 
= É Pp | -}. ep (x) der = — 4 
0 
2 
0 
and we obtain 
= 1 
Samael tt 
Thus, adding to this equation 
d, Po (1) == 
é 
we get finally the required relation 
1 
= Ap Pp (1) = 3. 
7. In this article we wish to give a second verification of the 
former expansion because this leads to a very interesting integral 
containing BessrL’s functions. This verification is obtained by direct 
summation of 
de + af, (z) + AP, (7) SE > 
‘where 
1 
a, = 1 Ee ane ap $ [Pn—1 (1) — Pn (1)]. 
It appears from the equation (10) that 
ne a 
Pal oss eter, (2Va) da 
0 
mg fer er J, VD de 
0 
therefore 
7 é a pn 
Pn- 1 (1) — Aer J, (2V a) d (e- 4a”) 
or, after partial integration 
Proceedings Royal Acad. Amsterdam. Vol. XV. 
