169 
Proceedings (Vol. XXII, N° 1) that they are expansible in series 
of the form (16). Consider a domain R(x) >c-+ 0, take a positive 
number Jd, < d and substitute «=c-+ d, + y in the second integral 
of the right-hand member of (5), then A(y) > d—d, and thus positive, 
so that we have 
| fornia g (t) dt 
where the latter ae! exists, since od is a limited function in 
the interval (0,1). Hence a(x) is in the whole domain considered 
of the form 
1 
z in | t*i—1 g (t) 0) di fer dt, 
0 
9 (é) 
a (x) = (@—b) u (2) 
where w(x) is a function remaining within finite limits and 6 a 
number lying without the domain. Suchlike functions, however, can 
always be expanded in series of the form in question. 
A second, more direct proof, is obtained by substituting t= 1—u 
in the same integral as considered before, and using the following 
reduction 
5 zg 
xr—c—1 — (1 —u)Be- (1u Blue! If m 
(lu) (1 — u) (1—-w) (1—u) Me n(— 1) ke )e 
where the series for A (w eee converges uniformly in the 
intervalO <u <1. Since, for R (3) > R(c) the integral 
1 
fo (1—wu) (lL—u)#—-¢—1 du 
0 
converges absolutely (on account of the continuity of g(l—u)), we 
may, after performing the substitution in question, integrate term 
by term, and then we find (replacing again 1—uw by ¢ in the partial 
integrals) 
1 1 
fo peel dt = Sm (—Ij" cm flames) des (IM 
0 0 0 
This expansion is, therefore, valid for R(x) > R(8) > Re). Since 
the product of this series with w--c can be Briana into a 
series of the same form, the required proposition has been proved 
again *). 
1) In NierseN's book (le. p. 125) we find an analogous proof ofthe proposition 
in question; this, however, does not part from the integral in the second member 
of (5), but from the original integral, so that the hypothesis must be made that 
the latter converges absolutely for limt=0. The reduction (5) makes this 
hypothesis superfluous. 
