316 
from tu to t=, and one from t=a@ tot=0. The latter gives 
an amount which is independent of u, and this amount gives zero 
for the final integration. Therefore, after changing the letters u and 1, 
we may write for the preceding expression 
erk (Cm LO 
zailin af e= (u—1)* dn Ja 
(1,1) 
and this, if we take (2) and (5) into account, is just equal to the 
first member of (7). The latter equation thus has been proved in 
case g <_ 0. 
If g>0 and, to begin with, O<g< 1, Pincuerie defines the 
coefficientfunction of g(t) by means of an auxiliary function 
Pp (t) In 
Re ae a ieee (n+1) (t—1)-1 
0 
(8) 
The order of ¢,(¢) is lower by unity than that of p(t) and thus 
negative, so that «, (7) = /p,‚(t) is defined by (2). By (4) we have 
Ipd==l ¢—1) gO S219, 0 — 6 219, (Dan sense) 
if @ be the operation defined by 6/(#)=/(v-+ 1). If we denote 
by g(t) the result of the operation (a) = (—1)* D- 
(t — 1) 
applied to g( and by pix (t), the result of the operation 
1 
(1) = — Det applied to p‚(t, we derive from the preceding 
equation 
Vip, #1 pig Sr [Lien ee ee 
Now, the operation (a) is commutative, as will appear in the 
simplest manner from the expansion (6). Hence gi, = Ya, if by 
the latter expression the result is denoted, which is obtained, if 
first the operation (1) and then («) is applied. But p,(t) is a 
function for which the equality (7) has already been proved; if this 
is taken into account, we may infer from (10), using the identity 
Fy +D=y rly), 
Ve) sl tap, (t) ee) 
lee = Teo Be —(#+a—1)I (11) 
(t—1)# (t—1)# 
Using again the relation (9) we may write 
tp, tp tp 
I= (eel == — J| ¢—1 —| . (12 
ee 
Further, in connection with (8) 
