315 
has been laid down. Passing to the second member we write 
l 
v it 
a dae ar 
where the argument of s and of 1 —s is zero. Further 
t=p(t) “tx +2—ly(t) 
cE dt, 
(t—1)+ T Oni (t— 1) 
(1,1) 
so that the second member in question is equal to 
1 
BN Ca city a | 
acne (t—1)¢ | Cr | i 
0 
(1,1) 
If we substitute in the second integral s =u: ¢, this expression 
passes into 
t 
: POUR A 
iG (t—1)¢ LS '¢— ue! in| ne 
(1,1) 
Since the argument of s was zero, the argument of w is equal to 
that of ¢; thus the variable under the sign of integration goes along 
the straight line from w=O0 to u=t in the v-plane. But it may 
go as well from u = —i), on the circumference of the circle (1, 4), 
along that circumference in the positive direction to the point u == t. 
On this supposition we consider the system of two integrations, to 
be performed in succession, as a double integral. Then in the corre- 
sponding aggregate (w, t) a definite value w, of « has to be associated 
with all those values of ¢ lying, in the ¢-plane, on the circumference 
of the circle (1,1) between tw, and the end-point t= + 20 of 
that circle. Hence the double integral may be replaced by the pair 
of two successive integrations denoted in the expression 
0 
dt | du, 
amin) ki ad 
where the Pee ah pen to ¢ has to be performed in the 
positive direction from t—=u to t= + id"). On account of the pro- 
perties of y(t) the latter integration may be replaced by an integration 
1) The here given argument is strong, in so far it is based upon known truths, if the 
functions under consideration are finite in the whole domain of integration. This 
is the case for R(x)>1 and R(x) > 1, but, since both endforms are analytic 
functions of x ‘and of « for R(x) >0 and Ria)>O0, they must also be equal for 
the latter values. 
21 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
