106 THE ROYAL SOCIETY OF CANADA 
Thus 
Way = 1 ES DID CE ROSE) 
Pie i D ee MINE PENSE 
The function Z, (x) was introduced by Siemon and studied by 
him for positive integral values of », zero inclusive; it has also been 
employed by Lord Rayleigh. The formula (16) was proved, subject 
to the above restrictions, by Siemon, Programm der Luisenschule, 
Berlin, 1890. (N. pp. 54, 292). The proof of the theorem in its 
general form (by a method which differs totally from that here adopted) 
was apparently first effected by Nielsen. 
es PT EE) ee (16) 
$ 7. In the present class of investigations arising from the study 
of the expansions known as Neumann series of the second kind, the 
following series 
142 a (a—1) a (a—1) (a—2) (a—3) 
(a+1) (a+2) (a+1) . (a+2) . (a+3) . (a+4) 
can be made to play a part of considerable importance. So far as the 
writer is aware, this series has not hitherto been evaluated. We pro- 
pose to show that 
a 2 An 0. 
a (a—1) a (a—1) (a—2) (a—3) 
mue) a GED EDGE) LEO à 
_ Na T (a+1) 
De Gees ue 
We shall assume a to be real and apply Gauss’s well-known test for 
convergence in the case of series where the test-ratio 
netan?-1+... 
Mea Pe eT NER 
Such a series is convergent when, and only when, a’—a<1. This 
test shows that it is necessary and sufficient for the convergence of 
the series in (17) that a shall be positive. 
E Has 
T (a+1)’ 
formula I («) I (x+3) =2-2*+1 Vr T (2x). 
We have to show that 
muCES ones) a(a—1) (a—2) (a —3) 
io Fame, Poivcrermerst 
5e Lie) 
Fa MMONT ACE. 
The expression on the left-hand side 
fs T(a) TO) 2G oy T'(3) +2 (4 Re M5) D 
Multiply both sides of (17) by 7 and make use of the 
T+) T(a-F3) Teno: 
es [ane [1+2 (59) p+2 (9) 
