329 
hence equation (9) shews completely the behaviour of M («x Op), 
when v—> 1. 
In fact, we may write 
; | hg …_ hp) | hp 
Lim} M(«6v)—M(«?)'= logg—— > shlog |2sin — tri (10) 
x—>1 Q hal | q | g 
and again we may notice that only the real part of M (x Op) becomes 
infinite, the imaginary part tending to a finite limit. 
Taking, for instance, p=1, g = 2, we shall find 
Lim { M(— «) — M(a*)} = } log 2, 
1 
or 
IT (1 + «@?n—1) 
Bid, 
ra ra 
1 
a known result in the theory of tbe 9-functions. 
4. Finally, I will state that the discussion of the fundamental 
equations (1), (2) and (3) furnishes the proof that the function 
N(z) = Eb, 
Teil 1—z" 
cannot be continued beyond the circle |z| =—=1 in each of the 
following cases : 
Beb B0. 
In this case we shall have 
A om 
2s lines 
q ie 1 
log 
Ne >< 
ee 
We bim-b,, A0: 
n= 
Now it will be seen that 
d—= A 
KiM 
2-1 if q 
log 
lx 
… br 
Il. Lim 2=A 40, s>0. 
no ns 
In this case the equation holds 
A 
x1 q +s 
