30G 
ME. E. AV. BARNES ON THE THEORY OE THE 
Again, it has no tinite zeros, for its zeros would be those of rj(^ + ojj) for 
nin = 0, 1, . . , CO . And its poles are given by 
z -f" 1)1 ^ 0 )^ —h — b 
wp = 0, 1, . . . 
CO 
m.. = 0, 1, ... CO . 
Thus^^^'' has no zeros or poles in the finite part of the plane. 
Change now z into z + and we have 
V {Z + ) 
Viz) 
— z U 
■m.,— 1 
GO 
X n 
rn., = 1 
1 -)“ I ® IHjWj 
- I “i) - '“'- ‘I'l eawgMa I Ml) + — 
Now the product last written must l)e convergent, for all other terms of the 
identity are Unite for Unite values of \z\, and this product is evidently of the form 
,>p- + 'I 
, where p and <[ are functions of o)y and oidy. 
Hence w'e must have 
P(;+ <.,) = r(i) rr‘(^ IM,) e-*», 
Now we have proved that 
U(m<»,) = ra2 )rr*C|Mi)- 
Sir \ ^ 2//(771 
(--0 
Thus if we put 
v e shall have 
and similarly we shall have 
/■(.) = 
^ '' i'(j') ’ 
,/ (•^ T W) ) n. • + fl 
-!_ — + Pi 
/(M 
/’(- + mP 
/(M 
Hence f{z) is a doubly periodic function of the third kind, with no finite zeros or 
poles. 
Thus w'G must have 
for in Hekmite’s expression of such a function the cr functions are each associated 
with a finite non-congruent zero."^^ 
To determine C we put z = 0, and obtain 
0 =u r 'Ap 1 = 1 . 
1 = 0 L H (~l "dj 
Hifierentiating logarithmically, the identity 
* t'orsYTii, ‘ Tlieoiy uf Eunctiuns,’ § TIZ. 
