Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

432 MR BAKER, ON THE THEORY OF FUNCTIONS
to represent the equation of a given complex (n-2)-fold extending to infinity; let us
apply this to find the equation of the (n -)-fold constituted when n=2 by any
enumerable system of discrete points f, 2,..., having infinity as a point of condensation, in regard to which it is assurned that for some positive integer, rn, the series
-n-2 + 2-m-2 +...
is absolutely convergent; this condition corresponds to the general one that the integral
jlm+i dSn_2 is convergent; for instance the points may be those given by a formula
a + 2kw + 2k'',
where k, k' are integers and o'/o is not real, in which case, as is well known, it is
sufficient to take m=1. Taking
P (x t) = log [(xi - tl)2 + (2 - t)2],
and, as in the general case (~~ 13, 14),
V = ifHm+ dSn-2 = H.m+,
(-m 1)
=2 ~ go(x 1-o( |0)+-t... + e(, + --- (t ~;) g |(0[o)J,
where the summation extends to all the points e = xi + ix,, and
i = j (Cdit ajm - dt2Hn) dsn2,
=, dt - _ dt, -W )
we find easily
r\ T 1 T2 1 nt^+1
T+ia= [ log l-)+j+ -+...+ ^^];
hence as r= 27r for n= 2 we have
o (T) = I [1 - ) "t (,+l1),n+l]
namely the theorem gives the general integral function of finite genre*; whereas in
Weierstrass's factor formula for a general integral function the number of terms it is
necessary to take in the exponential may increase beyond all limit as we take a more
and more advanced factor of the product, our theorem limits itself to the case where
the same finite value of m will suffice for every factor.
19. In the case of functions of one variable a simple case of functions of finite
genre arises for periodic functions, the value of m for the sigma functions being m==1.
And in the general case the fundamental (n-2)-fold of integration may be periodic;
* The usual exponential outside the infinite product being absent.