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.

434 MR BAKER, ON THE THEORY OF FUNCTIONS
wherein, in the latter, x,=a and x2=b, and the integration in regard to '3, x4 is to be
taken for each of them over the whole range - o to + oo. Hence we obtain
r, (1_ 1 )
which, as the general theory requires, is a function of the complex variables (in fact only
of T = tl + it2). Thence
) = | (Uldr, + rd7r) = 7r + ( -)]
and therefore, as w = 27r2 for n = 4,
27r 7T
which is precisely right.
20. Transcendental functions of one variable which have no essential singularity in
the finite part of the plane of the variable may be distinguished into two classes according
as, to speak first of all somewhat roughly, their zeros become indefinitely dense or not, as
we pass to the infinite part of the plane. If circles be described in all possible ways, each
to contain a certain definite number, say 1V, of the zeros of the function, N being at least
two, the areas of these circles may have zero as lower outside value as we pass to the
infinite part of the plane, or may have some quantity greater than zero as lower outside
value. More precisely, in the former case, however small A may be, and however great R
may be, among the circles described to contain N zeros whose centres are at distance at
least R from some definite finite point of the plane taken as origin, one or more can be
found whose area is less than A; in the latter case it is possible to assign a quantity A
finitely greater than zero, and a finite R, such that among the circles described to contain
N zeros whose centres are at greater distance than R from the origin, no circle can be
found whose area is less than A. The most obvious example of the latter possibility is
the case of a periodic function; here a period parallelogram necessarily contains only a
finite number of zeros; and this parallelogram is indefinitely repeated to however great
finite distance we pass. As example of the former possibility we may take the case of an
integral function whose zeros are the real quantities log 2, log3, log 4,.... The length of
the streak which contains the N zeros beginning with log R is at most
log (R + \T)- log R = log (1 +
which diminishes without limit as R increases.
21. Consider now an integral function of one variable of the former of the two kinds,
for which circles containing a specified number N of the zeros of the function are formed
of as small area as we desire, however great be the distance R of their centres from a
finite point of the plane. It is still conceivable that for proper choice of the constant m,
independent of R, and not less than unity, the product
Rm- C,