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.

OF A UNIFORM BRANCH OF A MONOGENIC FUNCTION. 9
at a finite number of determinate sections. Notwithstanding the remarkable researches of
M. Fuchs and M. Appell and others, this problem of finding a representation, which at
once is unique for the whole plane and is sufficiently simple, has not hitherto been
solved.
The beautiful researches of MM. Fabry, Hadamard, Borel and other French writers,
which have their origin in M. Darboux's memoir "Sur l'approximation des fonctions de
très-grands nombres " and which aim at the development of the criteria whether a point on
a circle belonging to the elements F(a), F(1> (a), F(2) (a),... is a singularity of the function
or not, are well known. My theorems make it possible to study this problem from a
more general point of view than these writers and to find the criteria which distinguish
the points of the star belonging to the elements F(a), F(1) (a), F(2) (a),... from other points.
It can be stated that, to each selection of the coefficients called c(), there corresponds
a special system of criteria.
For these investigations, the following theorem can serve as the point of departure:If x is a point within the star A belonging to the elements F (a), F(') (a), F(2) (a),...
and if e is a positive quantity sufficiently small, it is always possible to choose a positive
number S so that, o being a positive quantity as small as we please, a positive integer
X exists such that
\h,1() ()FP(1) (a) (1 + e) (x - a) + h,2() () F(2) (a) ( (1 + e) (x-a)}2... +~hx)(a)FÀ(a){(l +e) (x-a)} < a,
provided X\X.
If on the contrary, x does not lie within A, this property does not hold.
M. Poincaré has pointed out a certain substitution which is of great value in the study
of certain mechanical problems, particularly in that of n bodies. When this substitution is
used, a development of the function in powers of the time can be obtained which is valid
for real values of the time as far as the first positive or negative singularity nearest the
origin. But the mechanical problem requires in general a knowledge of the first positive
singularity, and not merely the nearest singularity, positive or negative. It is obvious that
the resolution of this problein can be brought within my theorem. In fact, knowing the
elements F(to), F(1) (to), F(2) (to),... at a given epoch to, we can obtain a development which
represents the function and is valid for all real values of t > to up to the first singularity of
the function.
Recently I had an opportunity of giving an account of a portion of my investigations
before the Academy of Sciences of Turin. My friend M. Volterra then made the following
interesting communication.
If in any dynamical problem the unknown functions be regarded as analytic functions
of the time, the problem will be solved completely froin the analytical point of view when
it can be shewn that the real axis of the time falls completely within the stars of the
' Liouville, Journ. de MIath., 31e Sér., t. iv. (1875), t The quantities 6 and h(À) (6) have the same significance
pp. 5-54. n
as in the formula (5).
VOL. XVIII. 2