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. 5
The formula
~n2 z42 n2nF 1 /+++ ( hl+h2+...+ha )
gn (x)=::_ o... h h)(, (............. (4)
h,=O h2=O h,l= h 2!.......
gives an expression for gn (x) which perhaps is the simplest of all as regards the mere
form. There are other forms in which the coefficients c(V are rational numbers, or are
numbers depending in a special manner upon the transcendents e and 7T, and which are of
great simplicity.
Upon this I shall not dwell: but I enunciate another theorem which is an almost
immediate consequence of my main theorem.
Denote by A the star which belongs to the elements
F (a), F()(a), F(2)(a),.......
and by FA(x) the corresponding functional branch belonging to the same elements. This branch
FA(x) can always be represented by a series
o=o
where the quantities G,(x) are polynomials of the form
G, (X) -= cv()F()(a) (X - a),
each coefficient t) being a determinate number (which can be taken as rational) depending
only upon jk and v. The series
=o
converges for every value of x within A, and it converges uniformly for every domain within
A. For all values within A we have
00
Gx,() = Lim gn(x),
iz=O ln= o
where gn(x) is the polynomial in my main theorem.
In what precedes, a definition has been given of the star belonging to the elements
F (a), F "()(a), F (2)(a),.........................................(1).
In accordance with this terminology, we can speak of the circle belonging to the
elements (1) which, in fact, is the circle of convergence C of the series
P (x | a) = I1 F(1)(a) (x - a).
/x=O j!.
It is evident that this circle is inscribed in the star which belongs to the same elements.
The circle may be regarded as a first approximation to the star. To the circle C corresponds an analytical expression P (x a) which has the property of representing FA (x)
within C, of converging uniformly for any domain within C, and of ceasing to converge outside