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.

XXII. On the Theory of Functions of several Complex Variables.
By H. F. BAKER, M.A., F.R.S., Fellow of St John's College.
[Received 9 February 1900.]
THE present paper is primarily a reconsideration of the paper of M. Poincaré in
the Acta lfMathematica, t. xxII. (1898), p. 89; and depends for its interest on the remarkable discovery of the expression of an integral function by means of the potential of
the (n - 2)-fold over which the function vanishes, which is virtually contained in
M. Poincaré's paper in t. in. of the Acta Mathematica (1883), pp. 105, 106. The
following points of novelty may however justify its publication. (i) By means of a
generalisation of the theorems of Green and Stokes, for the transformation of multiple
integrals, the imaginary part of the function of the complex variables is introduced concurrently with the real part; (ii) and thereby, as would appear, the coefficients in the
quadratic function used by M. Poincaré (Acta Mlath., t. xxii. p. 174) are shewn to be
zero. (iii) The theory is put in connection with Kronecker's formulae (Tferke, Bd. I.
p. 200), whereby it follows that the imaginary part of the logarithm of the integral
function is a generalised solid-angle, just as M. Poincaré has shewn the real part to
be a generalised potential. In general Kronecker's integral, unlike Cauchy's, does not
represent a function of complex variables unless the (n - 1)-fold of integration is closed;
in the present paper there arises a Kronecker integral which is an exception to this
rule (the integral,r-1, ~~ 1.2, 17). (iv) The definite formula here given for the integral
function is not limited to the case of periodic functions; though on the other hand it has
not that general application which belongs to the theory of M. Poincaré's earlier paper,
in the Acta Math. t. iI. In that paper there remains in the resulting formula an integral
function of which the existence is proved, for which however no definite expression is
given; in the present paper, in order to have a definite expression, I have hasarded a
limitation which may be regarded as a generalisation of the notion of the genre of
functions of one variable. This limitation arises by regarding the (nz-2)-fold integral
which enters here as a generalisation of the sum which is obtainable by taking the
logarithm in Weierstrass's general factor formula for an integral function of one variable.
The paper is divided into two parts, of which the former contains a formal proof of a
theorem constantly employed in the theory developed in Part II.