Colloquium publications.

178

THE MADISON COLLOQUIUM.

fi(x, y) meromorphic on one side of S and having 2 as a natural
boundary; and also a function f2(x, y) meromorphic on the other
side of I and also having 2 as a natural boundary, then must
(E( )= 0.
How far are these conditions sufficient? In the present memoir
Levi shows that this last condition is sufficient; namely: If
1(S() = 0 in every point of Z: ~o = 0, then there are functions
analytic on each side of 2, but having I as a natural boundary; —
all this, at least, when 2 is suitably restricted in extent.
In a later paper Levi* obtains the further result, that if
~(S() < 0 in all points of Z: p = 0, then there exists a function
f(x, y) analytic on the side of Z where p > 0 and having / as a
natural boundary;-all this, at least, when 2 is suitably restricted in extent.
~ 11. A THEOREM RELATING TO CHARACTERISTIC SURFACES
An analytic surface in space of four dimensions may be represented by a pair of equations:
(1) u(xi, x2, yl, Y2) = O, v(xl, x2, Y, y2) = 0,
where u and v are real functions of the four real variables, analytic
at the point in question, their Jacobian with respect to two of
the variables,-say yi, y2,-not vanishing there.
Levi-Civitat raises the following question. Suppose two real
functions, p and q, are given along such a surface, and are analytic
there. Thus p and q may be any functions of xj, x2 analytic
at the point in question, if these are the preferred variables.
Does a function of the complex variables exist:
V(x, y), x = x + ix2, y= y1 + iy2,
*Ann. di Mat. (3), 18 (1911), p. 69.
t Rendiconti Accad. Lincei (5), 14 (1905), p. 492. He prefaces his problem
by recalling the Cauchy problem for two independent variables, x and y, and
an analytic curve C in their plane; an arbitrary sequence of analytic values
being assumed along C.