Colloquium publications.

24

THE CAMBRIDGE COLLOQUIUM.

The coefficient of an' is called by L6vy the derivative with respect to an'.
Thus the derivative with respect to an' of 9O(AM)/On is - O4(AM)/Os, and
the derivative of 9O(AM)/Os is + O4(AM)/On. Hence we can write down
at once the derivative with respect to an' of the left-hand member of (33),
and since this must vanish, we have the further identity
(3) (AM) a 0(MB) _ O (AM) 9b (MB)
On On Os Os
Consider functionals ~4 which as far as concerns the point arguments, are
continuous with continuous derivatives in the neighborhood of the curve C.
It can be shown quite simply that this requires either (a), that 4b shall be
independent of C and shall be a function merely of a single point argument
A or B, or (b), that if it depends upon C, it shall, as far as its point arguments
are concerned, be an analytic function of x + iy and xi - iyi, where A = (xy),
B = (xly,), and i = i '- 1. In fact, in this last case, from the equations
(33), (33') follow the equations:
Oa (AM). 9O(AM) a O(MB) (MB) _ (
An s ' An ~ s '
from which the above conclusion can be shown to follow.
The value of the functional may be chosen arbitrarily for one curve Co,
and is then determined by the equation for the other curves C.
The Green's function does not remain continuous when A and B approach
the same point M on the curve, and hence is not subject to the above analysis.
Levy has the following theorem:
Consider a functional b[C j AB], equal to the function (1/2wr)g[C I AB] plus a
function which remains analytic when the point arguments lie in the neighborhood
of the curve. A necessary and sufficient condition that there exist such afunctional
which satisfies (32) and takes on, for a given curve Co given arbitrary values
4'o[C I AB], is that
(33') aco(AM) = Oo(MB) -
(33{) 's as 0;
Os - Os
that is, that (o shall remain constant if one point argument is held fast and the
other allowed to travel around the curve Co.
18. Variational vs. Integro-Differential Equations. If equation (28) is completely integrable, we can obtain its solution by means of any particular
family of curves we please, which leads to the curve C. In this way, by introducing a quantity X as a parameter which determines the curve of the family,
the equation under consideration becomes an integro-differential equation, or
perhaps reduces to a degenerate form of such an equation.
Consider the equation
(34) 64[C] = fCF[C |, M] n(M)ds,
in which the functional q4[C] has no point arguments. If we introduce a
second parameter t for the set of orthogonal trajectories to the X-curves, we
may write an(M)ds = r(Xt)dXdt, where r(Xt) is a known function, so that (34)
takes the form