the abscissae are current densities in 106 amp./cm2. The specimens varied in breadth from 0.06 to 0.22 mm. There was no correlation between the results and the breadth, although the microphone effect has a very

Extrapolated Difference Between A. 0. and D. C. Resistance, Per Cent.

strong connection. The results are scattering, but perhaps not more than would be expected when the magnitude of the current densities and the fact that different samples of gold leaf may differ in specific resistance by a factor of 2 or more is considered.

The curve drawn through the observed points is taken as the best mean of the experimental results. Let us denote the equation of this curve by (x). Then the departure from Ohm's law is given by the expression (x)dx/x. This will be proved in the detailed paper. This integral may be calculated graphically from the observed points.

In figure 3 is given the departure from Ohm's law, calculated in this way, for gold of two thicknesses and for silver. The departure is positive, that is, the resistance is greater at high density. For thin gold and silver the departure rises to something of the order of 1% at a density of 5X10-amp./cm2; for the thicker gold it is greater. It was not possible to reach as high current densities in the thick as the thinner gold. The accuracy is greatest for thin and least for thick gold.

The form of the curve is not that supposed as a first approximation by Maxwell, but the departure rises more rapidly than the square of the current. In fact the form of the curves seems to suggest infinitely high order of contact with axis at the origin. If however, for the purpose of numerical comparison we assume that below current densities of 106 amp./ cm2. the departure from Ohm's law is proportional to the square of the current within the limits of error, than the curve for thin gold shows that for it the resistance at 1 amp./cm2. cannot differ by more than 1 part in 1016 from the resistance at infinitely small density (against 1 part in 1012 of Maxwell).

Theoretically the existence of this effect at currents of the order of 106 amp./cm2. means that, granted a free path mechanism of conduction at all, the free path must be much longer than supposed on the classical basis, and accordingly the number of free electrons much less. Now on other grounds I have been led to the belief that conduction is by means of a free path mechanism,2 of a different sort than that supposed in the classical theory, and that the number of electrons is much less and their paths longer than according to that theory. These new results are now in accord with this point of view. I have not been able as yet to carry through a more exact discussion on the free path basis and to obtain the numerical value for the length of the free path which these results would involve.

The difference in results for gold of different thickness is also what would be expected. The results for thick gold are not nearly as accurate as for the thinner, but there can seem no question but that the departure is

really greater for thick than for thin gold. If the free path is of the order of 10-5 cm., that is, comparable with the thickness of the leaf, then the average path in the thicker metal will be longer than in the thinner, and a greater departure would be expected in the thicker metal, as found.

[I am much indebted to my assistant, Mr. J. C. Slater, for his skill in making the readings.]

1 Maxwell, C., Everett, J. D. and Schuster, A., B. A. Rep. 1876 (36–63).

2 Bridgman, P. W., Physic. Rev., Ithaca, (2), 17, 1921 (161–194).

AN INTEGRAL EQUALITY AND ITS APPLICATIONS

BY EINAR HILLE

HARVARD UNIVERSITY

Communicated by G. D. Birkhoff, May 15, 1921

The purpose of this note' is to deduce an integral equality adjoined to a linear homogeneous differential equation of the second order and to show some applications of such equalities to the question of the distribution of zeros of solutions of such differential equations in the complex plane.

Let G(2) and K(2) be two single-valued and analytic functions of z throughout the region under consideration and take the differential equation

We call relation (5) Green's transform of the differential equation (1). This relation can be used in many ways for obtaining information concerning the distribution of the zeros of a function w(z), satisfying (1). Formula (5) enables us to assign regions, below called zero-free domains Some of these ways are in

where there can be no zeros of w(z) or

dicated below.

The four differential equations (9)

dw
dz

.

dK1 = 0, dK2 = 0; dr1 = 0, dг2 = 0,

define four families of curves; the K1-family and the K2-family forming the K-net and the T1 and T2-families forming the T-net. The two families belonging to the same base net are orthogonal trajectories of each other.

Take a solution wa(z) of (1) such that

W。(2) = K(2) w。(z)'

dwa dz

vanishes at a regular point a in the complex plane. Construct the Riemann surface on which wa(z) is single-valued and mark the two base nets on the surface. Further, draw all curves on the surface, starting from z = a which do not pass through any of the singular points of the differential equation and which are composed of arcs of curves of the two base nets such that along the whole path one and the same of the following four inequalities is fulfilled, namely

We agree to smooth the corners of the path-curves, when necessary, in order to preserve the continuity of the tangent along the curve. Such curves we call standard paths and designate the four different kinds (distinguished by their characteristic inequalities) respectively

(11)

SK*, SK, SK* and SK ̧.

The points on the surface which belong to at least one standard path, emanating from z = a, together form the standard domain D(a) of a. A

point on the boundary of D(a) is counted part of the standard domain provided it is not a singular point of the differential equation and is different from a itself. In view of (6) and (7) we obtain the result: There is no zero of Wa(z) in the standard domain of a.

+

Here is another application of the Green's transform. Suppose G(z) and K(2) to be analytic and, furthermore, real on an interval (a,b) of the real axis. Then there are solutions of (1) which are real on the same interval. Draw all standard paths of the third and the fourth kind, SK, and SK,, which emanate from the points on (a,b). The points on these paths, not including possible singular points and the points on (a,b), form a zero-free region for all solutions real on (a,b) in virtue of formula (7). For this statement we have only used the fact that for real solutions Im w(2) w2 (2)] = 0 on (a,b). The generalizations are obvious.

The Standard Domain Is Covariant under Conformal Transformation.— By this we understand that a change of independent variable from z to Z by putting Z = F (2) which preserves the form of the differential system (2), carries the standard domain of a point zo in the z-plane over into the standard domain of the corresponding point Z。 in the Z-plane and these two domains are the conformal maps of each other by the transformation Z = F (2) and its inverse, provided F (2) is regular in D (zo) and F' (2) O there. This is obvious because Green's transform is an invariant under such a transformation and the base nets in the two planes correspond to each other by the conformal transformation.

The standard domain is not covariant under a simultaneous change of dependent and independent variable. By a fortunate choice of such variables it is often possible to determine zero-free domains of such an extent that there is comparatively little freedom left for placing possible zeros; thus one is able to obtain a fairly good qualitative description of the arrangement of the zeros in the plane. The transformation

often yields good service for investigation of the distribution of zeros of solutions in the neighborhood of an irregular singular point of the differential equation.

1 This note is an abstract of a paper offered to Trans. Amer. Math. Soc. for publication.