THEOREM V. In order that a bounded function f(x) shall be integrable from a to b as to a function u(x) of bounded variation, it is necessary and sufficient that the interval (ab) may be divided into partial intervals so that the total variation of u(x) in those in which the oscillation of f(x) is greater than an arbitrarily preassigned positive number w shall also be as small as one wishes.

THEOREM VI. If u(x) is of bounded variation and f(x) is bounded on the interval (ab), then a necessary and sufficient condition for the existence of the integral of f(x) as to u(x) from a to b is that the total variation of u(x) on the set D of discontinuities of f(x) shall be zero (Theorem of Bliss).

...

Of the four preceding theorems the only one needing further proof is the last. [For the definition of the total variation of u(x) on a set of points, see Bliss, 1. c., p. 633, ll. 12-19.] Let €1, €2, €3, .. be a sequence of positive numbers decreasing monotonically toward zero, and let D1, D2, D3, . . . be the closed set of points at which the oscillation of f(x) is 1, 2, 3, . . . Then the set D of discontinuities of f(x) is the limit of the set D, when n is indefinitely increased. Now if f(x) is integrable as to u(x) we have seen that the interval (ab) may be divided into partial intervals so that the total variation of u(x) on those in which the oscillation of f(x) is greater than an arbitrarily preassigned positive number shall be as small as one pleases; and this implies that the total variation of u(x) on D2, and hence on D, is zero. Again, if the total variation of u(x) on D is zero so is it on D, for every n; and hence ƒ(x) is integrable as to u(x) since it is such that the interval (ab) may be divided into partial intervals so that the sum of those in which the oscillation is greater than an arbitrarily preassigned positive number shall be as small as one pleases.

TRANSFORMATIONS OF CYCLIC SYSTEMS OF CIRCLES By L. P. EISENHART

DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY

Communicated by E. H. Moore, October 9, 1919

When two surfaces, S and S, are applicable, there is a unique conjugate system on S which corresponds to a conjugate system on S. Denote these conjugate systems, or nets, by N and Ñ respectively.

The cartesian coördinates x, y, z, of N and x, y, z of N are solutions of an equation of the form

which we call the common point equation of N and N.

Let M and M be corresponding points of N and Ñ. With M as center describe a sphere whose radius is the distance from M to the origin. Let 21 and 22 be the sheets of the envelope of the spheres as M moves over N, and let μ1 and μ2 be the points of contact with Σ1 and Σ2 of the sphere with center at M. The null spheres with centers at μ1 and 2 meet the tangent plane of N in a circle C. These ∞ circles from a cyclic system, that is they are orthogonal to ∞1 surfaces.1 If h and I are any pair of solutions of the system

με

με

the functions x', y', z', defined by the quadratures

are the coördinates of a net N' parallel to N, and the functions x', y', z',

are the coördinates of a net Ñ' parallel to N. Moreover, the nets N2 and N' are applicable, and consequently the function ' = x2 Σx'2 ΣΤ is a solution of the point equation of N' and Ñ'. By the quadratures

The functions x1, y1, S1, and X1, Yi, Z1, defined by equations of the form

are the cartesian coordinates of two applicable nets N1 and Ñ1, which are T transforms of N and Ñ1 respectively; a net and a T transform

are such that the developables of the congruence of lines joining corresponding points of the two nets meet the surfaces on which the nets lie in these nets.2

Since N1 and N1 are applicable nets, we can obtain a cyclic system of circles C1, in the manner described in the second paragraph. Hence each pair of solutions of (2) determines a transformation of the cyclic system of circles C into a cyclic system of circles C1, such that the nets enveloped by the planes of the circles C and C1 are in the relation of a transformation T. Moreover, it can be shown that corresponding circles lie on a sphere. We say that two such cyclic systems are in relation T.

Darboux has stated the results of the second paragraph in the following form: If a surface S rolls over an applicable surface S, and Q is a point invariably fixed to S, the isotropic generators of the null sphere with center Q meet the plane of contact of S and S in points of a circle C which generate the surfaces orthogonal to the cyclic system of circles C. Making use of these ideas, we give the following interpretation of the above transformations of cyclic systems:

If N and N are applicable nets, and N1 and N1 are respective T transforms by means of (5), where 0' = Ex'2-Zx'2, the cyclic systems in which a point sphere invariably bound to N and N1 meets the planes of contact, as N rolls on N and N1 on N1, are in relation T.

It can be shown that the two surfaces orthogonal to these respective cyclic systems which are generated by the points where an isotropic generator of the null sphere meets the plane of contact are in the relation of a transformation of Ribaucour, that is, these surfaces are the sheets of the envelope of a two parameter family of spheres and the lines of curvature on the two sheets correspond.

1 Guichard, Ann. Sci. Ec. norm., Paris, (Ser. 3), 20, 1903, (202).

2 Eisenhart, these PROCEEDINGS, 3, 1917, (637).

3 Leçons sur la théorie générale des surfaces, vol. 4, 123.

NEW MEASUREMENTS OF THE VAPOR PRESSURE OF

MERCURY

BY ALAN W. C. MENZIES

DEPARTMENT OF CHEMISTRY, PRINCETON UNIVERSITY

Communicated by W. A. Noyes, October 14, 1919

Writing in 1908, Laby1 remarked "The vapor pressure of mercury is intrinsically important; it has been determined for a wider range of temperatures than that of any other substance. Yet the greatest-and, it should be added, unnecessary-disagreement is to be found in the current values for this vapor pressure, nor is there any table combining all the existing observations." Laby thereupon proceeded to collate the best observations available at that time and published a weighted average table.

In 1910 Smith and I published the results of 43 direct observations of the vapor pressure of mercury by a static method over the temperature range 250° to 435°. The pressure measurement was immensely facilitated by the use of the newly-devised 'static isoteniscope;' and a good form of platinum resistance thermometry was available. The average divergence in temperature of an individual observation from the smooth curve given by the equation

log p

= 9.9073436 3276.628/00.6519904 log 0,

(R)

whose constants were chosen to fit our results, was 0.050°. Because of the rather high degree of consistency thus attained, we were encouraged to exterpolate the curve in both directions and found that the values so obtained agreed remarkably well with the average experimental findings of those who had worked either above or below our temperature region.* A critical discussion of our own and of the older work on this subject, with tabular comparisons, may be found in the paper referred to.2

In 1909, Knudsen developed a relationship' connecting the weight of gas passing through tubes containing pierced diaphragms with the difference of gas pressure at the ends, the density of the gas, the resistance of the tube and diaphragm and the time of flow. He applied this to measure the vapor pressure of mercury from 0° to 50°, above which temperature his relationship failed to hold. From the lower to the upper end of this range, his results run from 10.9 to 6.2% respectively in pressure lower than the values given by extrapolation of our 1910 results

by the above equation. This corresponds to temperature discrepancies of 1° at 0° and 0.8° at 50°. One regrets that such an elegant method is not more direct. Later, Knudsen' applied his admirable 'absolute manometer,' under certain difficulties, to measure this vapor pressure from -10° to +24.4°, and obtained results ranging from about 5% higher to about 10% lower in pressure, at the respective ends of this range, than the values furnished by our equation.

In 1913 Villiers obtained a series of values from 60° to 100° which ran higher than ours by 2.7% of our calculated pressure at 60° to 7.5% at 100°.

In 1914, Haber and Kirschbaum, adopting a suggestion of Langmuir's,10 measured the damping of a quartz fiber vibrating in mercury vapor at 20°, and obtained a single value by an independent method. This value lies 3.8% in pressure below that given by our equation, or, otherwise, differs by 0.41° for the same pressure.

In glancing over the results obtained by these various workers, here stated baldly without criticism, one is inclined to believe that our equation of 1910 continues to stand the test of time by averaging their scattered results as well as can be hoped for. At the same time one is impressed by the pre-war activity in this field, especially in the lower temperature ranges. It appeared, therefore, to be of interest to obtain, by a method as direct as possible, new measurements below 250°, the lowest point of the range studied in 1910; and, in the work here described I have extended that range from 250° to 120°.

As the static isoteniscope is not especially well suited for the measurement of the rather low pressures with which we are here concerned, an entirely new plan was adopted. Two McLeod gauges of suitable capacities were calibrated and sealed to a central pressure reservoir. The gauges, which were operated by gas pressure and without rubber connexions, contained purified mercury. The reservoir and connected gauges could be charged with dry hydrogen at any pressure desired, and sealed off by mercury. The smaller of the two gauges was completely immersed in a riotously stirred oil bath whose temperature was measured by a mercurial thermometer whose thread was all submerged. The corrections of this thermometer were known" to tenths of a degree. The other, larger, gauge and the reservoir were maintained in baths at room temperature. On operating the two gauges simultaneously, different pressure readings were obtained, due to the condensation of the mercury vapor that contributed part of the total pressure in the hot gauge; and from this difference in reading the vapor pressure of mercury at the