is about 20° Centigrade. The value of Vas measured by the potentiometer is 24,413 volts in each case. We estimate that the error of precision in the product V sin @ amounts to about one part in two thousand.

Column 6 contains the values of h calculated from the values of V sin 0. Since several constants occur in these calculations (the velocity of light, the grating constant of calcite and the charge on the electron), the values of h are not quite as accurate as those of Vsin 0. We estimate the probable error in h to be about fifteen parts in ten thousand. This gives us an average

This value of h agrees with that previously published by Blake and Duane. It is, however, a fraction of one per cent larger than the value recently obtained by E. Wagner in a careful series of measurements.

In the experiments described above the X-rays left the target in a direction at right angles to the line of motion of the cathode particles. An interesting question has been raised recently as to whether the limit of the continuous spectrum would be altered if the rays came off at some other angle. To test this point with the accurate method of measuring the voltage which we now have, we made a series of experiments with an ordinary Coolidge tube (tungsten target) placed so that the X-rays that passed through the spectrometer's slit left the target at an acute angle of about 45° to the direction of the cathode stream. The results of this series of measurements appear in table 2. As in the previous experiments the voltage applied to the tube amounted to 24,413 volts. The X-ray tube had no thin mica window, so that the accuracy appears to be somewhat less than in the previous series of measurements. The value of V sin 0, however, does not differ from that for rays at right

angles to the cathode stream. There does not appear to be a Doppler effect for the short wave-length limit of the spectrum that amounts to as much as one part in two thousand. This agrees with Wagner's results.1 A very much more detailed report of this research is being published in one of the Physical Journals.

1 Duane and Hunt, Physic. Rev., Ithaca, Aug., 1915, p. 166.

2 See a report in the Jahrbuch der Radioahctivität, etc., for 1919 by E. Wagner, and

also a report on "Data Relating to X-Ray Spectra" by William Duane, published by the NATIONAL RESEARCH COUNCIL.

Blake and Duane, Physic. Rev., Dec., 1917, p. 624.

'See E. Wagner, Jahrbuch der Radioahctivität, 1919, also Physik. Zeit., Nov., 1920, p. 621; and C. Zecker, Ann. Physik. Leipzig, Sept., 1920, p. 28, also a note by D. L. Webster, presented to the American Physical Society at the same meeting at which the authors presented a note on this research, April, 1921.

THE U-TUBE ABSOLUTE ELECTROMETER

BY CARL BARUS

BROWN UNIVERSITY

Communicated May 25, 1921

1. Electrical Condenser.-Adjusting the wide shank of the shallow U-tube heretofore described (these PROCEEDINGS, 7, 1921, p. 71) with the top plates removed, so as to admit a metallic disc above the earthed mercury surface and parallel to it, the device becomes an absolute electrometer. The disc is perforated at the middle so that the component rays of the interferometer may reach the mercury. This instrument is chiefly useful in measuring electrostatic potentials. If p is the electric-pressure below the disc charged at potential difference V and h is the head of mercury resulting

V=d√8πp=d √8πhpg=d √4πλpg.n

where d is the distance between disc and mercury, p the density of mercury and λ the wave-length of light when n fringes correspond to V. Hence if d=1 millimeter, V=.315 √n els, units; or =95 √n volts.

2. Improved Apparatus.-The electrometer eventually took the form shown in figure 1 which gives the apparatus in connection with the electrophorus

and a commutating key similar to Mascart's. To put the mercury M to earth, a steel screw, S, which also carries a flat clamp for fastening one end of the earthed wire, has been inserted. This screw, S, has the further important purpose of damping the oscillations of the mercury M or M', by adjustably closing the channel m. The deflections can thus be made quite dead beat, which is an advantage. To level the electrodes C, C' (using a small spirit level placed on them) each has connecting rod d, which carries a clamp at one end, allowing the rod a, a' to slide up and down, rotate

around a, a' and d, d' and admitting of small displacements along d. At the other end of each rod is a flat vertical plate which is received in a fissure at the top of the corresponding hard rubber post k, k' and clamped. This gives a horizontal axis at right angles to the preceding. The lower ends of the posts k, k' are suitably clamped to the tubes t, t', attached to the body B of the electrometer. Here further motion along the tubes t, t' and rotation around them is possible. In this way it is not difficult to place C, C' symmetrically above the mercury pools and parallel to their upper faces, for experimental purposes. It is not sufficient, however, if precision is required.

Figure 1 shows the Mascart key below on the right, which consists of the elastic brass strips l, l', the earthed cross bar n above them and the cross bar q below. Thus the whole U-tube is earthed when not in use. The bar q is connected by wires with the brush A of the electrophorus shown on the left, so that when lor l'are depressed into contact with q, C or C', respectively, receives a positive charge while the other electrode and the mercury is earthed. This affords a very satisfactory means of commutation; for since AV=C√n=C'√n', the electrodes are so adjusted that C and C' are nearly equal.

=

Tests were made with this apparatus and a known AV = 173 volts. For example, the scale readings in the ocular on commutation were x = 34, x' = 17 so that (x − x') .7 (n+n'), as the fringe breadth was .7 scale parts. Thus AV = A √24, or A = 35.3 volts per fringe, initially. With large fringes and under quiet surroundings 3 or 4 volts could have been detected.

The upper face of the electrophorus p is on a vertical micrometer screw, insulated by the hard rubber connector h. The distance apart of þ, r and p' (to be denoted by d' and d") or any change of this distance (Y) are thus closely measurable in turns (mm.) of the screw. In a dry room this apparatus retains its charge Q very well and a great variety of fields are producible.

3. Equations. If we treat the case of the electrophorus as a closed cylindrical field of cross section A, and if Vo is the potential of the charged hard rubber surface, we may write

where Q', V' are the positive charge and potential in the top plate at a distance d' from the charged rubber surface at potential Vo and K' the specific inductive capacity of the dielectric medium. A similar equation holds if Q", V" are the charge and potential of the lower plate at a distance d" from Vo with a layer of specific inductive capacity K" between. If the two plates are put in contact, V' = V".

If the two plates thus charged are then insulated and the top plate is

moved normally towards the lower, a distance of y, the equations reduce to - AV 4TQd"y/A(K"d' + K'd") = const. √n,

=

AV being the potential difference thus produced and measured at the Utube electrometer taken as small in capacity in comparison with the electrophorus.

Q = Q' + Q′′. Hence, as a first approach, the y, n locus is a parabola. For instance in the following example the insulation loss amounted to not more than 2 fringes in 10 minutes at full charge. The pitch of the micrometer screw being .1 cm., the upper plate was conveniently discharged when d' 1 cm. above the hard rubber surface. Large fringes (about 1.5 scale parts) were installed. The fringe displacements (n) observed on lowering and raising the plate are shown in figure 2. The outgoing and incoming series practically coincide.

=

4. Specific Inductive Capacity.—In equation (7) if the space d' is filled with air, K' 1. On the other hand if a plate of some insulator like glass is inserted of thickness d'g

=

a

d' = d'e + d'a

where d' is the thickness of the air layer. Moreover if K, is the specific inductive capacity of the insulator

d'/K' = d'a + d'g/Kg

If, therefore, in the absence of the insulator, y is the downward displacement of the upper plate which gives the same fringe displacement n, and hence the same V as the insertion of the insulator plate, the resulting equations eventually reduce to

To determine the specific inductive capacity of a given insulating plate, the electrophorus is discharged at a convenient distance, d', between plate and hard rubber face. The insulator (K) is then inserted (noting the fringe displacement n) and withdrawn. The fringes must return to zero, showing that no charge has been imparted by the friction of the insulator. The upper plate is now depressed (y) on the micrometer screw until the same fringe displacement n is obtained. The operation is quite rapid; nevertheless the results so obtained were usually too large. Dielectric hysteresis was looked for, but could not have exceeded a fringe breadth.

5. Absolute Values.-The comparison of the U-tube with three different Elster and Geitel Electroscopes, the latter all standardized in volts, is given in figure 3 and is as linear within the reading error. The U-tube results were computed by equation A, measuring d from the mercury surface M' in figure 1 to the electrode C', with allowance for the K of glass plate. They are about four times too large. When, however, the measurement of d was made from the top of the glass plate to the electrodes, the results of the two instruments practically coincided. Hence the thin glass plate here acts like a conductor. The charge is transferred to its top face.

6

GEOMETRIC ASPECTS OF THE ABELIAN MODULAR
FUNCTIONS OF GENUS FOUR (I)

BY ARTHUR B. COBLE1

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS

Communicated by E. H. Moore, June 21, 1921

1. Introduction. The plane curve of genus 4 has a canonical series g3 and is mapped from the plane by the canonical adjoints into the normal curve of genus 4, a space sextic which is the complete intersection of a quadric and a cubic surface. If we denote a point of this quadric by the parameters t, T of the cross generators through it the equation of this sextic is F = (ar)3 (at)3 = 0. For geometric purposes we may define a modular function to be any rational or irrational invariant of the form F, bi-cubic in the digredient binary variables 7, t; for transcendental purposes it is desirable to restrict this definition by requiring further that this invariant, regarded as a function of the normalized periods wij of the abelian integrals attached to the curve, be uniform.

There seems to be an unusually rich variety of geometric entities which center about this normal curve. Some of these have received independent investigation. It is the purpose of this series of abstracts to indicate a number of new relations among these various entities and to connect each with the normal sextic F. The methods employed are in the main geometric. Direct algebraic attack on problems which contain nine irremovable constants, or moduli, is difficult. However much information is gained by a free use of algebraic forms containing sets of variables drawn from different domains. Both finite and infinite discontinuous groups are utilized at various times.

2. The Figure of Two Space Cubic Curves.-White2 has introduced for other purposes the interpretation of the form F = 0 as the incidence condition of the point of the space cubic curve C1(7) and the planet of the space cubic C2(t). There is dually an incidence condition of plane 7 of C1(7) and point t of C2(t), expressed by a form F= (A7)3 (At)3 = 0. We call the sextics of genus 4 determined by F = 0 and F = 0 reciprocal. Each is the same covariant of degree three of the other.

3. A. Set of Four Mutually Related Rational Plane Sextics.-On each of the cubic curves C1(7), C2(t) regarded as a point locus there is a net of point quadrics Q1, Q2, respectively; on each regarded as a plane locus there is a net of quadric envelopes, Q1, Q2, respectively. The net Q cuts the curve C2(t) in ∞ 2 sets of six points which lie in an I2(t). An I1⁄2 on a binary domain may be visualized as the line sections of a projectively definite (but not localized) rational plane sextic S2(t). Thus the four nets determine. the four rational plane sextics of the array