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 I or 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 p, 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 g 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. GEOMETRIC ASPECTS OF THE ABELIAN MODULAR 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 gg 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, of the cross generators through it the equation of this sextic is F (AT)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. = The Figure of Two Space Cubic Curves.-White2 has introduced for other purposes the interpretation of the form F O as the incidence condition of the point of the space cubic curve C1(7) and the planet of the space cubic Ca(t). There is dually an incidence condition of plane 7 of C1(7) and point t of C2(t), expressed by a form F= (AT)3 (At)3 = 0. We call the sextics of genus 4 determined by F O 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 Q1 cuts the curve C2(t) in ∞ 2 sets of six points which lie in an I(t). An Ig 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 S1(T), S2(t) Two sextics in a row of the array will be called paired sextics; two in a column, counter sextics; and the other pairs, diagonal sextics. If any one of these sextics be given, its 12 spread out on a space cubic will determine the other space cubic and thereby the entire set of four. The nodal parameters of the paired sextics in the upper row are those of the ten common chords of C1, C2; in the lower row, those of the ten common axes of C1, C2. The equations of the sextics are = 0, (a a')2 (at) (a't) (ar)3 (a'r)3 = 0, (AA')2 (AT) (A'T) (At)3 (A't)3 (A A')' (At) (A't) (A7)3 (A'7)3 = 0, (a a')2 (a 7) (a'r) (at)3 (a't)3 = 0. Here the coefficients of the quadratics in t or 7 furnish three line sections of the respective sextic. The significance of the quadratic parameter appears in 6. 4. Two Birationally Related Quartic Surfaces.-The two nets Q1, Q2 of point quadrics on C1, C2, respectively, are apolar to a web of quadric envelopes Q; similarly the nets Q1, Q2 are apolar to a web of point quadrics, Q. The jacobians, J, J, of these respective webs are quartic envelope or surface, respectively; the first on the ten common chords, the second on the ten common axes of C1, C2. If we map by means of the web Qits space upon another space, the jacobian J, the locus of nodes of quadrics in Q, is mapped upon a surface Σ of order 16 and class 4, the Cayley symmetroid quartic envelope with ten tropes. The two cubic curves are mapped upon two paired rational space sextics R1(7), R2(t) which are conjugate to the paired rational plane sextics S1(7), S2(t), respectively, i.e., plane sections of the space sextic are apolar to line sections of the Conjugate plane sextic. The symmetroid Σ is the locus of planes which cut the sextic Ri in catalectic sections. Similarly the jacobian J counter to J is mapped by the web upon a point symmetroid counter to E, and C1, C2 upon rational space sextics R1(7), R2(t), counter to R1(7), R2(t), respectively, and conjugate to S1(7), S2(t), respectively. 5. References.-Meyer3 has discussed the relation of J to the sextic S2(t) and mentions the occurrence of counter sextics. Conner considers the mapping from J to Z and its connection with the paired rational sextics. The above introduction of the tetrad of rational sextics as defined by the sextics F, F of genus 4 is novel. Schottky, beginning with the abelian theta functions of genus 4, derives a set of ten points in space which are the nodes of a quartic surface and merely states a characteristic property of this surface by which it can be identified with 2. The writer has shown that can be transformed by regular Cremona transformation into only a finite number of projectively distinct symmetroids. These classes permute under the group (mod. 2) of integer transformations of the periods of the functions of genus 4. The analogous result for the plane rational sextic involves a subgroup of the group (mod. 2) of the periods of the functions of genus 5. This indicates a connection (which we seek) of the functions of genus 4 and those of genus 5. Proceeding the other way Wirtinger' obtains the plane sextic of genus 4 as the locus of vertices of diagonal triangles of a linear series gi upon a ternary quartic (p = 3). This transition will be discussed later. 6. The Covariant Conic R(T) of the Rational Plane Sextic S2(t).-From the existence in the net Q1 of a quadratic system of cones we conclude that the rational sextic S2(t) has a covariant conic K(7) such that the ten nodes of S2(t) determine upon K(T) the ten pairs of nodal parameters of the sextic S1(t) paired with the given sextic S2(t). This theorem furnishes the bond between ten nodes as a ternary figure and ten nodes as a binary figure on the rational curve. The equation of the sextic in Darboux coördinates referred to the norm conic K (7) is precisely that given in 3. 7. The Perspective Cubics of S2(t). The form (21). We denote by the symbol() an algebraic form of order i, in the variables of an Ski, of order is in the variables of an Sk2, etc. Unless explicitly restricted these sets of variables are digredient. Thus F = (ar)3 (at)3 is a form (33). By polarizing F into (atı) (AT2) (AT) (at)3 and replacing the pair of parameters 71, 72 by the point x which they determine in the plane of K(T) we obtain the (211) form T (πx) (dT) (dt)3 a general form of the orders indicated with nine absolute constants. For given this form determines a rational cubic envelope perspectives to the sextic S2(t), i.e., line t of the cubic is on point t of the sextic. The sextic is the locus of the meets of corresponding lines of any two of the ∞1 perspective cubics, and it has the equation (') (dd') (dt)3 (8't)3 = 0. Conversely given the sextic the family of perspective cubics is determined. Each cubic 7 has three cusps whose parameters are given by (π'") (dt) (d'r) (d"r) (ôô′)3 ("t)3 = 0. This is F = (AT)3 (At)3 whence the cusp locus, GC(7), is birationally general and of genus 4. The equation of the cusp locus is the determinant of the coefficients of (x) ('x) (dT) (d'T) (d') (dt) (ô't) regarded as a form bi-quadratic in 7, t. Thus GC(7) is a sextic whose six nodes are the points for which the first minors of the above determinant vanish and these first minors furnish the nine linearly independent quartic adjoints of GC(†). The curve of genus 4 has two special series gå, residual with respect to each other in the canonical series. These appear in the normal form as the triads on the two sets of generators of the quadric containing the sextic. One of these series on GC(7) is the triads of cusps of perspective cubics of S2(t). The web of adjoint cubics of GC(7) is furnished by the form (πx) (ñ'x) (π′′x) (dd′) (♪♪′) (8′8′′) (58′′)2 (8't) (d′′r) = 0, t and being variable with the cubic of the web. For fixed 7 and variable |