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 (a a')2 (at) (a't) (ar)3 (a'r)3 = 0, (AA')2 (A7) (A'7) (At)3 (A't)3 = 0, (A A')' (At) (A't) (A7)3 (A'T)3 = 0, (a a')2 (ar) (a'T) (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, 2. 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 R¡ in catalectic sections. Similarly the jacobian J counter to J is mapped by the web upon a point symmetroid counter to Z, and C1, C2 upon rational space sextics R1(7), R2(t), counter to R1(7), R2(t), respectively, and conjugate to Si(7), S2(t), respectively. 5 5. References.-Meyer has discussed the relation of J to the sextic S2(t) and mentions the occurrence of counter sextics. Conner1 considers the mapping from J to 2 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(r) of the Rational Plane Sextic S2(t).—From the existence in the net Q 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(1) determine upon K(7) the ten pairs of nodal parameters of the sextic Si(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 (111). We denote by the symbol (2) an algebraic form of order is in the variables of an Ski, of crder 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 (ar1) (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 (1) form T (x) (dr) (ôt)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') (ôt)3 (ô't)3 = 0. Conversely given the sextic the family of perspective cubics is determined. Each cubic has three cusps whose parameters are given by (ππ'π") (dt) (d'r) (d′′r) (♪♪′)3 (8") = 0. This is F = (AT)3 (At)3 whence the cusp locus, & C(7), is birationally general and of genus 4. The equation of the cusp locus is the determinant of the coefficients of (x) (π'x) (dr) (d'r) (88') (dt) (8'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(7). The curve of genus 4 has two special series gi, 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 (x) (π'x) ("x) (dd′) (ôô′) (8′5′′) (ôô′′)2 (8't) (d"τ) = 0, t and being variable with the cubic of the web. For fixed 7 and variable T t we have the pencil of adjoint cubics on the cusp triad of the perspective cubic T. = The form (x) (dr) (dt)3 for fixed x and variable 7 is a pencil of binary cubic. This pencil has two linear combinants: a = (πx) (π'x) (dd1) (ôô') (dt)2 (d't)2 and b (πx) (π'x) (dd′) (ô8')3. The invariants i, j of the binary quartic a also are combinants. The invariant j is a sextic curved, the invariant i is b2. Hence the discriminant of a factors and the two factors b3d and 3 - d furnish the equations of the cusp locus GC (7) and the rational sextic S2(t). We conclude further that there are 12 perspective cubics of S2(t) with flex points at the meets of b and d. The sextics osculate at these points with the flex tangents as common tangents. The 12 flex points are the branch points on GC(7) of the function t(7) defined by F= 0. Thus a projective (but not a birational) peculiarity of GC(7) is that the 12 branch points of one of its series gi lie on a conic b. The parametric line equation of the conic K(7) on which the nodes of S2(t) determine the nodal parameters of S1(7) is of degree four in the coefficients of (213). Its symbolic form is (ÃÑ'Ñ′′) (ñ'''x) (d't) (ôô′)3 (ô"ô''')3 { (dr) (d"d''') + 2(d"r) (dd''') } = 0. With reference to the cubic space curves C1(7), C2(t) the point x determines an axis l, of C1(7) on planes 71, 72, 7 is the third plane of C1(7) on a point y of l; and t a plane of C2(t) on y. Then to points x on GC(7) there correspond axes of C1 on points of C2 and to the nodes of GC(7) the six axes of C1 which are chords of C2; to points x on S2(t) there correspond axes of C1 on planes of C2, and to the nodes of S2(t) the ten common axes of C1, C2. If x。 is a node of S2(t) the form (É。) (dÃ) (ôt)3 factors into (loT) (Not). (qot)2 where (qot)2 is the pair of nodal parameters. The ten forms (lo) (Not) will appear later in connection with the symmetroid. Other covariants of the (1) form are easily interpretable with reference to C1, C2. Thus a furnishes the four parameters t of tangents of C2 which meet the axis l, of C1, and b determines the axes l, of C1 which are in the null system of C2. From the definition of perspective curves the line t' of the perspective cubic (x) (dr) (ôt')3 will cut the sextic (ππ'E) (ôt)3 (8't)3 (dd') in the point tt'. On forming the incidence condition of line and point, removing the factor (tt'), and setting t't, we obtain = (ÑÑ'Ñ”) (♪♪′) (dt)2 (d't)2 (8′′t)3 (d'd") (dr) 10 which furnishes the seven contacts1ot of the perspective cubic with the sextic. This is a form (1) of general type containing nine absolute constants which will appear later. 1 This investigation has been pursued under the auspices of the Carnegie Institution of Washington, D. C. 2 H. S. White, these PROCEEDINGS, 2, 1916 (337). Meyer, A polarität und Rationale Curven, pp. 320-47. 4 Conner, Amer. J. Math., 37, 1915 (29). 5 Schottky, Acta Math., 27, 1903 (235). • Coble, Amer. J. Math., 41, 1919 (243). 7 Wirtinger, Math. Ann., 27, 1892 (261); Untersuchung über Thetafunctioners, Leipzig (1895). 8 Coble, Amer. J. Math., 32, 1910 (333). 9 Shenton, Ibid., 372 1915 (247). 10 Cf. Coble, l. c., p. 352. MELANOVANADITE, A NEW MINERAL FROM By WALDEMAR LINDGREN Department OF MINING, METALLURGY AND GEOLOGY, MASSACHUSETTS INSTITUTE OF TECHNOLOGY Communicated, March 9, 1921 Late in 1920 Mr. W. Spencer Hutchinson, Consulting Engineer for the Vanadium Company of America, brought to my attention three specimens of a mineral collected by him at Mina Ragra, Peru. He suspected that it was a new mineral, and this opinion was proved correct by chemical and optical examination. The formula is 2CaO. 3V2O5.2V204 and I wish to propose for it the name of Melanovanadite, in allusion to it being practically the only vanadium mineral of a deep black color.1 The mineral occurs in acicular bunches on black brecciated shale, the individual crystals being at most 3 mm. long. The greater thickness of the needles is about 0.5 mm. ranging down to 0.1 and 0.01 mm. The color is black, luster almost submetallic, streak very dark reddish brown. The hardness is 2.5 the specific gravity 3.477 at 15° C. The habit of the crystals is prismatic, parallel to c, with monoclinic symmetry. The principal faces consist of a flat, striated prism, the longer diagonal being parallel to the b axis, minor pinacoidal faces, and usually well developed terminal faces of pyramids and smaller domes. The crystals have a perfect cleavage parallel to (010). Under the microscope the crystals remain black except in very thin prisms which are translucent with brown color. Flat cleavage pieces parallel to the clinopinacoid only become translucent when the thickness is about 0.003 mm. and then show maximum extinction of about 15°. Resting on the prism (100) the crystals become brown translucent with a thickness of about 0.03 mm. and then show lower extinctions of 12° to 13°, while these resting more nearly on the orthopinacoid extinguish at lower angles. The perfect cleavage being perpen1 The ending "vanadite" is an obsolete form of "vanadinite," but there can scarcely be any objection to using this form in the present case. dicular to the poorly developed orthopinacoid, extinctions of 0° are rarely seen. Exact optical measurements are difficult on account of the deep color. Obscure hyperbolae show on the prism faces and it is probable that the plane of the optic axes lies parallel to the perfect cleavage (010). The absorption is very strong. The a ray is visible with dark yellowish brown color through the prism faces and the orthopinacoid and has according to a determination kindly made by Professor C. H. Warren a coefficient of refraction of 1.74; while the ẞ and y rays are somewhat higher but cannot be measured exactly on account of the strong absorption. The ẞ and y rays therefore lie in the 010 plane of perfect cleavage and their absorption is so strong that such cleavage pieces only become translucent in extremely thin plates, with dark reddish brown color, B and y differing slightly in depth of tint. The double refraction is strong. The material was analysed by Professor L. F. Hamilton of the Massachusetts Institute of Technology, who reported the following composition: Assuming the alumina, iron oxide, magnesium and silica to be impurities from the admixed shale the analysis may be recalculated to: V,Os : V2O4 : CaO = 302 : 209 : 184 or approximately 3:2:2. The formula would therefore be |