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We consider the shape of two point Hermite interpolation on the sphere. A three-parameter family of spherical rational quartic curves has been derived. We derive the singularity conditions theoretically and discuss the shape of the solutions, which is characterized by the presence of loops and cusps on restricted as well as whole segment. Our results can be used to develop a practical scheme for spherical Hermite interpolation, which would generate quartic spherical C/sup 1/ Hermite splines.

The eigenvalues of the pure quartic oscillator bounded by infinitely high potentials are studied. Compact recursion relations for the power series coefficients of the odd and the even wave functions are obtained which allow numerical values of the eigenvalues to be computed via an iteration scheme.

The charge-renormalization effects of quartic scalar interactions are calculated in a general gauge theory at the three-loop level using the Gegenbauer series expansion technique. The result agrees with a previous calculation by Curtright.

It is to be expected that a finite number of plane curves of order four should have seven given lines as bitangents, because the number of conditions imposed is equal to the number of effective free constants in the equation of such a curve, viz., 14. Aronhold made the interesting discovery that one curve could be determined in which no three of the given lines have their six points of contact on a conic. The method, due to Geiser, of obtaining the bitangents as projections of the lines of a cubic surface leads to a simple proof of the existence of this quartic.

The object of this note is to correct an error in my paper “Extensions of a theorem of Segre's…,” the notation used being the same. The curve C4 dealt with is regarded as given by its canonical representationand at one point in the paper we sought the locus of the lines analogous to the line A2A4 of the figure of reference for each of the ∞2 representations of this type (p. 671, small print). In the space representation of the locus there is an additional principal curveand the order of the locus must be reduced by that of the form corresponding to the points of this conic. The locus sought is in fact none other than the cubic form, locus of chords of C4, the present system of lines being the directrix systemt†. This follows at once from the following results, which can be shown immediately using the above representation:(1) The space joining such a line g to any tangent cuts the curve again in coincident points, and thus contains a second tangent;(2) The line joining the points of contact of these tangents meets g, and the points give the involution

The following is an easy method of representing geometrically the Sextic Covariant of a Binary Quartic which I have not seen given elsewhere. It was suggested to me by my work on Bi-Circular Quartics. The usual method of representing this Covariant is to regard the Quartic as four points on a conic, when the Sextic is represented by the intersections of the conic and the sides of the Harmonic Triangle of the Quadrangle whose vertices are the points of the given Quartic. The present method has the advantage of regarding the given Binary Quartic as a Tetrad of points on a straight line.

It is proved that there are exactly 8 8 imaginary cyclic quartic fields with class number 2 2 .

In this paper, it is proved that there are just seven imaginary number fields, quartic cyclic over the rational field, and having class number 1. These are the quartic, cyclic imaginary subfields of the cyclotomic fields generated by the fth roots of unity, where f is 16 or is a prime less than 100. This completes the list of imaginary, quartic, abelian number fields with class number 1. There are 54 such fields, with maximal conductor 67.163. In [5] , Uchida proves the following about imaginary, quartic abelian number fields of class number 1: (A) If the field is bicyclic, there are 47 such fields, (with maximal conductor 67.163) with possibly one more. (B) If the field is cyclic, the conductor must be less than 50000. The existence of another field in (A), depends, however, on there being an imaginary quadratic field with class number 2 and discriminant

We study the transformation produced on optical phase space by an arbitrary refracting surface. We show that this factorizes into two root canonical transformations, one representing the propagation from a reference plane to the refracting surface in the first medium, and the other, the inverse propagation back to the reference surface in the second medium. This factorization allows for a simple parameterization of the effect of a refracting surface to some aberration order, illustrated for cubic surfaces and Dragt’s Lie-algebraic description of a quartic surface and its Seidel aberration coefficients.

Introduction. It is well known that the plane sections of the Steiner Quartic Surface include all types to within projection of the rational plane quartic curve with simple singularities and that the invariants of any curve cut from this surface may be expressed as symmetric functions of the coefficients of the cutting plane.f If we vary this surface so that two of its double lines become coincident the plane sections are rational quartic curves with tacnodes; t if the three double lines are made coincident the resulting surface has as plane sections rational quartic curves with osenodes.? It is the purpose of this paper to express the invariants of the plane sections of these two surfaces in terms of the coefficients of the cutting plane.

In an earlier2 paper of the author, the Hilbert subgroup series in a normal algebraic number field N of degree 24 whose group is S4, the symmetric group of degree four, were studied. The study of those subgroup series gave rise to a study of the characterization of those fields N = B(VII, V,.s2), where B is normal of degree six with group S3, the symmetric group of degree three, ju and MA2 are quantities in B, and N is the field described above. In the present paper necessary and sufficient conditions are given that N = B(VII, VM2) have the prescribed properties. In ?4 an interesting result is adduced concerning the extension of totalimaginary normal fields B with group S3 to normal fields N with group S4. In the final section the question of the realization of the various entries in the tables3 of our earlier paper, is considered and it is shown that many of those cases are realizable. It must be mentioned, finally, that in the entire paper, with the exception of ?4, the reference field may be taken to be an arbitrary algebraic number field instead of R, the field of the rational numbers, as we do.

where X1, X2, X3, x4 are the homogeneous co6rdinates of a point of a 3-space S3 and xi,y, * * , X34 are those of a 9-space Sg transforms the points of S3 inito the points of an octavic variety V138 of order 8 and dimension 3 in Sq. To a plane of S3 corresponds a Veronese quartic surface t lying on V38 and to a quadric surf ace of S, corresponds a section of V38 by a hyperplane of Sq. Of interest is the surface J12 of order 12 on V38 which corresponds to a general cubic surface F3 of S3. It has 27 conics whose relative positions can best be studied by reference to the relative positions of the 27 lines on F 3. By representing F3 upon a plane q by means of the cc3 cubic curves through six general points P1, P2, * *, P6 of 4, we see that 4112 can be represented uponl 4 by means of the oo9 sextic curves with six nodes at Px [X = 1, 2, . . ., 6]. If we let (yi: Y2: y3) be the coordinates of a point of 4 and (p1(X) P2 () p3()) be those of the six points Px, we may take the equations

SOME years ago you published some interesting communications in regard to the graphical solution of cubic and quartic equations (vol. lxi. p. 55, vol. lxiii. pp. 515 and 609, vol. lxiv. p. 5). The solutions then given give only the real roots of the equation. I therefore take the liberty of directing attention to the following method, which gives the roots of cubic and quartic equations whether the roots are real or complex, and may be applied to equations of higher degree, with more complicated results.

A description of the factorization of a quartic polynomial over the field GF(2n) is given in terms of the roots of a related cubic.

PERHAPS the following serni-graphic solution of a quartic equation may be of some interest to your readers.

In the present article, we solve the center-focus problem for a class of quartic Kukles-like systems with third-order nilpotent singularities and prove the existence of five limit cycles in the neighborhood of the origin.