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Let e be an integer greater than 1 and let pbe a prime such that p ≡ 1 (mod e). Criteria for 2 to be a residue of degree e modulo p have been obtained in various forms for e = 2, 3, 4 and 5. Thus, Euler and Lagrange proved that 2 is a quadratic residue mod p if and only if p ≡ ± 1 (mod 8). Gauss showed that 2 is a cubic residue mod p if and only if p is representable as p = l2 + 27m2 with integers l and m, and that 2 is a quartic residue mod p if and only if p is representable as p = 12 + 64m2. Lehmer (5) and Alderson (1) have found similar but more complicated conditions for 2 to be a quintic residue mod p. There are analogous results about 3. For example, it follows from quadratic reciprocity that 3 is a quadratic residue mod p if and only if p ≡ ± 1 (mod 12), and Jacobi(4) showed that 3 is a cubic residue mod p if and only if 4p is representable as l2 + 35m2.

The paper considers the analysis of shallow and deep arches by curved beam finite elements, the interest in this problem being that it forms a limiting case of the shell problem. The individual elements are assumed to be shallow with respect to a local base line and different types of strain-displacement equations are utilised. Particular attention is focussed on the choice of the displacement patterns for the elements and models based both on independently-interpolated displacement components, of up to quintic order, and on assumed strain distributions are considered. Numerical applications using the various models show that of those tested only quintic-quintic models and assumed-strain models are generally satisfactory.

SummaryA simple formulation is established for the determination of vibration characteristics of pre-twisted beams by the matrix displacement method. An important point in the analysis is the assumption of complete polynomial displacement distribution in the principal directions. In particular, two twisted elements which employ quintic polynomials are introduced and used in the calculation of natural frequencies of pre-twisted beams for several boundary conditions. The results demonstrate rapid convergence.

Quintic designs, a class of PBIB designs having five associate classes, are defined. Properties of NN', where N is the incidence matrix, are examined to present necessary conditions for the existence of quintic designs. The variances of elementary treatment comparisons are found. The block structure of two types of quintic design is studied. Nous definissons les plans ‘quintic’, une classe de blocs imcomplets partiellement equilibres ayant cing classes associees. Nous etudions les proprietes de NN' ou N est la matrice d'incidence, dnecessaires pour que les conditions d'existence de tels plans soient satisfaites. Nous derivons lea variances des comparaisons de traitement elementaire, et etudions la structure de bloc de deux types de plans ‘quintic’.

The paper is concerned with the performance of curved-beam, finite element models of circular centre line in the solution of circular arch problems. Attention is concentrated on the selection of the assumed displacement patterns and the comparative efficiency of some relatively high-order, independently-interpolated models and of previously formula: ed models is investigated. Results of a number of applications are presented in the form of convergence curves for central displacement and of moment and force distributions throughout the arch. These results illustrate the great improvement which is gained by increasing the order of the assumed displacement components for independently-interpolated models from cubic to quintic.

Free axisymmetric vibrations of annular plates of non-uniform thickness have been studied on the basis of classical theory of plates. The fourth order differential equation governing the transverse motion is solved using the quintic spline interpolation technique. Characteristic equations for plates of linearly varying thickness have been obtained for three combinations of boundary conditions at the inner and outer boundaries. Frequencies, mode shapes and moments have been computed for different values of taper constant and radii ratio, for the first two modes of vibration.

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Introduction. By making use of the fact that all the dollble ratios of n things can be exprx3ssed rationally in terms of a properly chosen set of n-3 double ratios, E. H. MOOREX has developed the "cross-ratio group,' Cn!, a Cremona group in X_3 of order n! which is isomorphic with the permutation group of n things. H. iIC. SLAUGHT§ has discussed the C5! in considerable detail. Botll C5! and C6! have been listed by S. KANTOR.|| I have already pointed outS that C>! defines a form-problem which I shall denote by ]'CX,. The solutioll of PCnt carries with it the solution of the n-ic equation, and I have worked out ill detail the application to the quintic equation. After the adjunction of the square root of the discriminant of the n-ic, a new form-problem, PCn,lS, can be enunciated whose underlying group is the even subgroup C!l2 of Cn!. It appears in the particular case, n= 5, that the solution of PC5!/2 can be expressed ratiollally in terms of the solution of KLEIN'S A-problem.*$ The transition from the one problem to the other is accomplished by a very direct process which is suggested by simple geometric considerations. Explicit formulae are always desirable, axld these have been lacking hitherto in the solutions of the quintic equation in terlns of the A-problem. By illtroducing PCa!/2 as an intermediate stage I have obtained sucll formule from the long known invariant theory of the l)inary quintic.

Quintic splines of various end conditions are characterized by a system of linear equations suitable for iterative solution by blockunderrelaxation. Optimal relaxation parameters can be determined for arbitrary knots. Contrary to the direct methods known, this iterative method is numerically stable also for a large number of knots and for strongly varying distances of the knots. An ALGOL procedure and test examples are given. Spline-Funktionen vom Grad funf mit verschiedenen Randbedingungen werden durch ein Gleichungssystem charakterisiert, das iterativ mit Blockunterrelaxation gelost werden kann. Die optimalen Beschleunigungsparameter konnen fur beliebige Knoten bestimmt werden. Im Gegensatz zu bekannten direkten Methoden ist diese iterative Methode numerisch stabil auch fur eine groβe Anzahl von Knoten und stark variierende Abszissendistanzen. Eine ALGOL-Prozedur und Testbeispiele werden angegeben.

This paper deals with the subharmonic oscillations which occur in systems with nonlinear restoring force. It is first shown that the order of the subharmonics has a close connection to the form of the nonlinear characteristics. The subharmonic oscillation of order ⅓, i.e., the oscillation whose fundamental frequency is one‐third that of the applied force, is then particularly investigated for the cases in which the nonlinear characteristics are expressed by (1) cubic and (2) quintic functions. In both cases the stability problem of the periodic solution is discussed in detail according to the stability criterion given previously by the author. The analysis reveals that in the latter case (2) the second higher‐harmonic of the subharmonic, i.e., the oscillation of order ⅔ builds up with negative damping. This is verified by experiments in which the oscillation of order ⅔ causes the collapse of the original subharmonic oscillation.

In the previous paper by A. Meir and A. Sharma, error bounds for lacunary interpolation of certain functions by deficient quintic splines are developed. In this note, we extend their results to a wider class of functions and indicate that the extended results are best possible. In addition, a stability result for such interpolation is also presented.

where S*H°(I2) denote the symmetric algebra of H°(f2), is surjective. (2) The kernel I of ~p is generated by its elements of degree 2 and of degree 3. (3) I is generated by its elements of degree 2 except in the following cases: (i) C is a non-singular plane quintic (g= 6) (ii) C is trigonal (i.e., C is a triple covering of PI). (We refer the reader to (4.11), (4.12) and (4.13) for a special study of those exceptions).

There are infinitely many formulas of the form \[\int_{ - 1}^1 {f(x)dx = a_{ - 1} f( - 1) + a_0 f(0) + a_1 (1) + b_{ - 1} f''( - 1)b_1 f''(1)} \] that are exact for quintic polynomials, although, in general, there is no interpolating quintic through the six pieces of data. On the other hand, there is no corresponding formula for \[\int_{0}^1 {f(x)dx} \] exact for quintics.This paper discusses general questions of this type.

is a binary quantic of order m whose coefficients are variables. We study the functions of the a's and x's which are invariantive under the operation of transforming fm by G. Sections 1, 2, 3 deal with methods of constructing such functions, especial attention being given to the case p = 2. In ? 5 fundamental systems of first degree concomitants of the quartic and quintic are derived. Section 4 is devoted to a proof that if the systems of concomitants modulo 2 of the forms of orders 1, 2, 3 and the simultaneous systems obtained by combining forms of these three orders are all finite, then the system of fm is likewise finite. The hitherto undemonstrated theorem on the finiteness of the formal concomitants modulo 2 is thus reduced to a comparatively simple problem. In ? 6 there is proved a theorem on the reducibility of any covariant, modulo 2, of the binary cubic in terms of a set of fourteen invariants and covariants.

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In considering the conditions for the existence of given systems of equalities between the roots of an equation, we obtain some very interesting examples of the composition of relations. A relation is either onefold, expressed by a single equation U = 0, or it is, say k -fold, expressed by a system of k or more equations. Of course, as regards onefold relations, the theory of the composition is well known: the relation UV = 0 is a relation compounded of the relations U = 0, V = 0; that is, it is a relation satisfied if, and not satisfied unless one or the other of the two component relations is satisfied. The like notion of composition applies to relations in general; viz., the compound relation is a relation satisfied if, and not satisfied unless one or the other of the two component relations is satisfied. I purposely refrain at present from any further discussion of the theory of composition. I say that the conditions for the existence of given systems of equalities between the roots of an equation furnish instances of such composition; in fact, if we express that the function (*)( x, y ) n , and its first-derived function in regard to x , or, what is the same thing, that the first-derived functions in regard to x, y respectively, have a common quadric factor, we obtain between the coefficients a certain twofold rela­tion, which implies either that the equation (* )( x, y ) n = 0 has three equal roots, or else that it has two pairs of equal roots; that is, the relation in question is satisfied if, and it is not satisfied unless there is satisfied either the relation for the existence of three equal roots, or else the relation for the existence of two pairs of equal roots; or the relation for the existence of the quadric factor is compounded of the last-mentioned two relations. The relation for the quadric factor, for any value whatever of n , is at once seen to be expressible by means of an oblong matrix, giving rise to a series of determinants which are each to be put = 0; the relation for three equal roots and that for two pairs of equal roots, in the particular cases n = 4 and n = 5, are given in my on the Conditions for the existence of given Systems of Equalities between the roots of an Equation,” Phil. Trans, vol. cxlvii. (1857), pp. 727-731; and I propose in the present Memoir to exhibit, for the cases in question n = 4 and n = 5, the connexion between the compound relation for the quadric factor with the component relations for the three equal roots and for the two pairs of equal roots respectively. Article Nos. 1 to 8, the Quartic. 1. For the quartic function ( a, b, c, d, e )( x, y ) 4 , the condition for three equal roots, or, say, for a root system 31, is that the quadrinvariant and the cubinvariant each of them vanish, viz. we must have I = ae - 4 bd + 3c 2 = 0, J = ace - ad 2 - b 2 e + 2 bcd - c 3 = 0.

A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.

Surfaces of order n in space of n dimensions, for 3 ≤ n ≤ 9, were discussed by Del Pezzo, who showed that the prime sections of such a surface are represented on the plane by cubic curves through (9 − n) base points.

1. In 1891, Castelnuovo suggested that there may exist canonical surfaces which consist of three sheets covering the same surface. In particular, he remarked that, if the canonical curves of a surface contain a then the canonical model of the surface is certainly a three-sheeted surface. An example is the quintic surface with a tacnode, in space of three dimensions; its canonical model is a triple plane, branching along a curve of order 12 with twenty-four cusps which lie on a quartic.

Consider the pencil of [4]'s through the [3] which contains l15, l25, l34, any [4] of the pencil meets the V25 again in a conic, q.

Let D be the discriminant of an algebraic number field F of degree n over the rational field R. The problem of finding the lowest absolute value of D as F varies over all fields of degree n with a given number of real (and consequently of imaginary) conjugate fields has not yet been solved in general. The only precise results so far given are those for n = 2, 3 and 4. The case n = 2 is trivial; n = 3 was solved in 1896 by Furtwangler, and n = 4 in 1929 by J. Mayer [6]. Reference to Furtwangler's work is given hi Mayer's paper. In this paper the results for n = 5, that is, for quintic fields, are obtained.

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