Dygogram is the name of a curve, invented by Archibald Smith and employed in his Admiralty Manual of the Deviation of the Compass, to give a graphic representation of the varying magnetic field of a compass as the ship is swung round in azimuth; a description is given of the Dygogram by Maxwell in Electricity and Magnetism, §441.

§1. Let (x)u denote the function

then

The concept of a laminated near-ring was introduced in [2]. We recall briefly what it is. Let N be a near-ring and let a∈N. Define a new multiplication on N by x * y = xay for all x,y∈N. With this new multiplication and the same addition as before we have another near-ring which we denote by Na. The near-ring Na is referred to as a laminated near-ring, the original near-ring N is the base near-ring and a is the laminator or laminating element.

In this paper we extend the results of Garba [1] on IOn, the semigroup of all partial one-one order-preserving maps on Xn = {1,…, n}, to POn, the semigroup of all partial order-preserving maps on Xn, A description of the subsemigroup of POn generated by the set N of all its nilpotent elements is given. The set {α∈POn:lim α/≦r and |Xn /dom α|≧r} is shown to be contained in 〈N〉 if and only if r≦½n. The depth of 〈N〉, which is the unique k for which 〈N〉 = N ∪ N2 ∪…∪ Nk and 〈N〉 ≠ N ∪ N2 ∪…∪Nk−1 is shown to be equal to 3 for all n≧3. The rank of the subsemigroup {α∈POπ|imα|≦/n − 2 and α∈〈N〉} is shown to be equal to 6(n − 2), and its nilpotent rank to be equal to 7n−15.

Let A be a complex Banach algebra with an identity 1. In this note we study the subset Λ of A consisting of all g ∈ A such that the spectrum of g, sp(g), contains at least one non-negative real number. Clearly Λ is not, in general, a semi-group with respect to either addition or multiplication. However, Λ is an instance of a subset Q of A with the following properties, where ρ(f) denotes the spectral radius of f (4, p. 30).

On donne un cercle et deux points P et Q situés sur un diamètre, on joint les points P et Q aux extrémités A et B d'ux diamètre du cercle par les droites I'A et QB qui se coupent au point M. On fait tourner le diamètre AB et on demande

I. D' étudier les variations du rappoet de construire la figure quand le rapport a unedonnée.

II. D' étudier les variations de l' angle AMB, et de construire la figure quand cet angle a une valeur donnée.

III. A′ et B′ étant les seconds points d' intersection des droites MA, MB avec la circonférence donnée, trouver le, lieu du centre du cercle circonscrit au triangle MA′B′.

In this paper we will study the properties of a natural partial order which may bedefined on an arbitrary abundant semigroup: in the case of regular semigroups werecapture the order introduced by Nambooripad [24]. For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1, 2] and further studied by Chacron [7]. Burmistroviˇ [6] investigated Sussman's order on separative semigroups. In the abundant case his order coincides with ours: some order theoretic properties of such semigroups may be found in a paper by Burgess [5].

and

denote two double binary (2–1) forms in (x, ξ). It is proposed to discuss the geometrical significance of their simultaneous covariant complete system, which is here quoted without proof.

It is easy to see (cf. Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of π-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be π-regular).

The Dickson polynomial Dn, (x, a) of degree n is defined by denotes the greatest integer function. In particular, we define D0 (x, a) = 2 for all real x and a. By using Dickson polynomials we present new types of generalized Stirling numbers of the first and second kinds. Some basic properties of these numbers and a combinatorial application to the enumeration of functions on finite sets in terms of their range values is also given.

1. This note gives an asymptotic evaluation of an integral of the form

as n tends to infinity, where is a sequence of real-valued functions. The theorem to be established is a natural extension of B. Levi's generalised Laplace-Darboux theorem (1, 341-51); it gives a rule for evaluating a wider class of asymptotic integrals.

The integrals of §§ 5, 6, and 7 of the following paper were first established by C. de la Vallée Poussin in a memoir Sur quelques applications de l'intégrale de Poisson (Ann. de la Soc. sc. de Bruxelles, vol. 17, 1892–3). An analogous integral to that of § 5 was discovered by A. Hurwitz, who seems not to have been aware of de la Vallée Poussin's memoir, and will be found under the title Sur quelques applications des series de Fourier in the Annales de l'École normale, vol. 19, 1902. In view of the value of these integrals for the theory of the Fourier series, the discussion now given, which follows different lines from those of previous proofs, may be of some interest. The discussion turns chiefly on the Second Theorem of Mean Value which is quite as applicable to Poisson's as to Dirichlet's Integral.

The question to be discussed is this:—

Can Aronhold's theorems on the bitangents of a quartic curve of genus three be extended to the tritangent planes of a space seztic of genus four?

The answer is “No”: decisiveness arises from the fact that the gist of Aronhold's results can be stated in a very simple form, namely:—

(a) Given seven lines in a plane it is possible to derive from them uniquely and symmetrically a quartic curve that has them for bitangents.

1. The six planes through the middle points of the edges of a tetrahedron perpendicular to the opposite edges are concurrent.

The sequence of orthogonal functions derived from Laguerre-polynomials is known to be complete, and hence closed, in L2 (0, ∞) if (a) > − 1. In a recent paper Dr Kober introduced a generalisation of this sequence, which enables him to extend the known results also for (a) < − 1. Kober's guiding principle seems to be the following one: The Laguerre orthogonal functions form, for (a) > − 1, a complete system of self and skew reciprocal functions of the Hankel transformation of order a. Now, if (a) < − 1, the ordinary Hankel transform has to be replaced by so-called cut Hankel transform. Hence the system of functions which has to replace-Laguerre orthogonal functions when (a) < − 1, should be a complete system of self and skew reciprocal functions of the cut Hankel transformation of order a, such that it reduces for (a) > − 1 (when the cut Hankel transform reduces to the ordinary one) to the sequence of Laguerre orthogonal functions. This, of course, is by no means a unique definition; nevertheless, together with what one would call the permanence of the Mellin transform, it enabled Kober to find a sequence of functions which (i) reduces to the sequence of Laguerre orthogonal functions when (a) > − 1, m = 0, (ii) is a complete set of self and skew reciprocal functions of the cut Hankel transformation with kernel Ja, m and (iii) has the required qualities of completeness and closedness.

The general equation of the second degree in two variables

can be brought by a direct process into the form

the determination of the constants ξ η l, m, n depending only on the solution of quadratic equations; so that the method is suitable for determining the foci, directrices, and eccentricities of conies with given numerical equations.

Recent research on aspects of distributive lattices, p-algebras, double p-algebras and de-Morgan algebras (see [2] and the references therein) has led to the consideration of the classes (n≧1) of distributive lattices having no n + 1-element chain in their poset of prime ideals. In [1] we were obliged to characterize the members of by a sentence in the first-order theory of distributive lattices. Subsequently (see [2]), it was realised that coincides with the class of distributive lattices having n+1-permutable congruences. This result is hereby employed to describe those distributive p-algebras and double p-algebras having n-permutable congruences. As an application, new characterizations of those distributive p-algebras and double p-algebras having the property that their compact congruences are principal are obtained. In addition, those varieties of distributive p-algebras and double p-algebras having n-permutable congruences are announced.