Direct geometical proofs of the addition theorem, and others allied to it, are often valid for only a limited range of values, and the constructions used are applicable to only one set of formulaæ. In the following paper a method is given which is valid for angles of all values and which is applicable to all the usual formulae involving sines, cosines, and their simple products. One of each set is proved: the others may be obtained either directly or by substitution in the usual manner.

1. The word contour is largely used in ordinary language, but its meaning, when so used, is in general very different from its meaning as a scientific term. We speak of the contour of a hill, a cloud, a country, and so on; meaning usually a profile or an outline,—sometimes a particular outline only. Yet, even in this popular use of the word, we have an indication of its more exact significance. Thus, we see that the visible horizon, if we consider it to be a contour line, is the curve in which the earth's surface is met by its tangent-cone the vertex of which is at the observer's eye. The tangent-surface has a constant characteristic; and it is this possession of a distinctive property by all surfaces which give rise to contour lines, which furnishes the reason for the peculiar applicability of the method of contours to physical problems.

In this paper I give some operational representations, according to the Carson-Heaviside Calculus, which, as far as I know, are new; and I deduce some properties of the functions thus introduced.

This paper considers the behaviour of a quotient map between Fréchet spaces concerning the lifting of bounded sets. The main result shows that a quotient map between Fréchet spaces that lifts bounded sets with closure (or equivalently such that its strong transpose is a topological isomorphism) must also lift bounded sets without closure.

We study the influence of the link structure of the prime spectrum of a Noetherian ring on the representation theory of the ring in the case that the ring satisfies the strong second layer condition and has exact integer Gelfand–Kirillov dimension. In particular, we show that Jategaonkar's density condition is satisfied and that the growth of an injective module is controlled by the growth of its first layer.

The number of the r-permutations of n things is the same as the number of ways of adding r things to a row which originally contains n - r other things.

For suppose the n letters A, B, C … N to be placed in a row, then each permutation consisting of r letters may be indicated by placing below those r letters in the row, the digits 1, 2, 3 … r, to indicate the order they have in the permutation. We may suppose zeros placed below all the remaining n - r letters. Thus it is clear that the number of the r-permutations of n things is the same as the number of ways in which r numbers can be added to a row originally containing n - r zeros.

Although only three of Whittaker's papers (12, 13, 14) are on the theory of automorphic functions, he retained his interest in it throughout his life. His son, Dr J. M. Whittaker, informs me that his last mathematical conversation with his father was on this subject. The standard English work on automorphic functions by L. R. Ford (4), and its precursor in the Edinburgh Mathematical Tracts series, owe much to Whittaker, and it was he who suggested the term “isometric circle” of which Ford makes such elegant use in his development of the theory. Professor G. N. Watson informs me that Whittaker had expressed his willingness to contributea a chapter on automorphic functions to the revised and expanded edition of Modern Analysis which is in preparation.

If a ∈ L(1, T) for every finite T>1, then we say that the infinite integral a(u)du is convergent with sum s if a(u)du = s. It is well known that a necessary and sufficient condition for a(u)du to be convergent (with some finite sum s) is that Cauchy's criterion,

holds. The object of this note is to obtain a similar result for summability (C, α) of a(u) du which reduces to Cauchy's criterion in the case of convergence. The corresti ponding problem for summable series has been treated by A. F. Andersen in (1).

The process of duplication of a linear algebra was defined in an earlier paper, where its occurrence in the symbolism of genetics was pointed out. The definition will now be repeated with an amplification. Although for purpose of illustration it is applied to the algebra of complex numbers, duplication will seem of no special significance if attention is fixed on algebras with associative multiplication and unique division; for duplication generally destroys these properties. The results to be proved, however, show that it is significant in connection with various other conceptions which appeared in the discussion of genetic algebras; namely baric algebras and train algebras (defined in G.A.), also nilpotent algebras, linear transformation and direct multiplication of algebras.

We suppose that f(t) is integrable in the Lebesgue sense m (π, π) and is periodic with period 2π. We denote its Fourier series by

Then the allied series is