The purpose of this note is to solve a problem of Dr A. M. Sinclair. Denote by Aw(I, T) the algebra with identity generated by a bounded linear operator T in the weak operator topology. We prove the following result.

In 1950, Wintner (11) showed that if the function f(x) is continuous on the half-line [0, ∞) and, in a certain sense, is “ small when x is large ” then the differential equation

does not have L2 solutions, where the function y(x) satisfying (1) is called an L2 solution if

When points and lines are not specifically designated in the course of the following pages it will be understood that the notation for them is that recommended in the Proceedings of the Edinburgh Mathematical Society, Vol. I. pp. 6–11 (1894). It may be convenient to repeat all that is necessary for the present purpose.

Our starting point is the differential equation

where A(z) is a transcendental entire function of finite order, and we are concerned specifically with the frequency of zeros of a non-trivial solution f(z) of (1.1). Of course it is well known that such a solution f(z) is an entire function of infinite order, and using standard notation from [7],

for all , b∈C\{0}, at least outside a set of r of finite measure.

In Vol. XXXV. (Session 1916–17), Part I., of the Proceedings of the Edinburgh Mathematical Society, I discussed in considerable detail the properties of the Apolar Locus of two tetrads of points. I showed there that, subject to certain defined conditions, a unique quartic curve would be obtained, which would be the Apolar Locus of the two given tetrads. I mentioned, however, in §7 of the paper, that in the case when the two tetrads lie on the same conic, the above-mentioned conditions are not independent, and that, in fact, not a unique quartic but a pencil of quartics is obtained.

Hagen's proof (1837), as described in the 8th edition of Mansfield Merriman's “Method of Least Squares,” is based on the assumption that the error may be supposed to consist of the algebraic sum of an infinite number of infinitesimal errors of equal amount ε, each one of which is equally likely to be positive or negative. Thus if 2m is the number of the infinitesimal errors, the probability of the error x ≡ 2p ε occurring is

and the maximum value of P occurs when p = 0, and is

.

Suppose we are given a solid of revolution generated by a conic section. Slice out a frustum of the solid [14, diagrams pp. 77, 80]. Then, construct a cylinder, with the same height as the frustum, whose diameter coincides with the diameter of the frustum at the midpoint of its height. What is the difference between the volume of the frustum and the volume of this cylinder? Does this difference depend on where in the solid the frustum is taken?

The beautiful theorems which answer these questions first appear in a 1735 manuscript by Colin Maclaurin (1698–1746). This manuscript [14], the only original mathematical work by Maclaurin not previously printed, is published here for the first time, with the permission of the Trustees of the National Library of Scotland. (An almost identical copy [15] exists in the Edinburgh University Library.) In this work, Maclaurin proved that the difference between the cylinder constructed as above and the frustum of the given solid depends only on the height of the frustum, not the position of the frustum in the solid. When the solid is a cone, Maclaurin showed that its frustum exceeds the corresponding cylinder by one fourth the volume of a similar cone with the same height. For a sphere, the cylinder exceeds the frustum by one half the volume of the sphere whose diameter is equal to the height of the frustum; this holds, he observed, for all spheres. He derived analogous results for the ellipsoid and hyperboloid of revolution. Finally, for the paraboloid of revolution, he proved that the cylinder is precisely equal to the frustum.

Let P be any point on a bicircular quartic having A, B, C for foci; so that l. PA + m. PB + n. PC = 0, where l, m, n are known. It will be shown how the fourth focus E (lying upon the circumcircle of ABC) may be found; and also the relations subsisting between any three focal distances.

The resolution problem of the system

where U(t), A, B, D and Uo are bounded linear operators on H and B* denotes the adjoint operator of B, arises in control theory, [9], transport theory, [12], and filtering problems, [3]. The finite-dimensional case has been introduced in [6,7], and several authors have studied the infinite-dimensional case, [4], [13], [18]. A recent paper, [17],studies the finite dimensional boundary problem

where t ∈[0,b].In this paper we consider the more general boundary problem

where all operators which appear in (1.2) are bounded linear operators on a separable Hilbert space H. Note that we do not suppose C = −B* and the boundary condition in (1.2) is more general than the boundary condition in (1.1).

In the present note we shall obtain the expansion in a series of Legendre functions of the second kind of an integral function φ (ω) represented by Laplace's integral

where f (x) is an analytic function of x, regular in the circle

where an are constants and qn (ω) = in+1Qn (iω).

It was proved by Salmon (Geom. of three dimensions (1882), p. 331) that the chords of the curve of intersection of two algebraic surfaces of order m and n. which can be drawn from an arbitrary point,meet the curve upon a surface of order (m — 1) (n — 1); it was proved by Valentiner (Acta Math. 2 (1883), p. 191), and by Noether (Berlin. Abh. (1882), Zur Grundlegung u.s.w., p. 27), that the surface of order (m — 1) (n — 1) is a cone, with vertex at the point from which the chords are drawn; and a converse theorem was given by Halphen (J. de l' école Polyt. 52 (1882), p. 106). But the proofs given by Valentiner and Noether have not the elementary character that seems desirable, Noether's proof in particular depending on the theory of the canonical series upon the curve.

A straight line KK′ meets the circumference of a circle at two real or two imaginary points K, K′, and H is the middle point of the real or imaginary chord KK′. If A, B, C, D be any four points on the circumference, and the pairs of straight lines AB, DC, AC, BD, AD, CB meet KK′ at the pairs of points E,E′, F,F′, G,G′; then if any one pair of points be equidistant from H, the two other pairs will also be equidistant.

If a differential equation with meromorphic coefficients has a certain form where the growth of one of the coefficients dominates the growth of the other coefficients in a finite union of angles, then we show that this puts restrictions on the deficiencies of any meromorphic solution of the equation. We use the spread relation in the proofs. Examples are given which show that our results are sharp in several ways. Most of these examples are constructed from the quotients of solutions of w″ + G(z)w = 0 for certain polynomials G(z) and from meromorphic functions which are extremal for the spread relation.

Throughout the paper, T will be a Markov operator on C(X) (X compact T2), i.e. a continuous positive operator such that Te = e (e the unit function). P will be the set of Borel probability measures on X, which we shall often think of as linear functionals on C(X), and , where T' is the adjoint of T. Let