The following problem is a slight generalisation of one posed partly solved by H. Nagler.1 We shall use A* for the conjugate transpose of a matrix A. A projection is an idempotent matrix, its latent roots consist of units and zeros.

The problem of designing cams or centrodes to produce any given motion in one plane is one of some practical importance; and it seems worth while to illustrate by examples some simple methods by which the solution can in certain cases be arrived at. These methods are founded, for the most part, on the use of the so-called Pedal Equation (or p-r-equation), which has great advantages in the present investigation, inasmuch as it depends on the form but not on the position of the ourve which it represents.

The curve in question is the non-singular intersection Γ of the n − 1 quadric primals

where it is presumed that no two of the n + 1 numbers aj are equal. Define

then it will be seen that the osculating prime of Γ at x = ξ is

In earlier volumes of the Proceedings of the Edinburgh Mathematical Society (cf. III., 104; IX., 83) it has been shown by researches of Dr. J. S. Mackay that the “discovery of the Wallace line … dates back only to about the year 1799 or 1800.” This result is reproduced in Cantor's Vorlesungen über die Geschichte der Mathematik (III., 542, 2te Aufl.). It was arrived at by considering the two following theorems given by Professor Wallace in the old series of Leybourn's Mathematical Repository:

Theorem A (Vol. I., p. 309 ; Vol. II., p. 54–5). If three straight lines touch a parabola, a circle described through their intersections shall pass through the focus of the parabola.

Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n,

where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4) Schönhage has shown that the constant ⅓ may be replaced by the larger number ½eγ, where γ is Euler's constant; this is achieved by means of a more efficient selection of the prime moduli used. Schönhage uses an estimate of mine for the number B1 of positive integers n≦u that consist entirely of prime factors p≦y, where

Herer x is large and α and δ are positive constants to be chosen suitably.

In the Appendix III. of his “Mechanics of Machinery,” Le Conte attacks the problem of rolling curves by an elegant analytical method in which the cartesian forms of the involute and the trochoids are derived from elementary differential equations. Weisbach(“Mechanics of Engineering and of Machinery,” Vol. III., chap. II.), while using mainly the geometrical method, discusses one application analytically (see §3 of this paper) in which polar forms are introduced. Many other writers, for example Barr (“Kinematics of Machinery,” chaps. III., IV.), deal wholly with the geometry of the subject.

It is shown that every spherically symmetric metric can be transformed into the isotropic form. As illustration an example is given.

The object of this note is to show that, in applying the methods of hypercomplex numbers to the theory of determinants, there is, for many purposes, no gain in using a particular number system.

1. If we write

where a is arbitrary, but s is a positive integer, then Vandermonde's Theorem is

Divide by , and put a for q, s + γ for p.

Let G be a locally soluble periodic group having a four-subgroup V. We show that if CG(V) is Chernikov then G is hyperabelian-by-Chernikov, if CG(V) is finite then G is hyperabelian.

When an elastic solid is subjected to a dynamical system of surface or body forces, not all of the work done by these forces is employed in deforming the material. The remainder is converted into heat energy producing a distribution of temperature throughout the body. Similarly the application of a surface temperature distribution, or the introduction of heat sources within the body, produces elastic as well as thermal effects. Thus we see that in the dynamical case there is a link between these two types of condition—thermal and elastic.

The initial-boundary value problem for the nonlinear heat equation u1 = Δu + λf(u) might possibly have global classical unbounded solutions, for some “critical” initial data . The asymptotic behaviour of such solutions is studied, when there exists a unique bounded steady state w(x;λ) for some values of λ We find, for radial symmetric solutions, that u*(r, t)→w(r) for any 0<r≤l but supu*(·, t) = u*(0, t)→∞, as t→∞. Furthermore, if , where is some such critical initial data, then û = u(x, t; û0) blows up in finite time provided that f grows sufficiently fast.

The following discussion of the analytical continuation of the hypergeometric function is believed by the writer to possess the advantages of brevity and simplicity.

Denote the integrals of Gauss's Equation

which are regular near 0,1, ∞, respectively by