This functional equation for Dedekind's

has been established in many different ways; however, it seems that the following proof, a straightforward application of the theorem of residues, has not been observed before. Since

it suffices to prove that

It is well known that every irrational number θ possesses an infinity of rational approximations p/q satisfying

It is also well known that there is a wide class of irrational numbers which admit of no approximations which are essentially better, namely those θ whose continued fractions have bounded partial quotients. For any such θ there is a positive number c such that all rational approximations satisfy

The following theorem is typical of a group of results which have been found to be of importance in the theory of Cesàro summability.

Theorem A. If 0 ≤ ρ ≤ κ (κ, ρ integer), p > −1,p+q −1 and

and ifsn = 0(np) (C, κ), thensn εn = 0(np+q) (C, ρ).

That these two subjects—the History of Science and the Psychology of Invention—are intimately connected with one another, is immediately evident and needs no explanation. Perhaps, however, it has not always been sufficiently appreciated. The recent Congress for the History of Science (Jerusalem, 1953) has given me an opportunity of trying to apply to the latter the data of the former.

Let q(x1; …, xn) be a positive definite quadratic form in n variables with real coefficients. Minkowski defined the successive minima of q as follows. Let S1 denote the least value assumed by q for integers x1 …, xn, not all zero, and let be a point at which this value is attained. Let S2 denote the least value assumed by q at integral points which are not multiples of x(1), and let x(2) be such a point at which this value is attained. Let S3 be the least value of q at integral points which are not linearly dependent on x(1) and x(2), and so on. We have

and it is easy to see that these numbers are uniquely defined, even though there may be several choices for the points x(1), …, x(n). The determinant N of the coordinates of the points x(1), …, x(n) is a non-zero integer. We denote by N (q) the least value of this integer (taken positively) for all permissible choices of the n minimal points, and by N′(q) its greatest value. Plainly N(q) and N′(q) are arithmetical invariants of q, that is, they are the same for two forms which are equivalent under a linear substitution with integral coefficients and determinant ±1.

The problem of the stability of a fluid rotating about an axis to an axisymmetric disturbance has been examined in the inviscid case by Rayleigh [1], who derived a simple criterion based on an analogy with the stability of plane stratified fluid of variable density. Later a complete discussion of the stability of viscous motion between rotating cylinders for small axisymmetric disturbances was given by G. I. Taylor [2]. More recently, the problem of magneto-hydrodynamic stability has claimed the attention of several workers, and, amongst other problems, the stability of a rotating fluid, when a constant magnetic field is applied in the direction of the axis of rotation, has been examined by Chandrasekhar [3]

As is well known the stability of viscous flow between two concentric rotating cylinders was first successfully treated both experimentally and theoretically by G. I. Taylor [1]. The mathematical problem underlying this classical investigation in hydrodynamic stability is the following:

The hydrodynamical equations allow the stationary solution

for the rotational velocity at a distance r from the axis of rotation, where A and B are constants related to the angular velocities Ω1 and Ω2 with which the inner and outer cylinders (of radii R1 and R2, R2 > R1) are rotated. Thus

where

In §§2, 3 a simple expression in finite terms is found for the small transverse displacement of a thin plane elastic plate due to a transverse force applied at an arbitrary point of the plate. The plate is clamped along, and is bounded internally by, the parabola CDE shown in Fig. 1.