The purpose of this note is to present new proofs of two classical theorems in ideal theory, both due to W. Krull: the intersection theorem (see Krull(1) for the original proof, and Northcott (2), Chapter III, where the theorem is stated in the form proved below), and the principal ideal theorem ((1), or (2), Chapter III, Theorem 6). Both proofs depend on a device introduced by the author in (3) in another connexion.

A new method is described for measuring the surface impedance of metals at low temperatures and at radio-frequencies. Using this method, the surface impedance of normal tin and the penetration depth λ in superconducting tin have been studied at about 9 Mc/s frequency as a function of impurity content. The measured surface impedances agree well with the values expected, and the penetration depths increase with impurity content, in confirmation of Pippard's observations (14) at 9400 Mc/s. The variation of λ with reduced temperature t is well represented by the equation λ/λ0 = (1 − t4)−½, in impure as well as in pure tin. The absolute value of λ found for pure tin is about 4 × 10−6 cm., significantly lower than the values previously found.

The complex variable method is applied to obtain solutions for the deflexion of a supported circular plate with uniform line loading along an eccentric circle under a general boundary condition including the clamped boundary , a boundary with zero peripheral couple , a boundary with equal boundary cross-couples , a hinged boundary and a boundary for which , η being Poisson's ratio. These solutions are used to obtain the deflexion at any point of a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre. Expressions for the slope and cross-couples over the boundary and the deflexions at the centres of the plate and the loaded patch are obtained.

In the expression of Taylor's theorem

there are three well-known forms of the remainder R(n, a, b). The first is the Schlömilch form

Let Λ denote the integral lattice in n-dimensional Euclidean space. A classical theorem of Minkowski's states that any bounded closed convex region K symmetrical in the origin 0 and with volume 2n contains a point of Λ other than 0. There will be a lattice point other than 0 in the interior of K except when K has certain forms, of which we will denote an arbitrary one by K*. An example of a K* is the cube |xi| ≤ 1 (i = 1, 2,..., n), and more generally a famous theorem of Hajós (3) states that if K* is a parallelepiped it is defined (except for integral unimodular transformations of the x's) by inequalities of the form |x1| ≤ 1, |a21x1 + x2| ≤ 1, …, |an1x1 + … + an, n-1xn-1+xn| ≤ 1.

The following theorem was stated and proved by Krull(l). We give a new and simple proof.

Theorem. Let K be afield with valuation υ, value group Γ and residue class field R.