The International Mineralogical Association's approved amphibole nomenclature has been revised in order to simplify it, make it more consistent with divisions generally at 50%, define prefixes and modifiers more precisely and include new amphibole species discovered and named since 1978, when the previous scheme was approved. The same reference axes form the basis of the new scheme and most names are little changed but compound species names like tremolitic hornblende (now magnesiohornblende) are abolished and also crossite (now glaucophane or ferroglaucophane or magnesioriebeckite or riebeckite), tirodite (now manganocummingtonite) and dannemorite (now manganogrunerite). The 50% rule has been broken only to retain tremolite and actinolite as in the 1978 scheme so the sodic calcic amphibole range has therefore been expanded. Alkali amphiboles are now sodic amphiboles. The use of hyphens is defined. New amphibole names approved since 1978 include nyböite, leakeite, kornite, ungarettiite, sadanagaite and cannilloite. All abandoned names are listed. The formulae and source of the amphibole end member names are listed and procedures outlined to calculate Fe3+ and Fe2+ when not determined by analysis.

In a former paper published in these Proceedings it was shown that an integral function of order less than 1 cannot have any asymptotic periods, and it was suggested that a function of order 1 can have at most a set Kω(k=±1, ±2, …). This was subsequently found to be the case. Meromorphic functions for which K, the exponent of convergence of the poles, is less than ρ, the order, behave in many ways like integral functions, so we should expect that (i) if 0≦κ<ρ1 there should be no asymptotic periods, (ii) 0≦κ<ρ1 either none or else a single sequence kω(k = ±1, ±2, …). It will be shown that this is so.

A common method of solving a linear differential equation consists in expressing the differential operator as a product of factors. The possibility of doing so has been studied extensively by Vessiot, following the work of Picard and Drach, on the lines of the Galois theory of algebraic equations. The analogous process of resolving a, linear differential system, consisting of an equation together with boundary conditions, into two or more systems of lower order does not seem to have been investigated. Such a resolution is not always possible, even in cases where the differential equation can be factorised. Thus the system

is equivalent to the systems

but the system

cannot be so resolved.

Let R be a (1 — 1) relation between the members of two similar classes A, B1. It correlates the members of a subclass X of A to the members of a certain subclass Y of B1 and thus defines a relation ρ connecting X and Y. It is clear that ρ is a (1 – 1) relation and that it has the property (M). If X1ρ Y1, X2ρY2, then implies

Theorem 1 needs very little explanation. It is the converse of the well known theorem that the indefinite integral F(x) of a function f(x) possesses a derivate on the right at every point at which f(x + 0) exists. If f(x + 0) does not exist, nothing can be said as to the existence or otherwise of F+(x); but in a general way we might expect that the integral of a function which oscillates comparatively slowly, say sin (log x) at x = 0, would be more likely to possess a derivate than that of a function which oscillates more rapidly, say . It appears from Theorem 1 that this is not by any means the case. In fact the integral of sin (log x) has not a definite derivate at x = 0 while that of has such a derivate.

The term “flat” is used to indicate that the minimum modulus of a function in a region is (in some sense) of the same order as the maximum modulus. Some properties concerned with this notion are described below. They came to light during an attempt to answer a question put to me by Professor Littlewood.

Theorem 1 of a recent paper “On the asymptotic periods of integral functions” can be replaced by the following more precise result.

If f(z) is an integral function of order ρ there is an integral function g(z), of order ρ, such that

The improvement consists in showing that, if ρ < 1, g (z) can be chosen to be of order ρ, not merely of order less than or equal to one.

A period of a function f(z) is defined to be a number w(≠0) such that

is identically zero; and it is not a difficult matter to show that an integral function may either have no periods or else a single sequence kλ, k = ±1, ±2, ….

1. The cardinal function is the interpolation function

which takes the values αr at the points α + rw. Its principal properties were discovered by Professor Whittaker, amongst others that

(A) When C (x) is analysed into periodic constituents by Fourier's integral-theorem, all constituents of period less than 2w are absent.

The generalised Riesz-Fischer theorem states that if

is convergent, with 1 < p ≤ 2, then

is the Fourier series of a function of class . When p > 2 the series (2) is not necessarily a Fourier series; neither is it necessarily a Fourier D-series. It will be shown below that it must however be what may be called a “Fourier Stieltjes” series. That is to say, the condition (1) with (p > 1) implies that there is a continuous function F (x) such that

An elementary proposition states that an absolutely convergent series is convergent, i.e. that if

then

The conception of the integral of one function with respect to another was introduced by Stieltjes in his classical memoir on continued fractions. He denned the integral as

and gave the formula for integration by parts.

UK legislation dictates that all pigs shall have access to straw or other material or object suitable to satisfy their behavioural needs (Welfare of Livestock Regulations, 1994). However, it is unknown how much straw must be provided to be behaviourally rewarding, and whether pigs’ prior experience of straw can impact upon their subsequent behavioural needs. Therefore, the aim of the current experiment was to investigate these issues by exposing pigs which did, or did not, have prior experience of straw to four levels of straw bed depth.