In the Euclidean theory of areas, where convex polygons alone are considered, there is no question as to the sign of an area. The element of area is the rectangle, and an area is signless, or always positive.

It is well-known that if v≧–½ and

then ø (x) is said to be self-reciprocal in the Hankel transform and may be described as Rv. If v = ±½(1) reduces to the Fourier sine or cosine transform. Functions of these two classes may be described as Rg and Re.

In a list of pairs of reciprocal functions, G. A. Campbell gives the example that

is Rc, but there does not seem to be any explicit reference to this function in the literature. It suggests that there is a corresponding function which is Rv.

W. H. Cornish (2) has investigated congruences on pseudo-complemented distributive lattices and has identified those ideals (resp. filters) that are congruence kernels (resp. cokernels). In this paper we show that many of the principal results concerning congruence kernels and cokernels hold in a semilattice and therefore do not depend on distributivity, nor on the existence of unions.

Three main aws regulate the treatment of ordinary algebraic quantities. These are the Associative Law, the Distributive Law, and the Commutative Law. If a, b, c, … , represent quantities dealt with in the algebra, the associative law of multiplication asserts that a(bc)=(ab)c, where the brackets have the usual meaning that the quantity within them is to be regarded as a single quantity: the distributive law of multiplication asserts that (a + b)(c + d)=ac + bc + ad + bd: and the commutative law gives ab = ba. With regard to addition, the associative law asserts that (a + b) + c = a +(b + c): and the commutative law gives a + b = b + a.

1. The question, “How, from a given function which is self-reciprocal for a transform of a particular order, can we construct other functions which are self-reciprocal for transforms of different orders?” was first raised by Hardy and Titchmarsh who gave some rules for constructing such functions. Following their method, I have shown, in a recent paper, that there are certain general theorems of the following type:—

If f (x) is its own Jμtransform, g (x) is its own Jv transform. In this note I add a few more such theorems, the interest lying mainly in the results themselves and not in a rigorous proof thereof; and hence only the formal procedure is given here.

Let R be a ring with identity and let Ω be a totally ordered set. Let Ω′ be a totally ordered set which is disjoint from and equipotent to Ω′ with ′:Ω→Ω′ an order preserving bijection. Define Ω1=Ω∪Ω′ and let Ω1, be totally ordered by inheriting the order from Ω and Ω′ and with ω<λ for all ω∈Ω and λ′∈Ω′. Let M be the free R-module R(Ω1).(We define the alternate bilinear form (*, *) on M by

In the theory of Electrostatics, or of the Newtonian potential, there exists between two systems of potentiating matter, a wellknown reciprocal relation, analytically expressed in the proposition known as Green's Theorrn. By applying his theorem to the case when one of the systems is of the simplest possible character, namely, a mass concentrated at a single point, Green deduced a general method of solving the equation for the potential. The idea of a similar general method of dealing with the equations of Elasticity is due to Professor Betti, of Pisa, who has proved a reciprocal relation between two states of strain of an elastic solid, analogous to the relation in Electrostatics referred to.

Ever since Mendel promulgated his famous laws, probability theory and statistics have played an important role in the study of heredity (9). Etherington introduced some concepts of modern algebra when he showed how a nonassociative algebra can be made to correspond to a given genetic system (1, 4). The fact that many of these algebras have common properties has led to their study from a purely abstract point of view (2, 3, 5, 6, 11, 12). Furthermore, the techniques of algebra give new ways of attacking problems in genetics such as that of stability.

§1. Appell's functions, P (θ, φ), Q (θ, φ) and R (θ, φ) are defined by the expansion

where j3 = 1, affording, both for the third order and the field of two variables, a very direct generalization of the circular functions, as

When a function ƒ(x) possesses an asymptotic series

this series provides a useful means of evaluating ƒ(x) for large values of x. The usual procedure is to sum all the terms in S(x) up to, but excluding, the term of smallest magnitude. The degree of accuracy obtained by this method cannot normally be improved by direct summation of S(x), but sometimes better accuracy can be obtained by using one of the familiar devices for accelerating the convergence of series. Simple δ2-extrapolation may be successful, and Rosser (1) and others have used the Euler transformation to some effect. The method given here provides, in suitable cases, a more effective means of evaluating ƒ(x) from the series for a wide range of values of x.

1. The Pole and Polar Theorem for the Circle may be proved as follows:—

Let P be any point, and C the centre of a circle.

Let AB be the diameter through P, and QQ′ any chord passing through P.

Let AQ, BQ′ meet in M; AQ′, BQ in O; MO, AB in N; and MO, QQ′ in P′.

There are certain well-known linkages for effecting the transformation of inversion, and, incidentally, by inverting a suitably situated circle, for producing straight-line motion. Reference may be made to the classic lecture by A. B. Kempe, “How to draw a straight line” (London: Macmillan, 1877). It is the object of this paper to call attention to the fact that these linkages have the corresponding property in non-euclidean geometry.