Birkhoff in [2] poses the following problem:

“Problem 73. Find necessary and sufficient conditions in order that the correspondence between the congruence relations and the (neutral) ideals of a lattice be one-one”.

This problem has been solved by Areškin [l] and Hashimoto [3]. Essentially the conditions reduce to the r e quirement that the lattice be a generalized Boolean algebra.

Hesse's theorem states that “if two pairs of opposite vertices of a quadrilateral are respectively conjugate with respect to a given polarity, then the remaining pair of vertices are also conjugate ”.

In the real projective plane there cannot exist such a quadrilateral, all four sides of which are self-conjugate [1, §5.54]. We shall show that such a quadrilateral exists in PG(2, 3), and that any geometry in which such a quadrilateral exists contains the configuration 134 of PG(2,3). We shall thus provide a synthetic proof of Hesse1 s theorem for a quadrilateral of this type, which, together with [1, § 5.55], constitutes a complete proof of the theorem valid in general Desarguesian projective geometry. We shall also show analytically that a finite Desarguesian geometry which admits a Hessian quadrilateral all of whose sides touch a conic must be of type PG(2, 3n).

This paper continues the work begun in [l] and examines the loss of power when using tests based on the assumption that the variable being sampled has a “complete” normal distribution, when in fact, sampling is from a symmetrically truncated distribution. The hypothesis considered here is the one - sided test for the variance of a normal distribution. Some tables have been computed and they show that appreciable losses in size occur. Some loss occurs in the power too, but this decreases with the alternative value of the variance and the degree of truncation.

In [1] G. Birkhoff stated an algorithm for expressing a doubly stochastic matrix as an average of permutation matrices. In this note we prove two graphical lemmas and use these to find an upper bound for the number of permutation matrices which the Birkhoff algorithm may use.

A doubly stochastic matrix is a matrix of non-negative elements with row and column sums equal to unity and is there - fore a square matrix. A permutation matrix is an n × n doubly stochastic matrix which has n2-n zeros and consequently has n ones, one in each row and one in each column. It has been shown by Birkhoff [1],Hoffman and Wielandt [5] and von Neumann [7] that the set of all doubly stochastic matrices, considered as a set of points in a space of n2 dimensions constitute the convex hull of permutation matrices.