The object of this paper is to introduce the differential operator, ▽, generalised for a Riemannian space Vn immersed in a flat space Vp, and then to discuss the general small deformation of Vn.

The object of this paper is to discuss conditions for a maximum or minimum of functions of integrals of the type

employing the methods of the Calculus of Variations.

When a given frequency distribution is to be graduated, it is customary to express the constants of the fitted curve in terms of the moments of the frequency distribution. The rth moment of the distribution, in which the relative frequency of a measure x is φ(x) or, in the case of a continuous variable, the differential of frequency is φ(x)dx, is defined in the respective cases by

the summation or integration being taken over the whole range of possible values of x. In the present paper we make use of another kind of moment, the factorial moment, which has already been considered by several writers, and which is specially suited to the case when the frequencies of the distribution are given for discrete, equidistant values of the variable. The (r + 1)th factorial moment, for the case where x, measured from some arbitrary origin, can increase by increments h, 2h, 3h,.…, will be defined to be

where the summation extends over all possible values of x; it will be denoted by m(r+1). By a suitable choice of scale the increments of x may be taken as equal to unity in any given case.

§1. Introduction. Care is needed in dealing with determinants whose elements are subject to experimental error, particularly when a determinant itself is small compared with its first minors. For, as these examples show, a relatively tiny error in one element may be responsible for a large error in the determinant

§ 1. Introduction. The algebra of quantum mechanics is characterized by the fact that the variables p, q obey all the laws of ordinary algebra except that multiplication is non-commutative and instead there exists a relation of the form

where c is a real or complex scalar constant and is thus commutative with both p and q.

§ 1. Consider the system of n first order linear differential equations:

together with the n boundary conditions

where aij, bij are constants and where we assume for simplicity of notation that gii = 0.

§1. A series

has been defined by J. M. Whittaker to be absolutely summable (A), if

is convergent in (0 ≤ x < 1) and f (x) is of bounded variation in (0, 1), i.e.

for all subdivisions 0 = x0 < x1 < x2 < . … < xm < 1.

It is well known that the square matrix, of rank n−k + 1,

which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.

The object of the present note is to show that a well-known theorem in the theory of non-linear partial differential equations, which is usually proved analytically, admits of a geometrical proof which exhibits the relations concerned in a more intuitive manner.

§ 1. Copson and Ferrar have proved the following theorem:

Ifis a divergent series of positive terms, and (pn + qn) un−pnun−1 = ynalways, where yn → 0, then un → 0.