The following is a generalization of a theorem stated by Professor H. S. Uhler and demonstrated by the writer in the American Mathematical Monthly of October 1927.

Let bmn be a positive function of m and n which decreases steadily with n, so that bmn ≥ bm, n+1 for all values of m and n. Assume also that |am1 + am+2 + … + bmn| < K for all values of m and n, K being finite. Denote bySmn the sum of the first n terms in the first m rows of the double series

Let umn, vmn be functions of m and n, vmn being real and positive for all positive values of m and n. Suppose that either vmn increases steadily to infinity with n, or that umn both tend to zero (the latter steadily) as n → ∞, for any fixed value of m. Denote by wmn, and assume that wmn exists for every value of m, being denoted by lm. Then from Stolz' extension of a result proved by Cauchy, and an allied theorem, we have , for all values of m. It follows from Pringsheim's Theorem that if the double limit of exists, being l, then lm → l as m → ∞.

The operations of differentiation, ordinary and central differencing, divided differencing, and forming linear combinations of the results of these, have in common the distributive, associative and commutative properties, and the further property that when performed on a general polynomial of the nth degree they produce a polynomial of the n−1th degree, while when performed on a constant they yield zero.

The derivation of the 4-nodal Cubic Surface and the Quartic Surface, of which it is a particular case, are well known: certain new results of interest from the point of view of symmetry, and extension to n-fold space, are provided by the symbolic algebra. In particular a simple proof is given, in § 2, of the theorem that a symmetry exists between the four vertices of the tetrahedron and the fifth point whose locus is the cubic surface, and this property can be extended to the case of n + 1 points in n-fold space with one additional point.

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.

The definition here used of the Stieltjes Integral is the same as that of a previous note, viz.:—

Let f (x), φ (x) be two real functions defined in (a, b) a finite interval on the axis of the real variable x. Let Δ1, Δ2, …, Δn, be a finite set of sub-intervals which together make up (a, b). Δrφ denotes the increment of φ (x) in Δr. Let ξr be any point of Δr, and form the sum

Introduction. In the present paper a formula will be obtained to express a Ferrers' Associated Legendre Function of any integral degree and order as a sum of a finite number of Associated Legendre Functions of an order reduced by an even number. When the order is reduced by unity, an infinite series of the functions of reduced order is required. Thus a Ferrers' function can be expressed as the sum of a finite or infinite number of zonal harmonics according as the order of the function is even or odd.

§ 1 Introduction. The formulae

where n, q, m are positive integers, are well known. I am not aware, however, of any paper dealing with the corresponding definite integrals of products of two Ferrers' Associated Legendre Functions of different integral orders and degrees. With the help of the formulae given in my previous paper in this volume it is possible to discuss these integrals. I have found in the case of the first integral that if the orders of the two associated functions are both odd or both even and if, in each case, the degrees are both odd or both even or equal, the integral has usually a non-zero value. If, however, the orders are both odd or both even and if the degree of the function having the greater order is less than that of the other, or if the degrees are one odd and one even, the integral vanishes. Further, if one of the orders be odd and the other even, and the corresponding degrees are also even and odd (or odd and even), the integral has a non-zero value; in all other cases it vanishes. The values of the second integral can be obtained in a similar manner. An example of the methods employed and the results obtained is given in the following section.

Turnbull : On differentiating a matrix, p. 126, line 12-