The paper is intended as an elementary introduction and companion to the paper on “Orthogonal Polynomials,” by G. J. Lidstone, J.I.A., vol. briv., p. 128, and the paper on the “Sum and Integral of the Product of Two Functions,” by A. W. Joseph, J.I.A., vol. lxiv., p. 329; and also to Dr. Aitken's paper on the “Graduation of Data by the Orthogonal Polynomials of Least Squares,” Proc. Roy. Soc. Edin., vol. liii., p. 54.

Following Dr. Aitken Σux is defined for the immediate purpose to be u0+…+ux−1.The scheme of successive summations is set out in the form of a difference diagram and is extended to negative arguments. The special point to which attention is drawn is the existence of a wedge of zeros between the sums for positive arguments and those for negative arguments.

The rest of the paper is for the greater part a study of the table of binomial coefficients for positive and for negative arguments. The Tchebychef polynomials are simple functions of the binomial coefficients, and after a description of a particular example and of its properties general methods are given of forming the polynomials by means of tables of differences. These tables furnish examples of simple, differences, of divided differences, of adjusted differences, and of a system of special adjusted differences which gives a very easy scheme for the formation of the Tchebychef polynomials.