The investigations of Professor Karl Pearson and his coadjutors on the applications of the theory of errors to statistics have greatly increased the importance and extent of the subject.

It is hoped that the following version of one of the methods of arriving at the so-called normal law of error may be of interest from an elementary point of view: affording, as it does, a simple application of the binomial theorem and some interesting graphical work.

The hopeless complexity, from the elementary didactic point of view, of the hitherto accepted accurate trigonometrical proofs of the power series for sin x, cos x has been so fully insisted upon in these pages (this volume, pp. 85, 86), that a more elementary method seems urgently called for.

Having established the expansion of functions without any reference to tests of convergence, we are able to enunciate the following simple test for the convergence of any series that may arise.

(We confine our attention to a series of which all the terms that we need to consider have the same sign.)