In the Proceedings of the National Academy of Sciences of the U.S.A., Vol. II. (1916), page 171, Professor F. Morley has established a theorem which both extends and simplifies the theorem of Feuerbach, viz., All curves of class three which (i) touch the six lines OP, OQ, OR, QR, RP, PQ joining four orthocentric points, O, P, Q, R, and (ii) pass through the circular points, also touch the common nine-points-circle of the triangles PQR, OQR, ORP, OPQ. Sixteen of these curves of class three break up into one of the four points aud a circle touching the sides of the triangle formed by the other three. Thus the sixteen instances of Feuerbach's theorem derivable from the four triangles are included as special cases, in Morley's theorem. A purely geometrical proof of the theorem may be worth consideration.

Many proofs of the Binomial Theorem have been given, and the proof which I give in this note has not the slightest claim to be considered new. My only object in giving it is to call attention to the fact that it depends merely on the rule of integration by parts, and provides a form of the remainder that in many applications is much simpler than any of the forms associated with Taylor's Theorem. Of its usefulness in this latter respect I give an example from the asymptotic expression for a Bessel function.

Gray and Mathews, in their treatise on Bessel Functions, define the function Kn(z) to be

We shall denote this function by Vn(z). This definition only holds when z is real, and R(n)≧0. The asymptotic expansion of the function is also given; but the proof, which is said to be troublesome and not very satisfactory, is omitted. Basset (Proc. Camb. Phil. Soc., Vol. 6) gives a similar definition of the function.

A curve in n dimensions may be taken as the limit to which a polygonal figure ABCDE … tends, when the sides AB, BC, CD, etc., all diminish towards the limit zero.

The curvature at A is .

Hagen's proof (1837), as described in the 8th edition of Mansfield Merriman's “Method of Least Squares,” is based on the assumption that the error may be supposed to consist of the algebraic sum of an infinite number of infinitesimal errors of equal amount ε, each one of which is equally likely to be positive or negative. Thus if 2m is the number of the infinitesimal errors, the probability of the error x ≡ 2p ε occurring is

and the maximum value of P occurs when p = 0, and is

.

Laguerre has shown that if be expanded in ascending powers of t,

where fn(x) is a polynomial of degree n, satisfying the differential equation

The general interpolation series, originated by Newton, has been studied mainly for its algebraic interest, only the special case of equidistant data being developed on the practical side. This is justified by the simplicity of this case, and by the numerous problems for which it suffices, but it may lead to undue simplification of data and to restrictions on experimental and computative methods. Thus tables of functions which are not in common use, or which are carried to many places, must often be limited to relatively few entries, and these might conceivably be not in arithmetical progression, with advantage both of easier tabulation and of more accurate interpolation. Data from experiment or statistics, again, are often fitted to a parabolic curve of arbitrarily chosen degree, and on rather inadequate grounds. The formation of a difference-table not only avoids the suppression of the original data, but supplies at a glance a useful analysis of them—indicating their consistency and regularity, showing with what accuracy a parabolic curve can represent them, and supplying its expression with minimum labour. For direct interpolation to new points Lagrange's formula, the usual alternative, fails in this respect and, when applied to unfamiliar data, is very apt to mislead. It is wasteful of labour and more liable to error, and cannot easily be extended to include fresh terms.