Formulae are established below for

for both odd and even integer values of r(r>1), and x≥0; also for

this series reducing to when x is an integer

In 1920 I established the formula, a most useful one,

where γ is Euler’s constant

0·57721 56649 01532 86060 6,

and in 1920 gave a simpler proof with extension to higher powers of x(x + 1) with formulae for Σ2n+1, but not for Σ2n.

If anything were calculated to make me anxious to do justice to my theme to-night it would be the association with your society of the men to whom I owe my earliest introduction to dynamics—at St. Paul’s School to Mr Pendlebury, your Secretary, and at Trinity College, Cambridge, to Professor Forsyth, your incoming President. The interest that they implanted has survived for half a century; and the applications to sport that I propose to describe are the immediate outcome of that interest.

In response to Dr. Lidstone’s letter in the May number of the Gazette, I have solved the equation

z 2−(2+i)z + (4+3i)=0,

a quadratic equation with complex coefficients chosen at random, by a process, or rather a succession of processes, of complete generality.

If P is a point of intersection of two conics which have one common focus, then e1 |PM1| =e2 |PM2|. It follows that if the two directrices cut in O, then OP passes through a second point of intersection, and that if directions of measurement perpendicular to the directrices are assigned, the equations e1 . M1P = ±e2M2P give two of the chords of intersection, we may say that these are the chords associated with the common focus.

Several proofs, based on projective considerations, have been given recently of the Petersen-Morley theorem. In this note a direct metrical proof is given, based on the dual vectors of Study, which seems simpler than any of these.

In Hilton’s Plane Algebraic Curves the functions F(t) and F′(t) are used in finding inflexions and cusps of plane curves. These functions are defined, respectively, by the determinants

where f(t), ɸ(t), ψ(t) are the values, in terms of a parameter t, of the homogeneous coordinates of a point on the curve. It is shown that F(t1) vanishes if t1 is the parameter of an inflexion or a cusp and F′(t1), the derivative of F(t1), vanishes if t1 gives a cusp.

From the first inception of vector analysis, the subject of notation has engaged the attention of many writers, but up to the present there has, unfortunately, been no sign of general agreement upon a standard notation.

It is proposed to discuss in this paper some properties of the general Frégier-point and point out its connections with well known conics like the Apollonian hyperbola and the nine-point conic of a quadrangle. Incidentally, we discover a pedal conic of a quadrangle which has very intimate relations with the nine-point conic.