We propose in this paper to develop equations which give the principal points and lines of the conic formed by the intersection of the general conicoid.

S=ax2+by2+cz2+2fyz+2gzx+2hxy+2ux+2vy+2wz+d=0,

with the plane lx+my+nz+p)=0, in a form convenient for numerical work. We discuss the general case in which special values of the constants do not lead to special cases the slight modifications necessary where such occur become clearly manifest in numerical work. For brevity, the expressions ax+hy+gz+u, etc., will be denoted by the usual symbols X, Y, Z, and for nY - mZ, lZ - nX, mX - lY we will write U, V, W where lU + mV + nW≡0. We can assume without loss of generality that n is not zero.

In a previous article I have outlined my conception of the rôle of history in the exposition of mathematical technique. In this article I attempt to provide my bare thesis with a respectable clothing of practicability The treatment of Number and Continuity which follows is a short and very sketchy historical supplement to the technical treatment of the same subjects in, say, the earlier chapters of Hardy’s Pure Mathematics.

The ease with which certain series involving finite difference operators can be summed by the method of separation of symbols has long been recognised. It does not appear to be universally known, however, that many forms of algebraic series which at first sight do not seem to lend themselves to this treatment can be summed very simply by the application of operators.

This paper is devoted principally to the exposition of a new calculating process which may be useful for what are called “natural” dynamical problems. A few examples, mostly trivial, are worked out to illustrate the process. But it is possible that the method may be used with advantage for several important problems, and a paper concerning its use for problems of gravitational astronomy will be shortly published elsewhere.

As a rule graphical methods are more tedious than an analytical treatment; but they are welcomed by the weaker student on account of the illumination which they shed upon relations that would otherwise remain somewhat obscure.

Two notes which have recently appeared in the Mathematical Gazette dealing with the graphical discrimination of the cubic tempt me to publish methods of discussing the roots of the cubic and of the quartic which have been in use here for some time and have stood the test of experience.

The following question was set in the University of London Inter mediate B.Sc. Applied Mathematics paper, 1922.

“Two masses of 2 lbs. and 3 lbs. are fastened at the ends of a light string of length 2 feet and placed on a smooth horizontal shelf 5 feet above the ground. The 2 1b. mass is placed on the edge of the shelf and the 3 lb. mass one foot away from the edge on a line perpendicular to the edge through the mass of 2 lbs. If the mass is gently pushed over the edge find the time that elapses before the 3 lb. mass strike the ground” (g=32 f.p.s2)