This note is intended to supplement Note 1754 and the article on pages 77 to 79 of Gazette XXVII. There is some danger that the reader of that note or article may get the impression that the writers are trying to prove some absolute truth about the signs when they are really only concerned with consequences of the definitions and conventions. There is not much difference between a definition and a convention. Often a convention is little more a an instruction about how a definition is to be interpreted. Neither a nition nor a convention can be right or wrong: it can be appropriate or appropriate.

If BPC, DPE are circular sections of the cone vertex A, a sphere can be drawn containing these two sections, and the cone will touch the sphere at the points P, P1 where the line of intersection of the two planes meets the sphere. Thus AP2 = AD AB. Taking BPC as the base of the cone, if AC is the least and AB the greatest of the generating lines drawn from A to the base, ABC will be a principal section of the cone and the planes of circular ection of the cone are perpendicular to the plane ABC. If AF is perpendicular to DE and AH perpendicular to BC,

That is, AP2 is the product of the perpendiculars drawn from P to the planes hrough the vertex A parallel to the planes of circular section, divided by in B sin C.

Starting with any triangle Δ ≡ ABC, let Δ1 ≡ A1B1C1 be its pedal, Δ2≡A2B2C2 the pedal of Δ1 and so on indefinitely Since the circumradius Rn of Δn is equal to 1/2Rn-1, we may expect a steady contraction towards some definite limiting point. It is proposed to inquire into the position of this point.

It is usually considered that the solution of isoperimetric problems is only to be achieved through the medium of the more or less advanced technique of variation calculus. The following theorem is derived using only the notions of pure geometry and elementary calculus, and is interesting since the method of proof is easily extended to several other problems of the same type.

What is centrifugal force? On selecting at random about a dozen elementary textbooks on Mechanics to find a definition, I was pleasantly surprised to discover (unless I have inadvertently missed it in any of them) that only four of them mention the name “centrifugal force”, and two only of these four mention “centripetal force” as well. No doubt if I had had easy access to textbooks written for and used by engineers, the results would have been different; but as I am concerned only about mathematical students, I have not troubled to pursue my search except in the kind of textbooks likely to be used in schools which teach elementary mechanics as a branch of mathematics.

Although it is difficult to think of a rational reason for being more interested in, say, the square number 698896 than in the square 700569, or for that matter in 4 or 49, there is apparently some psychological reason, and the pursuit of square numbers of particular digital forms has a long history. The publication of the attractive collection of digital curiosities in Note No. 1823 has suggested to me that there may be some interest in further figures from discoveries I have made in analogous fields. References in a few cases indicate previous publication in scattered problem columns.