This lecture grew out of an attempt to trace the consequences of putting together two examination questions. The subject-matter of both is the reflections X, Y, Z of a point P in the sides of a triangle ABC. In the first question we have to show that the perpendiculars from A on YZ, from B on ZX, from C on XY meet in P′, the centre of the circle XYZ, and that the relation between P and P′ is mutual.

Since AY = AP =AZ, the perpendicular from A to YZ bisects it. Hence this perpendicular passes through the centre of the circle XYZ, as do similarly the other two perpendiculars. Let X′, Y′, Z′ be the reflections of P′ in the sides of ABC. Since BC is the common perpendicular bisector of PX and P′X′, PX′ =P′X, and similarly PY′ = P′Y, PZ′ =P′Z. Hence P is the centre of the circle X′Y′Z′, and the two radii are equal.

Rouse Ball in his Mathematical Recreations tells us that there are no rules for the construction of bordered squares ; I have made the following rules, which are now published for the first time as far as I am aware.

It is remarkable that the rules for odd bordered squares are apparently not known as they are so simple. We use two diagrams, one a skeleton to make the method clear, and the other completed to show the full square.

Rule I. Place the letter M at the (n +1 )th square as shown in Fig. 1, where n denotes the order of the square (here n = 9).

Rule II. Now, commencing at the second column, write the numbers 1, 2, 3, … up to and under M. Then write the next number in the diagonal at the bottom right-hand corner ; the next number in the same column and in the second row, and continue until we arrive at the square above M.

Numerical integration, including the numerical integration of differential equations, is becoming increasingly necessary in technical applications. For such purposes, when a very high order of accuracy is not required, formulae which use successive values of the function to be integrated are more convenient than those using differences. Individual authors have given formulae of this type from time to time but a systematic list is not known to the present writer, and would seem to be of service.

The present list gives formulae of four types : (I) formulae for evaluating definite integrals ; (II) formulae for forward integration ; (III) formulae for integrating over a subrange ; and (IV) formulae involving values of both the function and its integral. The leading term in the error series is given. Finally, some comments are made upon the formulae and their uses.

In the Gazette No. 249, Vol. XXII, Mr. Gibbins has deduced some interesting formulae connected with the triangle ABC, of which the side BC is given by L = lx + my +n= 0 and AB, AC by the equation

The equations of the circles in § 3 of Mr. Gibbins’ note are very elegant indeed. But, in principle, it does not seem desirable to have recourse to pure geometry in dealing with problems in algebraic geometry. The algebraic analysis is not difficult. It is not necessary to resolve S into factors. Let us assume that S≡vw, where v ≡ l2x + m2y + n2 and w ≡ l3x + m3y + n3.

On a récemment signalé des simples constructions qui mènent d’un triangle plan tout à fait arbitraire à un triangle regulier. On peut établir que ce triangle dérivé jouit de cette remarquable qualité en s’inspirant à une profonde maxime d’Abel (” il faut énoncer un problème de manière de le pouvoir toujours résoudre “); on doit alors se proposer en général de déterminer de quelle espèce est le triangle résultant de l’application d’une des constructions citées. Dans plusieurs cas il arrive que l’expression de la longueur d’un quelconque de ses côtés est une fonction symétrique des éléments du triangle primitif ; alors la régularité du triangle d’arrivée est hors de doute. Il est aussi à remarquer qu’à l’expression cherchée on arrive très souvent par un procédé (qui rappelle la “ triangulation géodésique “) qui mérite d’être signalée à tous ceux qui auront à l’avenir à s’occuper de questions analogues à celles étudiées dans le mémoire qui suit.

Very often it is proved experimentally that two electrified bodies repel or attract each other, and from this fact is deduced the conclusion that there exist two kinds of electricity of such properties that bodies of the same kind of electricity repel and bodies of different kinds attract each other. This proceeding is not quite correct.

Example. If we consider two integral numbers, then their difference is divisible by 3 or not. But it is obviously wrong to deduce from this fact that we can divide the set of all integral numbers into two such classes, that the difference of two numbers belonging to one class is divisible through 3 and the difference of two numbers belonging to different classes is not. It is known that there exist three classes possessing these properties.

Multiply throughout by a2 and write y for ax+ b ; the equation becomes

where H ≡ac − b2, G ≡ a2d − 3abc + 2b3.

Since in an equation with real coefficients complex roots occur in conjugate pairs, (i) must have at least one real root; so if α is this root, (i) may be written

Accordingly the two remaining roots are also real if

But since α satisfies (i),

and so

Hence if (i) has three real roots, G2 +4H3 ≤ 0; and clearly, when G2+4H3 = 0, two roots are numerically equal to and the third to .