Cette note que j'ai l'honneur de presenter à la Société Mathématique d'Edinburgh, par 1'entremise aimable de M. J. S. Mackay, contient, ou des résultats que je crois nouveaux, ou des développementi sur des sujets que j'ai déjà souvent abordés dans la Géomé'trie et qui concernent: la transformation continue dans le triangle et dans le tétraèdre, les formulas entre les éléments du triangle, et la Géométrographie. Pour abréger, je passerai rapidement sur les points que j'ai déjà développés ailleurs, me contentant de renvoyer, si Ton desire plus d'explications, aux meémoires ou la chose a été faite.

I. Construction for the Brocard points.

Let ABC be a triangle. Describe a circle touching AB in A and passing through C; draw the chord AP parallel to BC. Join BP meeting this circle in Ω

1. If the coordinates of a point on a parabola,

he (am2, 2am), in which I call m the parameter, then the equations to the tangent and normal at the point are

and

and to the chord through (m),(m′) is

Let OQ, OR be two straight lines meeting at 0, and P any point. Required to draw through P a straight line cutting off a given area OAB from the two straight lines.

Draw PD parallel to OR cutting OQ in D.

Construct a ΔOPC equal to the given area, and such that OP is one of its sides, and that another of its sides, OC, lies along OQ.

When points and lines are not specifically designated in the course of the following pages it will be understood that the notation for them is that recommended in the Proceedings of the Edinburgh Mathematical Society, Vol. I. pp. 6–11 (1894). It may be convenient to repeat all that is necessary for the present purpose.

The following formulae may be added to the list given in the Proceedings of the Edinburgh Mathematical Society, Vol. XII., pp. 86-102 (1894).

and also on.

(1) If O be a given point in the plane of a given conic, the mutual relationship between point and conic is marked, first and foremost, by the existence of a certain determinate straight line (which is always real) known as the polar of O with respect to the conic. Next following the polar in natural order of sequence, come a certain pair of determinate straight lines:

The single real pair of common chords of the conic and a point circle at O.

The investigation given in the following Note was suggested by a passage in the paper by Mr Leraoine, presented to the Society by Dr. Mackay at a recent meeting. The main subject of that paper is what he terms the “Transformation continue dans le triangle et dans le tètraédre”; for the explanation of that phrase and other terms connected with it, the reader is referred to the paper just mentioned. In the notation for the quantities connected with the triangle, however, I shall follow Dr. Mackay''s system as explained in his paper in Vol. I. of our Proceedings.

§ 1. The object of this Note is to prove the following theorems:

in which and the series subject to conditions for convergence.

Let the straight line be denned by the coordinates (α β γ) of a point on it and by its direction cosines l, m, n. It may be referred to as the line (α β γ l, m, n). "Write, for shortness, the equation to the quadric surface in the form F(x, y, z)=0.

The chief object of this Note is to develope some simple and rather interesting properties of the operators α β γ referred to in my Note read at last meeting. I take for brevity the symbol μ. to denote the compound operator

The convergence of the series is assumed.

The method consists in assuming that the function is equal to a certain power series with undetermined coefficients, substituting these series in the addition formula. This gives an identity.

Picking out coefficient of y we get

§ 1. 1 have ventured to bring the formulae of this paper before the Society, as I have been unable to find reference to them in any text-book or any original contribution to mathematical literature which I have come across. I confine my attention completely to the central surface, as the corresponding formulae for the paraboloids are very readily deduced by a similar process.

1. To designate the line which bisects at right angles the join of two points A, B, the term axis of A, B is proposed. Reasons:—

(1) brevity; (2) avoidance of the suggestion that the line joining AB is necessary in constructing it (important in teaching Geometrical Drawing); (3) two points, like any other pair of circles, have a radical axis which is the line in question.

If in Tables such as Chambers' the differences were given for a second instead of for a minute, they might be arranged (and used) follows: