NOTES.

PASCAL'S THEOREM, Page 235.

M. STEINER was the first who (in Gergonne's Annales) directed the attention of geometers to the complete figure obtained by joining in every possible way six points on a conic. M. Steiner's theorems were corrected and extended by M. Plücker (Crelle's Journal, Vol. v., p. 274), and the subject has been more recently investigated by Messrs. Cayley and Kirkman, the latter of whom, in particular, has added several new theorems to those already known (see Cambridge and Dublin Mathematical Journal, Vol. V., p. 185). We shall in this note give a slight sketch of the more important of these, and of the methods of obtaining them. The greater part are derived by joining the simplest principles of the theory of combinations with the following elementary theorems and their reciprocals: "If two triangles be such that the lines joining corresponding vertices meet in a point (the centre of homology of the two triangles), the intersections of corresponding sides will lie in one right line (their axis)." "If the intersections of opposite sides of three triangles be for each pair the same three points in a right line, the centres of homology of the first and second, second and third, third and first, will lie in a right line."

Now let the six points on a conic be a, b, c, d, e, f, which we shall call the points P. These may be connected by fifteen right lines, ab, ac, &c., which we shall call the lines C. Each of the lines C (for example ab) is intersected by the fourteen others; by four of them in the point a, by four in the point b, and consequently by six in points distinct from the points P (for example the points (ab, cd), &c.) These we shall call the points p. There are forty-five such points; for there are six on each of the lines C. To find then the number of points p, we must multiply the number of lines C by 6, and divide by 2, since two lines C pass through every point p. If we take the sides of the hexagon in the order abcdef, Pascal's theorem is, that the three p points, (ab, de), (cd, fa), (bc, eƒ), lie in one right line, which we may call Sab.cd.ef either the Pascal abcdef, or else we may denote as the Pascal a form de.fa.bcs' which we sometimes prefer, as showing more readily the three points through which the Pascal passes. Through each point p four Pascals can be drawn. Thus through (ab, de) can be drawn abcdef, abfdec, abcedf, abfedc. We then find the total number of Pascals by multiplying the number of points p by 4, and dividing by 3, since there are three points p on each Pascal. We thus obtain the number of Pascal's lines 60. We might have derived the same directly by considering the number of different ways of arranging the letters abcdef.

Consider now the three triangles whose sides are

The intersections of corresponding sides of 1 and 2 lie on the same Pascal, therefore the lines joining corresponding vertices meet in a point, but these are the three Pascals,

The notation shows plainly that on each Pascal's line there is only one g point; for

given the Pascal

Lab.de.cf

the lcd. fa.be g point on it is found by writing under each term

Since then three Pascals
If we take the triangles

the two letters not already found in that vertical line. intersect in every point g, the number of points g = 20. 2, 3; and 1, 3; the lines joining corresponding vertices are the same in all cases: therefore, by the reciprocal of the second preliminary theorem, the three axes of the

(ab.cd.ef·
cf.be. ad)

three triangles meet in a point. This is also a g point de.fa. bc

}

and Steiner

has stated that the two g points just written are harmonic conjugates with regard to the conic, so that the 20 g points may be distributed into ten pairs. The Pascals which pass through these two g points correspond to hexagons taken in the order respectively, abcfed, afcdeb, adcbef; abcdef, afcbed, adcfeb; three alternate vertices holding in all the same position.

be.ac.
cd.bf.ae

ef.bd.ac
ad.ce.bf'

(5).

The intersections of corresponding sides of 1 and 4 are three points which lie on the same Pascal; therefore the lines joining corresponding vertices meet in a point. But these are the three Pascals,

We may denote the point of meeting as the h point, cd.bf.ae

ef.ac. bd)

The notation differs from that of the g points in that only one of the vertical columns contains the six letters without omission or repetition. On every Pascal there are three h points, viz., there are on

where the bar denotes the complete vertical column. We obtain then Mr. Kirkman's extension of Steiner's theorem:-The Pascals intersect three by three, not only in Steiner's twenty points g, but also in sixty other points h. The demonstration of Art. 268 applies alike to Mr. Kirkman's and to Steiner's theorem.

In like manner if we consider the triangles 1 and 5, the lines joining corresponding vertices are the same as for 1 and 4; therefore the corresponding sides intersect on a right line, as they manifestly do on a Pascal. In the same manner the correAAA

sponding sides of 4 and 5 must intersect on a right line, but these intersections are the three points,

Moreover, the axis of 4 and 5 must pass through the intersection of the axes of

In this notation the g point is found by combining the complete vertical columns of the three h points. Hence we have the theorem, "There are twenty lines G, each of which passes through one g and three h points." The existence of these lines was observed independently by Mr. Cayley and myself. The proof here given is Mr. Cayley's.

It can be proved similarly that "The twenty lines G pass four by four through fifteen points i." The four lines G whose g points in the preceding notation have a common vertical column will pass through the same point.

Again, let us take three Pascals meeting in a point h. For instance,

We may, by taking on each of these a point p, form a triangle whose vertices are (df, ac), (bf, ae), (bd, ce), and whose sides are, therefore,

Again, we may take on each a point h, by writing under each of the above Pascals af.cd.be, and so form a triangle whose sides are

ac.bf.de

cf.ae. bdr df.ab.ce

be.cd.af' be.cd.afs' be.cd.af

But the intersections of corresponding sides of these triangles, which must therefore be on a right line, are the three g points,

I have added a fourth g point, which the symmetry of the notation shows must lie on the same right line; these being all the g points into the notation of which be.cd. af can enter. Now there can be formed, as may readily be seen, fifteen different products of the form be. cd. af; we have then Steiner's theorem, The g points lie four by four on fifteen right lines I. Hesse has noticed that there is a certain reciprocity between the theorems we have obtained. There are 60 Kirkman points h, and 60 Pascal lines H corresponding each to each in a definite order to be explained presently. There are 20 Steiner points g, through each of which passes three Pascals H and one line G; and there are 20 lines G, on each of which lie three Kirkman points and one Steiner g. And as the twenty lines G pass four by four through fifteen points i, so the twenty points g lie four by four on fifteen lines I. The following investigation gives a new proof of some of the preceding theorems and also shews what h point corresponds to the Pascal got by taking the vertices in the order abcdef. Consider the two inscribed triangles ace, bdf; their sides touch a conic (see Ex. 4, p. 308) therefore we may apply Brianchon's theorem to the hexagon whose sides are ce, df, ae, bf, ac, bl. Taking them in this order, the dia

ce.bf. ad)

gonals of the hexagon are the three Pascals intersecting in the h point, df.ac.be

ae.bd.cf

And since, if retaining the alternate sides ce, ae, ac, we permutate cyclically the other three, then by the reciprocal of Steiner's theorem, the three resulting Brianchon points lie on a right line, it is thus proved that three h points lie in a right line G. From the same circumscribing hexagon it can be inferred that the lines joining the point a to {bc, df} and d to {ac, ef} intersect on the Pascal abcdef, and that there are six such intersections on every Pascal.

More recently Mr. Cayley has deduced the properties of this figure by considering it as the projection of the lines of intersection of six planes. See Quarterly Journal, Vol. IX., p. 348.

SYSTEMS OF TANGENTIAL CO-ORDINATES, Page 275.

Through this volume we have ordinarily understood by the tangential co-ordinates of a line la + mß+ny, the constants l, m, n in the equation of the line (Art. 70); and by the tangential equation of a curve the relation necessary between these constants in order that the line should touch the curve. We have preferred this method because it is the most closely connected with the main subject of this volume, and because all other systems of tangential co-ordinates may be reduced to it. We wish now to notice one or two points in this theory which we have omitted to mention, and then briefly to explain some other systems of tangential co-ordinates. We have given (Ex. 6, p. 128) the tangential equation of a circle whose centre is a'f'y' and radius r, viz.

(la" + mp" + ny')2 = r2 (l2 + m2 + n2 — 2mn cos A – 2nl cos B-2lm cos C):

let us examine what the right-hand side of this equation, if equated to nothing, would represent. It may easily be seen that it satisfies the condition of resolvability into factors, and therefore represents two points. And what these points are may be seen by recollecting that this quantity was obtained (Art. 41) by writing at full length la+mẞ+ny, and taking the sum of the squares of the coefficients of x and y, I cosa +m cosẞ + n cos y, I sin a + m sin ß + n sin y. Now if a2 + b2 = 0, the line ax+by+c is parallel to one or other of the lines y √(− 1) = 0, the two points therefore are the two imaginary points at infinity on any circle. And this appears also from the tangential equation of a circle which we have just given: for if we call the two factors w, w', and the centre a, that equation is of the form a2 = r2ww', showing that w, w' are the points of contact of tangents from a. In like manner if we form the tangential equation of a conic whose foci are given, by expressing the condition that the product of the perpendiculars from these points on any tangent is constant, we obtain the equation in the form

(la' + mẞ' + ny') (la” + mß" + ny") = b2ww',

showing that the conic is touched by the lines joining the two foci to the points w, w' (Art. 279).

It appears from Art. 61 that the result of substituting the tangential co-ordinates of any line in the equation of a point is proportional to the perpendicular from that point on the line; hence the tangential equations aßkyd, ay hẞ2 when interpreted give the theorems proved by reciprocation Art. 311. If we substitute the co-ordinates of any line in the equation of a circle given above, the result is easily seen to be proportional to the square of the chord intercepted on the line by the circle. Hence if Σ, 'represent two circles, we learn by interpreting the equation Σ= 2 that the envelope of a line on which two given circles intercept chords having to each other a constant ratio is a conic touching the tangents common to the two circles.

Lastly, it is to be remarked that a system of two points cannot be adequately represented by a trilinear, nor a system of two lines by a tangential equation. If we are given a tangential equation denoting two points, and form, as in Art. 285, the corresponding trilinear equation, it will be found that we get the square of the equation of the line joining the points, but all trace of the points themselves has disappeared. Similarly if we have the equation of a pair of lines intersecting in a point a'ß'y', the corresponding tangential equation will be found to be (la' + mß' + ny')2 = 0. In fact, a line analytically fulfils the conditions of a tangent if it meets a curve in two coincident points; and when a conic reduces to a pair of lines, any line through their intersection must be regarded as a tangent to the system.

The method of tangential co-ordinates may be presented in a form which does not presuppose any acquaintance with the trilinear or Cartesian systems. Just as in trilinear co-ordinates the position of a point is determined by the mutual ratios of the perpendiculars let fall from it on three fixed lines, so (Art. 311) the position of a line may be determined by the mutual ratios of the perpendiculars let fall on it from three fixed points. If the perpendiculars let fall on a line from two points A, B be λ, μ, then it is proved, as in Art. 7, that the perpendicular on it from the Α +ημ point which cuts the line AB in the ratio of m: 1 is and consequently that > 1+ m if the line pass through that point we have A+ mμ = 0, which therefore may be regarded as the equation of that point. Thus λ + 0 is the equation of the middle point of AB, X-μ= 0 that of a point at infinity on AB. In like manner (see Art. 7, Ex. 6) it is proved that lλ + mμ + nv = 0 is the equation of a point 0, which may be constructed (see fig., p. 61) either by cutting BC in the ratio nm and AD in the ratio m +n: 1; or by cutting AC::1:n and BE::1+n: m, or by cutting AB:: ml and CF ::1+m:n. Since the ratio of the triangles AOB: AOC is the same as that of BD: BC, we may write the equation of the point 0 in the form

:

μα

BOC.λ+COA.μ+ AOB.v=0.

Or, again, substituting for each triangle BOC its value p'p" sin 0 (see Art. 311)

Thus, for example, the co-ordinates of the line at infinity are λ=μ since all finite points may be regarded as equidistant from it: the point A + mμ + nv will be at infinity when l+m+ n = 0; and generally a curve will be touched by the line at infinity if the sum of the coefficients in its equation = 0. So again the equations of the intersection of bisectors of sides, of bisectors of angles, and of the perpendiculars, of the triangle of reference are respectively λ+μ+v = 0, λ sin A + μ sin B + v sinC = 0, λ tan A + μ tan B + v tanC= 0. It is unnecessary to give further illustrations of the application of these co-ordinates because they differ only by constant multipliers from those we have used already. The length of the perpendicular from any point on la + mẞ + ny is (Art. 61)

the denominator being the same for every point. If then p, p', p" be the perpendiculars let fall from each vertex of the triangle on the opposite side, the perpendiculars λ, μ, v from these vertices on any line are respectively proportional to lp, mp', np"; and we see at once how to transform such tangential equations as were used in the preceding pages, viz. homogeneous equations in l, m, n, into equations expressed in terms of the perpendiculars λ, μ, v. It is evident from the actual values that λ, μ, v are connected by the relation