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, Sab.de.ch, Jef.be .adr This is Steiner's theorem (Art. 268); we shall call this the g point, Sed fa.ber Sab.de.cf The notation shows plainly that on each Pascal's line there is only one g point; for given the Pascal fab.de.cf cd.fa.be Since then three Pascals the g point on it is found by writing under each term 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) three triangles meet in a point. This is also a g point de.fa.bc and Steiner cf.be.ad 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. 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 *For a proof of this see Staudt (Crelle, LXII, 142). a right line, as they manifestly do on a Pascal. In the same manner the corresponding sides of 4 and 5 must intersect on a right line, but these intersections are the three h points, Moreover, the axis of 4 and 5 must pass through the intersection of the axes of ab.cd.ef 1, 4, and 1, 5, namely, through the g point, de.af.bc cf.be.ad 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 Prof. Cayley and myself. The proof here given is Prof. 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, ab.ce. de.bf.ac de.bf.ac' cf. ae.ba}, cf.ae.bd 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 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 h 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, Art. 355); therefore we may apply Brianchon's theorem to the hexagon whose sides are ce, df, ae, bf, ac, bd. Taking them in this order, the dia. 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 Prof. 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. Still more recently the whole figure has been discussed and several new properties obtained by Veronese (Nuovi Teoremi sull' Hexagrammum Mysticum in the Memoirs of the Reale Accademia dei Lincei, 1877). He states with some extension the geometrical principles which we have employed in the investigation, as follows: I. Consider three lines passing through a point, and three points in each line; these points form 27 triangles which may be divided into 36 sets of three triangles in perspective in pairs, the axes of homology passing three by three through 36 points which lie four by four on 27 right lines. II. If 4 triangles a1b11, ab2c2, &c. are in perspective, the first with the second, the second with the third, the third with the fourth, and the fourth with the first, the vertices marked with the same letters corresponding to each other, and if the four centres of homology lie in a right line, the four axes will pass through a point. III. If we have four quadrangles a,b,c,d1, &c. related in like manner, the four points of the last theorem answering to the triangles bcd, cda, dab, abc lie on a right line. Considering the case when all four quadrangles have the same centre of homology, we obtain the corollary: If on four lines passing through a point we take 3 homologous quadrangles a1111, a,b,c,d1⁄2, α‚b ̧¤ ̧d ̧; then we have four sets of three homologous triangles, a,b,c1, &c. the axes of homology of each three passing through a point and the four points lying on a right line. IV. If we have two triangles in perspective a,b,c1, a2b22, and if we take the intersections of b12, b2c; c12, ca1; a12, ab1, we form a new triangle in perspective with the other two, the three centres of homology lying on a right line. It would be too long to enumerate all the theorems which Veronese derives from these principles. Suffice it to say that a leading feature of his investigation is the breaking up of the system of Pascals into six groups, each of ten Pascals, the ten corresponding Kirkman points lying three by three on these lines which also pass in threes through these points. It may be added that Veronese states the correspondence between a Pascal line and a Kirkman point as follows: Take out of the 15 lines C the six sides of any hexagon, there remain 9 lines C; out of these can be formed three hexagons whose Pascals meet in the Kirkman point corresponding to the Pascal of the hexagon with which we started. After the publication of Veronese's paper Cremona obtained very elegant demonstrations of his theorems by studying the subject from quite a different point of view. From the theory of cubical surfaces we know (Geometry of Three Dimensions, Art. 536), that if such a surface have a nodal point, there lie on the surface six right lines passing through the node, which also lie on a cone of the second order, and fifteen other lines, one in the plane of each pair of the foregoing; by projecting this figure Cremona obtains the whole theory of the hexagon. It may be well to add some formulæ useful in the analytic discussion of the hexagon inscribed in the conic LM-R2. Let the values of the parameter μ (Art. 270) for the six vertices be a, b, c, d, e, f, and let us denote by (ab) the quantity abL - (a + b) R + M, which, equated to zero, represents the chord joining two vertices. Then it is easy to see that (ab) (cd) — (ad) (bc) is LM – R2 multiplied by the factor (a — c) (b − d), and hence that if we compare, as in Art. 268, the forms (ab) (cd) — (ad) (bc), (af) (de) — (ad) (ef) we get the equation of the Pascal abcdef in the form The same equation might also have been obtained in the forms, which can easily be verified as being equivalent, The three other Pascals which pass through (bc) (ef) are (a−c) (b − d) (ef) = (a − ƒ) (e − d) (bc), (ab) (cd) (ef) = (a − e) (ƒ — d) (be), these being respectively the Pascals abcdfe, acbdef, acbdfe. (ac) (bd) (ef) = (a − e) (ƒ − d) (bc) = (b −ƒ) (c − e) (ad); these evidently intersect in a point, viz. a Steiner g-point; but the three (a−c) (b − d) (eƒ ) = (a − e) (ƒ − d) (bc) = (b − e) (c − ƒ) (ad) intersect in a Kirkman h-point. Mr. Cathcart has otherwise obtained the equation of the Pascal line in a determinant form. It was shewn (Art. 331) that the relation between corresponding points of two homographic systems is of the form Aaa' + Ba + Ca' + D = 0. Hence, eliminating A, B, C, D, we see that the relation between four points and other four of two homographic systems is aa', a, a', 1 BB', B, p', 1 γγ', γ, γ', Số, ô, ổ, 1 = 0,* and the double points of the system are got by putting & d, and solving the quadratic for d. But we saw Art. 289, Ex. 10, that the Pascal line LMN passes through K, K' the double points of the two homographic systems determined by ACE, DFB the alternate vertices of the hexagon. And since, if d be the parameter of the point K, we have M, R, L respectively proportional to d2, d, 1, it follows that the equation of the Pascal abcdef is M, R, R, L ad, a, d, 1 be, b, e, 1 cf, c, f, 1 = 0. SYSTEMS OF TANGENTIAL COORDINATES, Art. 311. Through this volume we have ordinarily understood by the tangential coordinates 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 coordinates may be reduced to it. We * On this determinant see Cayley, Phil. Trans., 1858, p. 436. 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 coordinates. We have given (Ex. 6, Art. 132) the tangential equation of a circle whose centre is a'B'y' and radius r, viz. 2mn cos A (la' + mẞ' + ny')2 = p2 (12 + m2 + n2 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. 61) by writing at full length la+mẞ+ny, and taking the sum of the squares of the coefficients of x and y, I cos a + m cos ẞ+n cos y, 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 xy√(-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 showing that the w, w' (Art. 258a). (la' + mp' + ny') (la" + mß" + ny") = b2ww', conic is touched by the lines joining the two foci to the points = It appears from Art. 61 that the result of substituting the tangential coordinates of any line in the equation of a point proportional to the perpendicular from that point on the line; hence the tangential equations aß kyd, ay kß2 when interpreted give the theorems proved by reciprocation Art. 311. If we substitute the coordinates 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 = k2' 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'B'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 coordinates may be presented in a form which does not presuppose any acquaintance with the trilinear or Cartesian systems. Just as in trilinear coordinates 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 ι + ημ point which cuts the line AB in the ratio of m: lis and consequently that 1 + m " if the line pass through that point we have +mμ-0, which therefore may be |