LIBRARY OF THE University of California. RECEIVED BY EXCHANGE Class I. The Syzygetic Pencil of Cables with a new Geometrical Development of its Hesse Group, Gg,^. II. The Complete Pappus Hexagon. DISSERTATION SUBMITTED) TO THE BOARD OF UNIVERSITY STUDIES OF THE JOHNS HOPKINS UNIVERSITY IN CONFORMITY WITH THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY BY CHARLES CLAYTON GROVE BALTIMORE, MD. June, 1906. I. The Syzygetic Pencil of Cnbics with a new Geometrical Development of its Hesse Group, Ggig. II. The Complete Pappns Hexagon. DISSERTATION SUBMITTED TO THE BOARD OF UNIVERSITY STUDIES OF THE JOHNS HOPKINS UNIVERSITY IN CONFORMITY WITH THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY BY CHARLES CLAYTON GROVE BALTIMORE, MD. June, 1906. Z^t Jgorb (gaftttnore qpnee BALTIMORE, MD., U. S. A. 1907 ,^ INTRODUCTION. 1. The course of lectures by Prof. Frank Morley during the winter of 1903-4 on cubic curves suggested this dissertation and prepared me to carry on the research. The trend was largely determined by an incidental question by Prof. A. Cohen as to the groups involved in the system of conies which I had just presented to the Mathematical Seminary. The interest and valuable sug. gestions of Dr. A. B. Coble in the carrying on of the work are gratefully acknowledged. 2. The close connection between the Hesse group and the syzygetic pencil of cubics makes it necessary to say at least something about this pencil of curves. Without attempting even an outline of the theory, I present in Section I. only such matter as is needed later, besides some new facts concerning the pencil and a figure showing the appearance of some noteworthy and specially related cubics of the pencil. No figure seems ever to have been published except that in con- nection with the paper of Prof. Morley in the Proceedings of the London Math- Society, Ser. 2, Vol. 2, Part 2, which shows arbitrarily selected cubics. The initial and all but the closing work leading to that figure was done by me. Therefore, I present a figure of the pencil herein, also one of the corresponding polar-reciprocal range of line cubics. Section II. shows how to derive a closed system of thirty-six conies analogous to the conic of Section I. as to which the pencil and range are polar-reciprocal. It also discusses the action of the polarities of these conies upon the four inflexional triangles, and presents some history of similar considerations. In Section III. there is given a brief history of the attempts to determine all finite groups of transformations, and in particular an account of the Hesse Group 165264 4 Introduction. of 216 collineations. Further, we derive and write down the matrices of these collineations by means of the closed system of thirty-six conies, which are here differently defined than in Section II. and are given accordingly. All the sub- groups are found and discussed. The collineations are finally classified as to periodicity. Section IV. treats of triangles in other. perspective forms than six-fold as are the inflectional triangles above. As a second way to secure triangles in three- fold perspective, also some in two- and one-fold perspective, we develop what we call the Complete Pappus Hexagon, in its dualistic forms and deduce a number of theorems connected with it. I. THE SYZYGETIC PENCIL. DRAWINGS OF THE PENCIL AND RANGE. 1. The name stzygetic^ is given to the pencil of cubics determined by a cubic/ and its Hessian A, x/4-;iA = 0, (1) which has the same nine inflexions for all its cubics, the intersections of the two cubics /and A, and so has the same four inflexional triangles. It is well known that any non-singular cubic may be brought in four ways, as shown for example by Weber, ^ into Hesse's canonical form xl-\- xl-\- xl-\- 6mxiXciXs= 0. (2) This simply means that the cubic has been referred to one of its inflexional triangles as reference triangle. The Hessian covariant of the form (2) is A = — m^xf + xl-^xl)-^{l ■^2m^)x,X2Xs. (3) Its vanishing gives the Hesse Cubic or Hessian. Thus if m is the parameter of the cubic (2) and m' is that of its Hessian, we have 6 m' = — ^ — ; and form (2) for all values of m from — oo to + <» gives the pencil as well as form (1). The relation between w and m' shows that each cubic of ike pencil has hut one Hessian hut is Hessian to three cubics of the pencil. Choosing one of the inflexional triangles as reference triangle, we readily calculate in turn, The coordinates of the nine inflexions, Their arrangement on the sides of each of four triangles, ^ The equations of the sides and opposite vertices of these triangles, 1 Clebsch-Lindemann : Lecons sur la Geometric, II, p. 230. "Lehrbuch der Algebra, 3. Aufl., II, §§ 106, 107; ss. 399-404. 8 C.-L. : II, (7) p. 233. 6 I. The Syzygetic Pencil. Drawings of the Pencil and Range. The polar conies of the inflexions, which are in each case^ two right lines, viz., the inflexional tangent and the harmonic polar of the inflexion. For future reference we give the sides of the inflexional triangles : A. B. C. D. jCj = 0. £Ci -f- a^g + ajg = 0. o^ Xi -f- ccg + iCg := 0. co aji -f- ajg + aJg = 0. 052=0. a^i + a>^iC2 H-G)iC3= 0. JCj + C0^iC2+ JC3= 0. «! -f G) ^2 + iCs = 0- (4) Xg = 0. jcj + G) ir2 H- o^ iCa = 0. iCj + a;2 + "^ s-g = 0. iCi + a^ + o) JC3 = 0. The equations of the vertices respectively opposite are given by exchanging ^ for X and interchanging a and o^. a and o^ are the complex cube roots of unity. 2. Some Particular Cubics of the Pencil. (a) The cubics whose parameter m is respectively 00, — ^, — \a?^ — ^co are the four inflexional triangles in the order as obtained above, of which ^ two are real and two imaginary. The polar line^ as to these triangles of any pointy has coordinates respectively, «! ^ X3 A: ^2^8 ' Vzyi yiy2' B: y\ — yzyz yl y%y\ yl — yiy2' C: y\ — ^^y%yz y\ — ^^yzy\ yl — ^^yiy^ D: yl — ^y^y-i yl — (^yzy\ y\ — ^yxyi' The determinant formed from any three rows of these coefficients vanishes identically, therefore the four polars pass through a common point, so we say. The four polar lines of any point as to the four inflexional triangles meet in a point, or also Any two of the inflexional triangles are apolar as seen from any point of the plane, and are thus syzygetic. {h) The first of two covariant cubics of the pencil is the Hessian, equation (3), p. 5. The second is the Cayleyan of the cubic (2). The polar conic as to this cubic of a point y is yy {x\ + 2 wa-ga-g) -f ^g (a^l + 2 w x^x^ -^^ y^[x\-\- 2m x^ x^ = 0. Considering the y's as parameters, this is a net of polar conies. By inspection, we see that the conic w^j — ^2^8 is apolar with the three conies of the net; >C.-L.: II, p. 227; Salmon: Higher Plane Curves, 8. Ed., §§ 74, 170, pp. 69, 146. « C.-L. : II, p. 230, also pp. 239, 310. »H, P. C, 8165, p. 143. I. The Syzygetic Pencil. Drawings of the Pencil and Range. 7 likewise the conies m^l — ^3^1, and m^l — ^1^3. So the net, has a corre- sponding weh of apular conies, given by the equation where the jy's are parameters. The contravariant or Jacobian of the web is the Cayleyan : 2 w ^1 — ^3 — ^2 -^3 2771^3 —^1 =0, — ^2 —^1 2 771^3 or wa? + ^i + ^i) + (l-4m^)^i^3^3 = 0, which is a cubic of the range^ enveloped by the oo^ lines composing the degenerate conies, the first polars of points along the Hessian of the cubic. Here as in the case of the Hessian, we see that each cubic of the range has but one Cayleyan but it is Cayleyan to three cubics. (c) The simplest invariant is found by operating with the Cayleyan on the cubic. ^ It is the quartic invariant denoted in Salmon by S. By operating thus and dividing^ by 24 we have S = m{l — m^). The vanishing of this invariant gives the parameters of four curves of the pencil called the equianharmonic cubics : jS^: xl-\- xl+ xl = 0. Ssi xl-{-xl + xl + 60^x^x^X3= 0. ^Sg : ccf + x| + aj| -f- 6 Xi ccg 2*3 = 0. S^: a:? + a^l + xi + 6 Xj Xg ccg = 0. This name is given because the constant anharmonic ratio of the four tangents* drawn from a point of the curve tangent to the curve itself is equianharmonic in these four cases. (d) By operating with the Cayleyan on the Hessian we have the sextio INVARIANT, T of Salmon.^ It is T=l — 20m^—8m^. iC.-L.: II, p. 344. 2 By operating with a line-form, as of the Cayleyan, upon a point-form, as of the cubic (2) called/, we mean that the f's are taken as partial differential operators. The equation of the Cayleyan means in this process 3 Salmon: H. P. C, §220, p. 191. *C.— L.: II, pp. 325, 326. 5H. P. C, §221, p. 193, also Weber: II, s. 406 (7). 8 I. The Syzygetic Pencil. Draxmngs of the Pencil and Range. The curves whose parameters are given for 7=0, are the six harmonic cubics of the pencil, called so since the ratio of the four tangents to the curve from points on it is harmonic. ^ With this amount of introduction and number of references necessary to prepare the student to read the whole most profitably, and with the equations at hand to which reference must later be made, we pass now to The Drawing of the Syzygetic Pencil. 3. As a far more convenient form of the cubics of the pencil for purposes of construction, we transform the equation of the pencil so as to have inflexions at the circular imaginary points / and /, by putting aa = a; + y + 1, x^ = — {x—\), xs = — (y—l), (6) where x and y are conjugate coordinates. By this transformation, equation (2) of the pencil becomes or it is of the form xy{x-\-y) + (i{a^-\-xy-\-y')-{-l= 0, (7) , 2(1— m), 2 — u where u = / . ^ — <- , and m = ^ . , ^ , . ^ 1 + 2m ' 2(1 + ^) 4. The Special Cubics of § 2. (a) The four inflexional triangles (4) become in conjugate coordinates, A: {x + y+l){x — l)(y-l) = 0. B: a? + xy-\-f — (x — ciy){x — (J'y) = 0. C: xy{x-{-y)-(^ia^-^xy'hf)-\-l = 0. ^ ^ D: xy{x-^-y) — ci^3^-{-xy-\-y^) + l = 0. Our former reference triangle A is seen thus to be now the line through the points G) and o^ on the unit-circle, and the point x= 1 taken twice. Since the cube terms are lacking in B, the cubic consists of the line at infinity, also, as readily seen, of the lines through the origin and o and u>^ respectively. The representable parts are shown in the figure of the pencil. The three dash lines are the real harmonic polars, the full lines are the degenerate cubics. » C.-L. : II, p. 326. I. The Syzygetic Pencil. Drawings of the Pencil and Range, (h) The Hessian by direct calculation on (7) is 1 2y-{- 2(1 2{x + y)-^li (i{2x + y) 6 2(a; + 2/)+i^ 2x-\- 2^ ii{2y-\-x) li{2x + y) (i{2y-{-x) 6 or E=^ii'xy{x + y) — (^^^-^){a?-^xy-\-y'')^Zii^z=0^ (9) which is of the form of the cubic (7) with its parameter ^'givenby^' = — ^^-ti. Therefore equation (7) likewise gives the syzygetic pencil for all values of ^ from — CO to + 00 as well as equation (1). The Cayleyan is calculated as follows : The polar conic as to the cubic (7) of a pointy is Regarding the jp's as parameters this is a net of polar conies. To find the weh of apolar conies we take the general line-conic «f + hr^ + c^ + d^n + en^ +/f ^ = 0, and operate separately on the point conies of the net. Thus we obtain 25+ 2c?+ 2^/-(- ^e = 0, 2a + 2d-\- y.f+ 2^e = 0, 2(ia + 2(ih -\- ^d + ec = 0. We desire apolar conies analogous in form respectively to those of the net and so we first take a = 0, b=zd=zl^ whence, by substituting in the three equations above and in the general line-conic, we get the conic 2(1^71+ 2fiy! — ^^ — [jp=z0. Second, put 5 = 0, a = c? = 1, and we get the conic 2fi^^+2iz^yi^4yj—fi^ = 0. Third, put a = l,e=/=0, and the corresponding conic is 2(P-^>7H->7')-i^ = 0. By these three we may write the apolar web of line-conics. Hi l2(z^yj+2fiyi'-4^-i/] +n,[2^f + 2fi^yj-4y!-(,^ -f n3[2 {^^-^yj+yj^)-^-] = 0. 10 I. The Syzygetic Pencil, Drawings of the Pencil and Range. The Jacobian of this web is by direct calculation tin— 2 /t*(^+2>7) —(2^ + ^2) 2^ — >7 — ^4-2>7 —ii or 2^(f+>7^)-3^^,7(^+>7)H-(^^-2)(f-^,7+>7')-i^'(^->7)-i^=0, (10) which is the Cayleyan of the cubic (7). (c) By operating with the Cayleyan (10) on the cubic (7) we have the qnartic invariant o 4 j, whose vanishing gives the parameters of the four equianharmonic cubics of the pencil, viz., /tz = 0, 2, 2{o, 2(a^ That two of these are real is seen from the parameters. (d) By operating with the Cayleyan on the Hessian (9) we obtain the sextic invariant T=S — 20(1^ — fi\ whose vanishing gives the parameters of the six harmonic cubics, viz., the roots of |it^ = — 10 ± 6 Vs. For the positive and the negative sign, there are in each case one real and two imaginary roots, or there are two systems of three values each of the harmonic ratio. The two real roots are — (1 — V3) and — (1 + V3). 5. The nine inflexions and harmonic polars mentioned in §1, pp. 5, 6 are in conjugate coordinates as follows, with the positions on the figure as indicated respectively — Inflexions. Point on figure. Its Harmonic Polar. 1. At 00 on 6)0)^ X — y=0 2. / 3. J 4. a' 6. 6. 7. a 8. 9. The Line on the Figure. Axis of reals Circular imaginary rays from the origin Line 1, — a^ x+ 2y = 0\ 2x + y = 0} x — cd^y + uP — 1 =0 X — u^y — o 4-1 = A cir. imag. ray through o X O^y — 6)^ + G) = X — iny +0 — 1 = X 6) Line 1, — q 0)// — (j2_|_i=o A cir. imag. ray through o^ X — wy — a)4-(o^ = <^ . (11) I. The Syzygetic Pencil. Drawings of the Pencil and Range. 11 By projecting the pencil into this form, we have the line at infinity as one real side of an inflexional triangle, and one real inflexion at infinity. That leaves^ to appear on the figure three sides of inflexional triangles, two inflexions and three harmonic polars. Towards a better understanding of the syzygetic pencil and the drawing of it we analyze it further and present first amongst our findings. 6. The Asymptotes. Take the line a: + y = X, which is perpendicular to the axis of reals at a distance ^7^ from the origin; that is, the line in which the reflexion of the origin is the point 7.. It cuts the cubic (7) where Xx(a — a) + /^(a;2 — XiB + X^) + 1 = 0. Since the cube terms vanish, the line meets the cubic at infinity. If next we put X = /[i, we have only ^3+1 = 0. Hence, since the square terms also vanish, the line x -\- y = ^ is tangent to the cubic at infinity, in direction perpendicular to the axis of reals and is the asymptote. The point at infinity on the line uuF, perpendicular to the axis of reals is as above noted one of the real inflexions, hence the asymptote just found is an inflexional tangent as well. Thus when ^i is given, the asymptote of the cubic of the pencil for that particular ^ as parameter is also given as the perpendicular to the axis of reals at the distance ^^ from the origin. 7. By taking the first polar of the inflexion q as to the general cubic (7) we find it breaks up into the two linear factors \x — G)Z/-f-t^ — If jw^(ic — uiy — 6)-|- \) ■\- ^ {x — o^2/)[ =^ ^) the harmonic polar and flex-tangent respectively. The latter cuts the asymptote of the cubic where l/^ — (1 + |ti)(G)x + oV — "^+ 1) = 0. The harmonic polar of uf' [see equations (11), p. 10] cuts the asymptote where — (g)£c + g)^^ — o^ -f- 1) = 0. Therefore, we may draw the flex-tangents to any cubic of pencil hy drawing from either inflexion, 6> or u^, to the point where the harmonic polar of the other, w^ or o, cuts the asymptote of that cubic. 1 C.-L. : II, p. 235 on the real parts of the figure. 12 T. The Syzygetic Pencil. Drawings of the Pencil and Range. 8. Intersections of the Cubics with the Axis of Reals. The general cubic (7) cuts the axis of reals where 2ic^+ 3|Ma^4- 1 = 0. The discriminant (Weber: I, s. 273) of this equation is D = — 108(^^4- 1), which shows that if 1. ft <; — 1, then Z> ]> 0, and the equation has three real distinct roots. 2. /u = — 1, then D = 0, " *^ " " one real repeated root. 3. fi'p- — 1, then i> <:^ 0, " " " " one real, two imaginary roots. As to the cubics this says that 1. foTfK^ — 1, the cubic cuts the axis of reals in three points and hence is, in general, of the bipartite type. 2. for /^ = — 1, there is one actual intersection and an acnode on the axis, as was noted in § 4 (a), p. 8. 3. for ^ >- — 1, there is but one real intersection and the cubic is of the unipartite type. The study of these intersections thus enables us to classify the cubics from the parameter. 9. Besides the aids in constructing the syzygetic pencil furnished by these facts as to the asymptotes, the inflexional tangents, and the intersections of the cubics with the axis of reals, we present, as the final means of facilitating the construction, the drawing of a number of circles concentric with the unit-circle and the calculation of the intersections of the cubics with these circles. The cubic (7) cuts the circle xy = p^ where Since x and y are conjugate complex coordinates, a: + «- = c is a right line perpendicular to the axis of reals at a distance ^ c from the origin. Therefore, the cubic cuts the circle where this perpendicular does, for values of c which are the roots of the equation in fic + ^ J . The location of the perpendicular is given by 4fi ^ ' I. The Syzygetic Pencil. Drawings of the Pencil and Range. 13 10. Two interesting matters of analysis will be noted before presenting the drawing. The Hessian cubic whose parameter is ^i' is Hessian to three cubics whose parameters are the roots of ^3 + 3^^3 + 4=0. (13) Their asymptotes cut the axis of reals at i /[^ resp. The Hessian itself cuts the axis of reals at points given by the roots of 2a;3 + S/tz'cc^ + 1 = 0. If we put in this equation x=-\^i, we obtain identically the former equation. Therefore, the Hessian cuts the axis where the asymptotes of its curves do, and these asymptotes are its tangents on the axis of reals. Or, for this form of the pencil, we may say, the Hessian is tangent on the axis of reals to the asymptotes of its curves. Since the axis of reals is the harmonic polar of the real inflexion at infinity and the asymptotes are flex-tangents at this inflexion, we may state the theorem projectively : The three flex-tangents of the three cuhics with a common Hessian, at any one inflexion, touch this Hessian in three points on the harmonic polar of the inflexion considered. This is an extension of theorems by Alfred Clebsch^ and Peter Muth.^ The discriminant of equation (13) is Z> = — 432 (^'^ + 1). Thus, if the Hessian is bipartite, i. e., if /u'<] — 1, there are three real unipartite cuhics of which it is Hessian; for the roots of equation (13) are then all real, two positive and one negative but greater than — 1. If (i' = — 1, its cubic is Hessian of the cubics whose parameters are — 1, 2, 2. That is, it is Hessian of itself and of one of the two real harmonic cubics twice over. If the Hessian is unipartite it is the Hessian of one real, bipartite cubic. Since our names unipartite and bipartite, following Salmon, are actually names of the cubics von Staudt calls resp. odd- and even circuit cubics, and since 3 by no projection does a non-singular cubic change its class of odd- or even- circuit, these facts as to the Hessian and its cubics remain for all real projections. 11. The two real harmonic cubics are readily seen to be mutually Hessian and cubic, for the parameters — (1 + V3) and — (1 — \/3) may be interchange- ably (J, and fi' and satisfy equation (13). 1 Ueber die Wendetangenten der Curven dritter Ordnung ; Crelle Journal 58, s. 232. ^Ucber ternare Formen, u, s. w. ; Inaug. Diss Giessen, 1890, s. 15. SC.-L. : II, end of foot-note p. 223. 14 I. The Syzygetic Pencil. Drawings of the Pencil and Range. Therefore, the flex-tangents of each touches the other and so, in our form, each cuts the axis of reals at the asymptote of the other. Compare Clebsch in Crelle Journal, Bd. 58, ss. 238, 239. 12. For the form (7) of the pencil, the two invariants are got very nicely as the invariants of the quartic ^ giving the intersections of the four tangents from the real inflexion at infinity. As shown in § 6, p. 11, the flex-tangent at infinity cuts the axis at a; = ^jw, and in §8, p. 12, the cubic cuts the axis at 2ic^ + 3//a:^ -f- 1 = 0. Therefore, the intersections of the four tangents from the real inflexion at infinity are given by the quartic (2 a^ + 3/1^0^ -|- 1) (2x — - /i^) = 0, or 4ic* + 4^a^ — 3^^0:2 -f 2a; — /[i = 0. The invariants^ of this quartic are those of the cubic, given in § 4, (c) and (cZ), to within a numerical factor. The Syzygetic Range of Cubics.^ 13. The invariant parts of the syzygetic pencil just studied are well known and are known to correspond dualistically throughout. In view of this corre- spondence it is observed that the polarity arising from the conic a;SH-«i + «i=o, (14) namely, «i = ^i, 3^2 = ^2, «3 = ^8, sends each part into its corresponding part, also the point-cubics of the pencil into the line-cubics of range, ^f+e + ^i+6m^i^3J3 = 0. (15) The conic (14) transformed into conjugate coordinates becomes 2(a^-f a;2/ + 2/^) + 3 = 0, which equation shows the conic to be an hyperbola with the inflexional lines (see -B, (8), p. 8) x — (ay = and x — id^y = as asymptotes, with vertices at ±: hi^/Q. Transformed to rectangular coordinates with the axis of reals as the X-axis, it becomes 6X^ — 2 F^ -|- 3 = 0; and with the asymptotes as axes it is XF= — '. From this equation we easily construct the conic by considering the equality of segments of chords contained between the curve and its asymptotes. 1 Salmon: H. P. C, §228, p. 199. « Weber: I, $ 70, 8. 230; II, § 108, s. 406. •C.-L.: II, p. 244 sq. I Fig. 1.— The Syzy'getic Pencil of Cubics. Fig. 2. — The Syzygetlc Range, polar reciprocal of Pencil as to Conic II. On a Closed System of Conies. 15 By the polar- reciprocal process as to this conic we deduce the line-cubics of the syzygetic range from the point-cubics of the pencil. The harmonic polars are seen to become the cusp-tangents common to all the curves of the range. Explanation op the Figures. The Syzygetic Pencil. The inflexional lines are solid. The harmonic polars are dash lines. All cubics are marked with their parameters. Specially related ones are the same kind of lines. The cubics represented are : 1. Two equianharmonic, ^ ^ and 2. 2. Two harmonic, ii=. — (1 — V^S) and — (1 + VS). They cut the axis of reals respectively at — i (1 + VS) and — J (1 — V3); the former is uni- partite, the latter bipartite. Their inflexional tangents from u and o^ are drawn in fine dotted lines. 3. The cubic for which /[^ = — § is Hessian of three for which ^ is resp. 4, 1.132f, — 0.88 2^. These four are drawn alike. 4. The cubic ^ =: ^ is Hessian of but one real cubic for which ^^=^ — 2. 5. The equianharmonic cubic, ff = 0, is also Hessian of but one real cubic for which ^ = — \/4. It is draw different from the others. The Syzygetic Range. The majority of the above cubics were reciprocated in the conic shown on the figure and the polar-reciprocal line-cubics appear drawn in the same kind of lines as the corresponding point-cubics from which they were derived and are named with the same value of ^ respectively. Clebsch has a drawing (C.-L.: II, p. 243) of the range but does not profess it to have been constructed. Its form is very considerably diff*erent from the accompanying drawing. II. ON A CLOSED SYSTEM OF CONICS. 14. The conic (14) of § 13, p. 14, is referred to triangle A of equations (4), p. 6, as reference triangle with the first line of B (ibid.) as auxiliary line. Its effect on the parts of the syzygetic pencil as given in § 13, raised the question as to the effect of the analogous conies as to all the possible reference frames in the four triangles (4). The equations of these analogous conies may be got by considering separately each of the four triangles as reference triangle with the nine remaining lines in 16 II. On a Closed System of Conies. turn as the auxiliary line. Thus, there are thirty-six such conies, nine for each triangle. As an example, say we wish to consider triangle B and the first side of triangle G as reference frame. Then, Sci = &) a^i + Ci) ^2 + " ^3 = 0, x'2 = oci + o^a-g 4- oajg = 0, Xs = Xi -\- (dX2 + (0^ Xg =z , whence x[ -\- X2 -\- Xs = (o^ +2 6)) (cj^ccj + iCg + a^) = is auxiliary line as we desired. The conic x[^ -f- x^'^ -f jcg^ = has therefore as to the original reference frame the equation a xl -i- xl -^ xl -\- 2 {0x^X3 -^ XqXi -\- X1X2) = . Similarly by inspection all the others may be deduced. The equations of these conies will be given in the next chapter where they are derived differently and their properties are defined. Their development as above was published in The Johns Hopkins University Circular, January 1905, pp. 16 ff. These thirty-six conies form a closed system for by operating two times with a polarity arising from any conic of the system we get a polarity of the system ; or, the product of three polarities of the system is a polarity of the system. It is easily shown that when a conic reciprocates a triangle into another, the two triangles are in perspective, and conversely. For two triangles in n-fold perspective there are n such conies, so for the four inflexional triangles, mutually in six-fold perspective, there should be six times ^Gz or thirty-six conies, as there are. 15. As to their effect upon the inflexional triangles, the polarities divide into sets in two ways. First, by nines Ai_q, 5i_9, etc., they send the vertices respectively of triangles A, B, etc., into the sides opposite, and at the same time reciprocate another triangle into itself and the vertices of each of the other two triangles into the sides of the other of these two triangles. Second, they divide into sets of six each, operating as indicated in the table : Polarities. Triangles. Triangles. into themselves send A and B respectively, and G and D into each other -^1-8 J B^_^ sen Ai-ij ^4-6 A^-9, A-3 *' ^l-B, ^7-9 A-6, ■B.-. " ^1-3, A-» " A " G A " D B '' G D " B G " B D B B G A D A G A D II. On a Closed System of Conies. 17 The classification shows also that besides the nine polarities which send a triangle into itself, vertices into sides opposite, there are nine others sending it into itself, but vertices into sides in anti-cyclic order as to the order of the vertices. We shall speak of the three cyclic and the three anti-cyclic forms of per- spective, meaning as indicated here. Cyclic Forms. Anti-cyclic Forms. V 2' Z' ' V 2' Z' 12 3 13 2 2 3 1 2 13 3 12 3 2 1 As analysis readily shows, the projecting rays are the nine harmonic polars with four vertices of the triangles on each. The centers of perspective are the vertices of the triangles, and the axes their sides. Further, for any two of the triangles, the vertices and sides of one of the remaining two triangles are respectively the centers and axes in cyclic order for the three cyclic per- spectivities, and those of the other triangle respectively in anti-cyclic order are centers and axes for the three anti-cyclic perspectivities. 16. "We would call attention to two very neat papers on perspective triangles by J. Valyi.^ He makes a slight error as to 6-fold perspective triangles by saying, "Unter den 6 Kegelschnitten giebt es hochstens vier reelle, die beiden Dreiecke sind immer imaginar," whereas our equations (4), p. 6, show one whole triangle and one side of the other may be real. After publishing the article in the J. H. U. Circular (§ 14, p. 16), I found the following papers by S. Kantor. In 1895 he notes ^ the 36 collineations, in connection with two triangles in 6-fold perspective. The following year he speaks^ of 36 conies in connection with the four Hesse Triangles. He uses these conies, which are the 36 herein presented but defined differently as stated in Th. XIII and used to operate on the collineations of types 5, 6, 7 of Jordan * to produce the most general group of correlations which contain these types. By finding the "intermediate^^ or Salmon's contravariant conic Xi X2) = 0. D1S2: a)'^x\-\-xl-^xl-\-2{w^X2Xs-{-XiXi-\-XiX2) = 0. D2S2: xl-\-a)^xl-\-xl-{- 2{x2X3-\-(o'^x,iXy-^XiX2) = 0. DaSi : xl-{-xl-^Q)^xl-\-2 (jCj x^ -\-XiXi-\- o? x^x^ = 0. A-y8i\ x\-\-2x2X^z=.(i. A2S2: tcl -|- 2 a^g a;i = 0. A3S2: xl-{- 2x1X2=: 0. III. The Hesse Group of 216 Collineations. 21 D1S3'. a/ Xi -\- xl -{• xl -{- 2 o)^ {co^ Xi x^ + a^s «! -}- a^i a^a) == 0- DiS^: xl -}- io^ xl -{- xl -{- 2 0)^ {x2 x^ + co^ x^ x^ -\- x^ X2) = 0. 1)3/^3: xl-\-xl-\- o)^ ajg + 2 ft>^ {x2 a*3 + a^g cci + «>^ ^i ^2) = 0. -4i /S'3 : x\ -\- 2 0/ X2X3=z 0. AiSzi a^H- 2w^a;3a;i = 0. AsSs'. »! -f~ 2 a^l 4" ^^ ^H~ 2 o>^ (a^a a^s + w a^s a^i + w^ x^ x-i) = 0. BiS^: x\ -\- 0)^ xl -\- CD xl •\- 2 o? [x^ x^ -\- o? XzX-i-\- m x^ a^) = 0. Ax^i'. a;? -f- 2 a> a^a a^a = 0. Aa?sa;i = 0. A^H^'. a;3-|- 2tt>a;ia;2:= 0. -Si 'S'4 : x\-^x\-\-x\-\;-2oi (ajg x^ + a^s a^i + ^1 ^2) = 0. BzSi'. a;J-|- wa;! 4-^^2;3-j- 2w(a;2a;3 + coa;3a;i4- w^a^ia^g) = 0. B^Sii xl -{- 0)^ xl -\- a) xl -\- 2 0) {x2 Xs -\- 0)^X3 Xi-{- (o x^ x^ = 0. C\ ^4 : o)x\ -^ x\-\- xl -\- 2 (1) {(o XiXz-\- XzXx-\- XiX.^ =z 0. (72/8'4: a;i4-(wa;2 + a;3 4- 2ft>(a;2a;3 4-a;ia;2) = 0. These conies form a closed system as explained and are in the same order except ^2^3 ^^^ BiS^ as if derived as in § 14. Further, the line-forms are given by interchanging o and o^ and writing ^^ for x^. We name point-conics with Roman caps; the same letters in script name the corresponding line-conics. 20. The Conics as Source op the 216 Collineations. The product of a point-conic on a line-conic is a collineation. With this in mind we form a multiplication table with the point-conics along the left side and the line-conics along the top. The 216 collineations are written in matrix form and are numbered. The number is put in the square on the table corre- sponding with the two conics which produce the collineation of that number. Thus arranged the collineations were readily classified, from their actions on the inflexional triangles (and for simplicity on triangle A) into the tetrahedral subgroups. In the multiplication of matrices remember that the separate terms of the ROWS UPON the terms of the columns of multiplicand give rows of product. In numbering I put ajh, where a is the number of the collineation given and h is that of the one having g) and a^ interchanged. 22 III. The Hesse Group o/ 216 CoIIineations. 21. The 216 Collineations. 1 1 10 79/94 10 1 w2 85/103 2/3 CO 11 4 1 12 10 10 10 1 1 10 1 CO 1 10 o^'^ CO 1 co"" 10 10 CO a>2 5 1 13/16 10 1 10 10 1 10 10 CO 10 10 1 ft>3 6/9 ' CO 14/17 1 CO co^ 73/91 74/92 75/93 co^ 1 10 1 6>2 1 10 1 w^ 80/95 a.2 10 10 86/104 81/96 10 co^ 1 87/105 76/97 10 co^ 10 82/JOO a>2 10 1 88/106 77/98 1 10 w2 83/101 10 a>' 1 89/107 7/8 1 15/18 10 CO co^ 19/28 111 20/29 co^ 1 a 21/30 CO^ CO 1 22/34 111 23/35 CO co^ 1 24/36 \ O? CO 1 co^ a) 111 CO cu^ 1 CO 1 (O^ 1 1 1 \ CO CI? 1 CO Cl? CO 1 co^ 111 CO^ 1 CO CO^ CO 1 Ill 25/31 111 26/32 1 CO co^ 27/33 CO 1 co^ O? CO \ 111 CO^ 1 CO 0) CI? \ 1 CO^ CO 38/56 CO CO 1 111 37/55 ^2 1 1 39/57 CO 1 CO 40/58 \ CO CO 41/59 CO CO 1 42/60 1 ft»2 1 CO \ Oi w' 1 1 CO CO 1 1 ft>2 1 1 CO CO CO CO \ CD CO 1 CO X CO G>2 1 1 CO CO 1 1 co^ 1 1 CO CO 43/61 1 CO CO 44/62 1 1 cd" 45/63 CO 1 CO 46/64 co^ I 1 47/65 1 1 ft>2 48/66 1 a>' 1 CO 1 CO 1 CO CO 1 1 co^ CO CO 1 CO 1 CO 1 CO CO 1 1 a^' CO 1 CO 1 CO CO CO 1 CO 1 CO CO CO CO 1 49/67 1 CO CO 50/68 CO CO 1 51/69 1 ^2 1 52/70 «2 1 1 53/71 1 1 co^ 54/72 1 a>2 1 1 1 co^ 1 co^ 1 CO a?' co^ 1 1 co^ 1 ' 1 ft>2 1 1 CO 1 CO 1 CO CO 1 1 co^ CO^ CO co^ CO CO^ 0? CO^ CO^ CO 78/99 fu^ 10 1 84/102 10 10 a)2 90/108 1 10 10 co"" 1 cu^ 10 a>2 1 10 co"" 1 w« 1 co^ 10 10 10 III. Tlie Hesse Group o/" 216 Gollineations. 23 s % ^3 ^. ^. ^.A ^. 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V SI 7«^ ^ Zl3 Xiii ^ ^ 6>l 103 lOS lOX lis ISI I8f L jL 1 X07 "H XOi 88 9b 84 n ^1 is- 191 193 19? ^^ Ti,s, I3L /•^i I'Ji 7W '91 -m Ulo so £4 10 II li. lOtf Tie III IX-J a& M 111 W3 I07 181 iss 189 / 7 L> 37 40 75 118 114 IX 1 XOS XII X'U B. B.S. m m ii4H '91 lis 111. U% uf Sb IS llo n m IIS lib I3S 133 /JWl X 19 91 91 94 10 /3 It ^7 uu UX iiti lift qi Si. 8T fC 7^ is 180 145 IS3 I4<{ A 5. A.^. 14 77 ?o qs 9^ 9S '1 II l4 IfS 140 38 Ib-S^ /to* no ?6 83 ^9 iSb tfS Ibo los 101 lOi St h8 ■so 8 1 f nk ISO n*. /S3 14^ I4S A /lA 1^ yy SI 99 9b 93 /vT IS IX, Ul ^9 f> /*v Its IH g'f 81 84 /sr 1^ IbX lOh 104 /ox SI S4 'Tf 9 8 / ISO I7x nie lU/j /4^ IS3 A> B,S. lit /iz IXk U4 hS 72- /St^ /J9 l(.x /*7 IX,g hi 10 II /z lOf no III /ts IbO fbf SS Si bi 131, /i? 141 I4S I4f Af3 1 8 f '7^ nil, 180 B. 7^,Ss I'f 113 liM U 1^1 7/ iss 1^ /6/ (3i- l3o 131 IS li 14- nv Hi. IIA /i7lo 180 nx B3 OA W i.4 T? m 2jOLt rLoi /« M, no 181 irj 1*4 /sy '•n 11.0 2,8 TT 34 To IS 14 :u8 XlO Xoif 190 '91 19^ nx ISO '7l> I4S IS3 14^ 1 8 f a GA zo i2 :lIo iei Itfi zos i("^ lU m /»g ISJ- IfX is-s ISg llol 3S ^9 iX IS II /^ i/2 ill 113 '94 lis 'P lU lyt 1^ 14/f lis- IS3 f 1 8 a CA 2.1 2.3 ^7 Z« xn "7 lu K-l M I9f, /M 19^ I'fh •^9 ibi- si \ '1 13 a X/b XM XI4. l2 1 a>2 1 1 1 1 CO^ 1 c6^ 1 CO co~ co^ CO^ CO^ CO 115/130 1 (0 CO 116/131 1 1 co^ 117/132 1 ce>2 1 118/136 co^ 1 1 119/138 1 1 co' 120/137 1 co^ 1 1 1 co^ CO^ CO co^ CO co^ co^ 1 2 1 1 CO CO CO^ CO^ CO CO^ CO co^ 1 CO CO CO CO 1 1 1 w^ co^ CO co'^ 1 CO CO 121/142 1 CO CO 122/143 1 1 w^ 123/144 1 co"" 1 124/139 1 CO CO 125/140 1 1 co'' 126/141 1 co' 1 CO^ CO co^ CO CU^ Cl? 1 1 co^ 1 W2 1 co' 1 1 CO CO 1 1 1 co"" 145/181 1 1 1 O) 1 CO 146/182 CO co^ 1 co^ 1 1 CO^ CO^ CO 1 co' 1 149/185 1 CO co^ CO CO^ CO^ 150/186 co"^ CO 1 147/183 CO 1 w^ 148/184 1 1 1 CO 1 co^ 1 1 1 CO co^ 1 CO^ CO 1 1 1 1 1 CO co^ CO co^ 1 CO 1 w^ 1 1 1 1 CO co"^ co^ CO 1 1 1 1 151/187 111 152/188 CO^ 1 CO 153/189 1 CO^ CO 154/190 1 1 1 155/191 1 CO^ CO 156/192 1 CO co^ 1 CO^ CO 1 1 1 CO^ 1 CO CO co^ 1 1 1 1 CO^ CO 1 CO^ 1 CO 1 CO^ CO 1 1 1 CO 1 co'^ CO^ 1 CO 1 1 1 157/193 1 1 1 158/194 CO^ CO 1 159/195 CO co^ 1 160/196 1 1 1 161/197 CO 1 co^ 162/198 CO^ 1 CO co"^ 1 CO 111 CO 1 co^ 1 CO co^ 1 1 1 1 CO^ CO 1 CO^ CO 163/208 1 co"^ co^ 1 CO CO^ 164/209 1 1 CO 1 1 1 CO^ CO 1 cu co^ 1 167/212 co'' co^ 1 1 1 1 168/213 1 CO 1 165/210 1 w 1 166/211 w 1 1 1 CO 1 CO 1 1 co' co^ 1 w^ 1 co^ CO 1 1 1 1 CO 1 1 w w^ 1 co^ CO 1 1 1 1 CO 1 CO 1 1 co^ co^ 169/214 CO 1 1 170/215 1 1 CO 171/216 CO^ 1 fO^ 172/199 1 io'' co' 173/200 Cl/ co^ 1 174/201 co^ 1 co^ 1 CO 1 1 co^ co"^ 1 1 CO 1 1 CO 1 CO 1 CO 1 1 W2 ^2 1 I CO 1 CO 1 1 1 w 1 CO 1 1 1 1 CO 175/202 w 1 1 176/203 1 1 CO 177/204 1 CO 1 178/205 CO 1 1 179/206 1 1 co' 180/207 1 CO 1 co'' co'' 1 Co' 1 f«>2 1 W2 ^y2 1 1 CO 1 CO 1 CO 1 1 1 CO 1 CO 1 1 1 1 <«> co^ 1 co' 1 co' co' CO^ CO 1 22. From the system of forms for the G210 as given by Maschke (Math. Ann. 33:324) we see that all the collineations can be deduced from six particular ones of the conies in line-foi'm with two of these in point-form also. These six are Ill, The Hesse Group 0/ 216 Collineations. 25 Ai Si , A2S1, J-g Si , AiS-z, A2 S2 , and Gi S^ ; with the point-conics Ai Si and Ai S2 . All the 216 collineations can be deduced from these conies and no other collineations arise therefrom. The Tetrahedral Stjb-Groups. 23. The sub-groups arising from the isomorphism with the 6^13 in the plane are these : to the identical transformation corresponds 1 Gi^ , " " 3 sub-groups G2 correspond 3 G^ss's, " '* 4 " " Gs " 4 G^'s, " " 1 '' " Gi corresponds 1 6^73. The four flex-triangles are regarded as the four faces of the regular tetrahedron. The collineations naturally then transform the four triangles thus : The triangles AB CD are sent respectively into " A BCD by collineations 1-18, " " BADC '' " 19-36, " " CDAB '' " 37-54, " " DCBA " " 55-72. The first eighteen form the identity Gi^-, the other three sets of eighteen each, each with the collineations of the Gi^, form the three G^s which inter- change the triangles by twos. All these form the one 6r^3, which permutes all four triangles. Next the triangles are permuted as follows : triangles AB CD are sent respectively into triangles AD B C by collineations numbered 73-90, 1. 2. 3. 4. ACDB DBAC CBDA { DACB \bdca CABD BCAD 91-108, 109-126, 127-144, 145-162, 163-180, 181-198, 199-216. The collineations of each of these four sets in connection with those of the 6^13 form a sub-group G^i , leaving one triangle unaltered and permuting the other three cyclically. 26 III. The Hesse Group of 216 OoUineations. The collineations of the Gig of course send each respective inflexional triangle into itself, but not point for point in each case. A comparison of the G^^ with respect to triangle A at once shows that, since A is, therefore, each triangle is sent by three collineations identically into itself. The collineations doing this are in each case identity and two of those of period three. Thus the action of identity and the eight of period three is accounted for. The remaining nine collineations of period two divide, differently for each triangle, into sets of three. Those of each set act the same on the triangle considered, sending one vertex into itself and interchanging the other two; that is, in other words, as is seen by comparing the equations of the harmonic polars, one of the three harmonic polars on the fixed vertex of the triangle remains, being the fixed line of the collineation, and the other two harmonic polars on that vertex are sent the one into the other. Other Sub-Groups. 24. These sub-groups are those observed by noting the effect of the collineations on the 36 conies. Notice that the conies naturally divide into sets of nine, each set composed of three sets of three. By examining the collineations of the G^^ we readily see that numbers 1, 4, 5, 10, 11, 12 form a dihedral group Gq, sending conic B^Si into itself. Hence from symmetry, there are 12 dihedral G^s, one for each set of three conies. These sub-groups each contain the identity collineation, two of period 3 and three of period 2 ; there is thus in each dihedral G^ a cyclic G^ which is an invariant sub-group. So the transform of one of the collineations of period two by one of the G^ is one of those of period two. The 6^18 contains identity, eight collineations of period 3 and nine of period 2. The former ones compose four cyclic G^s ; the latter with identity compose nine cyclic G28. Next, we observe that the collineations of the three sets of 18 each which with the 6^18 ^o^m the (r^g, are all of period four. These by sixes have a common second power. See Valentiner loc. cit. p. 69. These nine second powers are the nine collineations of period two. So one transformation of second order, the two of fourth order which produce it, and identity form a sub-group of period four. Thus there are twenty-seven cyclic G^&. It seems strange that Valentiner should have overlooked and even practically denied the existence of the G^2i6> when he knew there must be 54 transformations of period four, and states that this number is \ N, where N represents the total number of transformations. III. The Hesse Group o/ 216 Collineations. 27 He further gets the (r^g with its sub-groups. I am inclined to question the complete accuracy of the French resume for at the end of §45, pp. 173, 227, the Danish and the French are exactly contradictory. The cyclic G^b divide the conies into sets of two and four, half of a set belonging to each of two natural sets of three. In each of the eight sets of 18 each, which remain to be considered, 9 collineations are of period three and 9 of period six. These former of each of the four sets (1, 2, 3, 4, p. 25) of thirty-six with identity form 9 cyclic 6^3's, or in all 36 more cyclic (rg's making the total number 40 cyclic G^n. These 6^3's separate the conies into sets of three and permute them in the set. Further, the collineations of period six in each set of 36 appear by twos in 9 cyclic Gq9,. A cyclic (rg , different from a dihedral (rg , contains two collineations of period 6, two of period 3, one of period 2, and identity. There are thus from each set of 36 collineations above mentioned, together with the 9 of period two and identity each time, nine cyclic G^b or in all 36 cyclic Gq&. These separate the conies into sets of nine. Two of the collineations of a G^ send one of the nine conies into itself, two more send the same conic into each of the other two of its natural set of three, and the remaining two, which are of period six, send it into one of the remaining six conies of the set of nine. Jordan says, loc. cit. p. 18, that a group G belonging to this (Hessian) type (of order 24*94)) contains a group H of order 27. "The substitutions of G are permutable with ^." In each G^ there is such a group 6^37 consisting of the 26 collineations of period three and identity. So further we have 4 6^27's. Three of the collineations of period three from each lot of 36 above referred to, together with two of the eight of order three in (rig, and identity, form an Abelian 6^9. Thus finally there are four Abelian 6^/s. 25. To recapitulate, the Hesse group contains besides the sub-groups of § 23, these others : 9 cyclic G^s, 40 cyclic 6^3's, 27 cyclic G^s, 12 dihedral G^s, 36 cyclic (rg's, 4 Abelian G^'s and 4 G^^b. These arrange themselves in interesting form on the multiplication table as the heavy rulings help to indicate. The classification as to periodicity is 1 collineation identity, 9 collineations of period 2, 80 ii (I (I 3, 54 t( 11 ii 4, 72 f( 11 11 6. 28 IV. Perspective Triangles. — Complete Pappus Hexagon. IV. PERSPECTIVE TRIANGLES.— COMPLETE PAPPUS HEXAGON. 26. Having treated so fully the relations of the four inflexional triangles, which are in six-fold perspective, we should show how to obtain sets of triangles in the other perspective forms. Historically, it is of interest to note some papers. In 1870, H. Schroter (Math. Annalen 2:553-562) set for himself the question whether a triangle can be in more than single perspective with another triangle, and if so, in what other forms. His paper is characteristically clear and easy of reading, its method is synthetic, and it presents a construction for triangles in all possible forms; viz., one-, two-, three-, four-, and six-fold perspective. J. Valyi, whose paper was referred to in § 16, p. 17, set for himself in 1882 the same problem and reached the same results analytically without reference to Schroter. Some other papers directly or indirectly presenting perspective triangles are simply noted : Rosanes: Ueber Dreiecke in persp. Lage. Math. Ann. 2:549. Hess: Beitrage z. Theorie d. mehrfach persp. Dreiecke. Ibid. 28 :167. Third: Triangles triply in Persp. Proc. Edinburg. Math. Soc. XIX, p. 10. L. Klug: Desmische Vierseiten-Systeme. Monatshefte (1903) XIV, s. 74. M. Pasch: Ueber Vier-eck und seit. Math. Ann. 26 : 211-216. Caporali: Memorie, pp. 236, 252. Veronese : Sull' Hexagrammum mysticum. Lincei Mem. II, 1 (1877), p. 649. Triply Perspective Triangles in Circular Coordinates. 27. It is well known that two concentric equilateral triangles are triply perspective. We take two such, first as point-triads, with coordinates of vertices resp., 12 3 1' 2' 3' 1 o 0)^ at i^at oat These two point-triads are in perspective thus, 12 3 having Center, and Axis of perspective a, A, /?, B, y, r. (17) 1' 2' 3' 2' 3' 1' 3' 1' 2' IV. Perspective Triangles. — Gcmiplete Pappus Hexagon. 29 We name the vertices of the triangle of axes opposite correspondingly named axes A, B, G. The line 11' has coord. 1 — a 'J' at— 1, " 4. ^-at, (f (( 00/ '< '< o^- aa t ' G)^ at — 6), (d^a — a at, t (t 11 00/ 11 11 0) — t ' 0) a< — G)^, G>a . — cjrat. z The determinant of these coordinates vanishes identically for the sum of each column is zero. Therefore the lines meet in a point a. Similarly we show the points /? and y, and find then the coordinates of the three points to be ''-^~ t{l-a') ' ^-^- t{\—a^) ' r-""- t{\-a') ' ^^^^ From these coordinates we see that these three centers of perspective are on a circle concentric with the circles about the given triads, equally spaced and ordered as 1, 2, 3. The radius of their circle is 1 — aW (at^^\)(a — t') ^jg^ With radii of the given circumcircles resp. a and h this derived circle has radius ah \ (at^ — b){a — ht^) The coordinates of sides opposite the vertices 1, 2, 3 resp. are proportional to (1, 1, 1), (1, G)^ G)), (1, G), G,^). (20) Those of sides opposite 1', 2', S', resp. are (1, t^, at), {1, (ot^, cy'at), (1, (^H\ (oat). (21) Taking these sides in the same order of perspective as above we deduce the coordinates of the axes. From coordinates (20) and (21), we write off at once _ at — t^ __(^at — GiU"- _Gi^at — c}t^ where x^, is the point of intersection of sides i and i'. These three points lie on a line for we have identically at—t^ 1—at t^—1 a at — a^t^ a — cd^at ot^ — g)^ =0. Q^at — ot^ Gj^ — a at a^t^ — g) 30 IV. Perspective Triangles. — Complete Pappus Hexagon. Thus we get the coordinates of axis A to be [a<3_i, t{a—t% t^{l — a^)]. B " " [«<' — !, i^H{a—t% o<2(i_a2)]. (22) r " " [«<' — 1, i^t{a—t% GiH^il—a^)-]. From these we get the coords, of the vertices of this triangle of the axes thus, a,H{a — t^) 6)^2(1— a^) The meet of BF is J. = x = at^—1 G>H{a—t^) at^—1 at(a — t^) {l—a^)t' at^ — 1 o _ G) (1 — a^) t'^ _ G>^1 -a^)t' (23) C = x = Similarly, 'oF— 1 ' ^ ^ otF— 1 Clearly these points form an equilateral triangle, concentric with the other three, and ordered as 1, 2, 3. The radius of its circumcircle is ^^—')^ (ae-tia- {at^—l){a — ty The radii of the four circumcircles are seen to form the proportion. (24) a \{al-V^{a-f)_ , 1 t^ {at^-l){a — t^) or, in other words, the derived circles are a pair mutually inverse as to the same circle as to which the original circles are inverse. Therefore, we may state the theorem, Two concentric equilateral triangles are in triple perspective with their centers 0/ perspective and the three axes also equiangular triads concentric with the original two; and the radii of the circumcircles of the latter two triads are functions of the radii of the original circumcircles and of the clinant of the angle between the given triads. The product of the radii of the latter two circles equals the product of those of the former two; hence, the circles are by pairs mutually inverse in the same circle. The point-triads 1, 2, 3, and 1', 2', 3', are each in triple perspective with a, ^, y, thus, 12 3 with perspective centers 1' 2' 1' a 7 P /? a Y r 8 a a r /3 /3 a 7 7 a centers 1 2 3 So, as is known, two point-triads in triple perspective with their triad of centers IV. Perspective Triangles. — Complete Pappus Hexagon. 31 of perspective are a set of triads two and two in triple perspective with the points of the third triad as centers. Triangles 1, 2, 3 and A, B, G. 28. These point-triads as here written are in anticyclic triple perspective with centers of perspective a', ^', y'. From coordinates (17) and (23) we write the coordinates of the perspecting rays : a—t^-.(l — a^)t 1 — q<3-f (l_a»)f2 t{l — a'){t*-{-at^ — at—l) 1^= ^Z^T' ' at' — l ' {a — t'){at'—l) -^ ai^{a — f) — oi{\ — a^)t 6^(1 — a<^)4- q>^(1— o^)<'^ oit{\ — a^){t^-\'Oiat—ai—cS) ^^' ^r=^F ' at^—\ ' {a — t^){at^—\) ai{a^t^) — 0,^1— a?)t ai^{l—at^)^ai{l—a?)e ftj''<(l— a^)(<^ + tt>^a<— a<— q>») ^^' ^"Z::? ' a<8 — 1 ' [a — e){at^—\) The determinant of these coordinates vanishes identically, therefore the three lines meet in a point. From these and similar coordinates we find, "" -^-a(a^3— 1)' ^ -""^ a(at'-l)' ^ - ^ a (a<« - 1) * ^^^^ The radius of their circumcircle is -* ~^-— « — ^yt lar* (26) a \ {af^ — l)(a — T) The axes of perspective are found by taking the intersections of the sides thus, 12 3 A r B giving side of triangle 1' 2' 3' opposite 1'. •p A p (( II II H it tt ol p T> A (f H H (I tt tt ol By the same steps we show that Triangles 1', 2', 3', and A, B, O, are in cyclic triple perspective with centers of perspective a", P", y", and axes of perspective the sides of triangle 1, 2, 3 in order 1, 3, 2. The radius of the circumcircle of a", ^", y" is From a comparison of the radii of a^y (19), of a'^'y' (26) and of a"^"y'' (27), we may summarize thus: The circumcircle of a, /?, y, which are the perspective centers o/|i/ o' 3'P "^^ 32 IV. Perspective Triangles. — Complete Pappus Hexagon. {1 2 31 A O b\^ u'Aoee axes are 1', 2', 3', are inverse to the circle of 1, 2, S; also the circle of a, (3, y, and th/it of a'', /3", y", the centers {V 2' 3'1 A T> n\i whose axes are 1, 3, 2, are inverse to the circle of 1', 2', 3'. The reader can very readily draw the figure for all the steps above outlined, so we leave that to him. 29. As a more interesting source of triply perspective triangles, and one involving some considerations of the cubic, we present what we shall call THE COMPLETE PAPPUS HEXAGON. In Pappi AlexAndrini Mathematicae Collectiones^ a Federico Commandino Urhinate as Prop. 138, p. 368, we read: Si parallelae sint AB, CD, atque in ipsas incidant quaedam rectae lineae AD, AF, BC, BF, & ED, EC, jungantur, rectam lineam esse, quae per GMK puncta transit; and as Prop. 139, p. 368: Sed non sint ABCD parallelae, & in puncto N conveniant. Dico rursus rectam lineam esse, quae per GMK puncta transit. These theorems Pappus proved by proportion, the equal ratios being respectively between two lines and between the rectangles of two pairs of lines. Salmon^ has the same theorem stated thus: "If ABC are three points of one line and A!B'0 are three points of another line, then the intersections BC'IEC, CA'/CA, AB/A'B lie on a line.'' The most important mention of the simple case is by Rudolf Boger,^ who gives it as a simple form, free from the perspective relations, of Das Sechsech in der Geometrie der Lage. 30. The complete figure is constructed thus : The numbers 1, 3, 5, and 2, 4, 6, are regarded as the names of points or of lines, each set of three lying on a line or a point resp. We consider the cross- joins as follows: 1 i 2 1 3 > 1 4 1 6 Ui (1) (2) (3) '1' 3 2 5 4 1 e (4) (6) (6) [2] 5 2 1 4 3 6 (7) (8) (9) [3] 3 2 1 4 5 6 (10) (11) (12) V 5 2 3 4 1 6 (13) (14) (15) '6' 1 2 5 4 3 6 (16) (17) (18) [«] on 2i or Di resp. on ^2 or Z>2 resp. (28) (29) iProm U. 8. Cong. Library; also by F, Hnltsch In 3 Vols, giving the Greek text also, see Vol. 2, p. 885. « Conic Sections (6th Ed.) §268, p. 246, Kx. 1. *Sechseck and Involution, Mitteilangen d. Math. Ges. in Hambarg, Bd. Ill, Feb. 1899, s. 887. IV. Perspective Triangles. — Complete Pappus Hexagon. 38 This means, when the six numbers are points ^ that the line of 1, 2 intersects that of 4, 5 in the point (l); the line 2, 3 cuts line 6, 6 in point (2); and 3,4/6,1 is the point (3); further, that point (1), (2), (3) lie on a line [1] which is one of three lines similarly got and lying on point 2i . Observe there are nine lines, such as 1, 2, connecting the two sets of three points each, which have not been named. In the dual figure we shall call the nine points ai_3, /?i_3, 71-3. On the figures, the six points shall be marked Tt^ and the six lines P^ ; the points or lines (n) shall be marked simply n. For convenience in the analytical work slight changes are there made as to names but these will be readily followed on the figures throughout. Following the names given to points and lines in the Pascal hexagon, the points 2i and ^2 are called Steiner points, whose line is S) the lines P^ are called Pappus lines ; and we call the lines Di and D^ Hessian diagonals, which intersect in h. 31. As a convenient projection of the hexagon, we take the points nz^rii^y Ttg on a line considered the axis of reals, and the points Ttj, Tta, 7^5 on the line at infinity so that the lines from the three points on the axis to these three are equispaced lines, parallel respectively to x-=.ty x^iidty x^=-{^ty going resp. to n^ Ttg itf, Lines on ;r2 = a: a!= = 1, 2, , 9 — are in general thus, .,, . a—ba — (a—b)o>*+U Tti^a With % b IS x r= -^— 1 ; and the points {q) — g- = 10, 11, .'..., 18 — of § 30 are thus, rti,o with %6 is a; = ^ _ 3 ' ; or points {q) are got from points {p) by interchanging b and c in the equation where p-=.q— 9. In the above, a and b each permute for a, &, c, but a is never b ; and t, j are each 1, 3, 6, but i is never j in any one equation, a, 5, c are the general points Ttg, 7t4, Tte along line D^. These 18 points in 6 sets of three each, as indicated in §30, lie on six lines P^. From the equations of points (^) and {q) we get the coordinates of the lines P. 34 IV. Perspective Triangles. — Complete Pappus Hexagon. For lines Pi, where i = 1, 3, 6, we find coordinates are P,: a-{-hQ)-\-ooi^, — {a-^ha}^-\-co})or*tf o)'^{ab-{-bca)-\-caaP) — {ab-^hca)^-{-ca(o)or*^H. The coordinates of lines Pj, where y= 2, 4, 6, are Pj: a-\-ho?-\-ciOy — {a-\-bo)-^cii?)ari^H, a) {ah -\-hcu)^ ■\- caio) — {ah-\-bcco-\-caa)^)arH. The three lines P< lie on the point 2i , which is X=. — 6) ah -f &CO H- cao)^ a -\- hdi ■\- cu)^ and the three lines P, lie on the point Sg, which is a + &w^ + CO) jc = — a These two points are evidently conjugate and therefore symmetrical as to the axis of reals. Further, since the axis of reals, or Z>2, bisects the line between Si and Sg, the two points 2 are harmonic as to the two lines D. They are the Hessian pair of the three points Ttg, 714, Ttg. Since the points 2 are independent of t, these points remain the same for any three equispaced lines on a, h, c resp., mutually parallel ; or keeping one triad of points fixed the triad on the other line may move all along their line subject only to the condition that the angles between the lines on the fixed points remain constant. That is, 2i and ^2 are the same for all triads on the second line having the same Hessian pair. Thus along the one line may be generated a pencil of triads with the same Hessian pair by turning the equispaced triad on X, each three points cut out at any instant being a triad of the pencil. Reversing the Process. 32. Starting with the three lines P< on 2i and the three Pj on Sg, the lines Pij intersect in nine points as follows : n j3 , CL^ — he -\- (a — h){a — 6)0^ t 2a — — c p D . ^^(^ — ci^ + {o — a){c — b)G)t Ac — a — o p p . ^^ h^ — ca -{- (b — c)(h— a)t 26 — c — a IV. Perspective Triangles. — Complete Pappus Hexagon. 35 The remaining six may be written at once by comparing these three with the following table : Pi P, P, Pi a\ o" x\ 1 .2 W, 1 c", o" a", 0) The lines joining these intersections in pairs as indicated in §30 meet by threes in points along D^ and Z^j, which are not in general the original points on these lines. The lines {q) to the three new points along B^ are given by the second lot of three permutations (29) and have slopes respectively, ^^, w^^, o^^^; so they turn just twice the angle from the axis as the original pair and are likewise equispaced. The lines (^) intersect by threes on three new points, TtJ, Tts, Tt^, along D^,, corresponding respectively with a, h, c, in order along the external segment of the line. They are, a — h W — ca x„ = — a (? — ah , where a, 6, c, permute cyclically. a — h 2b — c — a c — a 2 c — a — b 33. Thus far the origin on the axis has been arbitrary. Now consider it the centroid of the three given points, so that a + 6 + c = 0, whence also a^ — bc=.b^ — ca-=:. c? — ab and be -\- ca -\- ab =^ — X, 2 a — b- The three new points, TtJ, then become resp., a'k b'k = \ say, c = 3a, etc. X =■ X := 2bc -\- ca -\- ah' *" he — 2 ca + a5 ' The counter-triad of the three points a, 6, c, is — 2bc -\- ca i- ah X =■ cX be -)r ca — 2 a6 * be 3a 2ca -{• ah 36 be -\- ca — 2ab 3c from (xa/bc) =: — 1 ; call it a'. from {xh/ca) = — 1 ; call it h'. from (xc/ab) = — 1 ; call it c'. The cubic along the line a, 5, c is x^ — Xx — abe = 0. 36 IV. Perspective Triangles. — Complete Pappus Hexagon. Differentiating this as to a;, we have 3 a:^ — X = 0, the roots of which are the polar pair of infinity, the intersection of D^ and D^. Calling the roots /and/', we have / + /' = 0,and//'=-/2 = -/'«=_|. ,./2 3^/'2 = |. Then by comparing the above values of this paragraph, we see n[a' = n',h' = nid=\; so the new triad and the counter-triad of the original triad are in an involution whose double-points are the polar pair of the intersection of the lines Di and D^ as to the original triad. Further, starting with the new triad as we did with the triad Ttj, Tta, 7t^^ we shall secure a third triad nl^ Ttg, ni, etc. continuously, all these triads having the same polar pair with respect to h and the same Hessian pair. Thus, there is constructed, as it were, a syzygetic pencil of triads of points along a line. For all the above there is of course its dual, giving a corresponding pencil of line triads on each of two points. 34. From the Complete Pappus Hexagon we have the following theorems with their duals : I. Three lines P^ on each of two points Three points Tt^ on each of two lines 2i, ^2, joined by the line S, intersect Z^j, D^, meeting in the point 5, are cross-wise in nine points ai_3, ft_3, cross-joined by nine lines which meet yi_3, which join by 18 lines, which are in eighteen points, 1-18, which are the the sides of two sets of three point- vertices of two sets of three line-triads triads each. each. The triangles of each set are inter se The triangles of each set are inter se in triple perspective, having as centers in triple perspective, having as axes of of perspective the points 2i and Sg each perspective the lines D^ and D^ each three times, and three points Tt^ on the three times, and three lines P^ on the line D2 or D^ resp.; and having as axes point X^ or 2i respectively; and having of perspective the line D^ or D^ three as centers of perspective the point 2i or times each for the sets resp., and twelve ^2 three times each for the sets resp., other lines all of which pass through a and twelve other points all of which point e, which is the pole of the line S lie on a line E^ which is the polar of as to any of the six triangles. The 18 the point h as to any of the six triangles, lines above lie by three on six points The 18 points lie by three on six lines Ttj, which are three and three on the P^ which are three and three on the lines Z>i and D^ above. points 2i and Sg above. IV. Perspective Triangles. — Complete Pappus Hexagon. 37 II The lines Z^j, D2 and S are the false The points 2i, ^2 ^^^ ^ ^^^ the false sides of the complete quadrilateral of vertices of the complete quadrangle of the Hessian pairs of the line-triads P^ the Hessian pairs of the point-triads on 2i and on ^2 • Tt^ on D^ and on D^ . From these follows the general theorem, as also from the demonstration of §§31, 32. III. Three lines on each of two points give rise to three points on each of two lines, and the latter by reciprocating the process give rise to three lines on each of the original two points. The derived three lines have the same Hessian pair, or are inclined to each other at the same angle as the original three. 35. We now give a proof of the theorems on the left. Take the two Steiner points 2i , ^2, to have coordinates 0*1, Cg, Cs and $1, $3, $3 resp.; the triangle aia2a3, formed by the intersections of the Pappus lines as will be shown, as reference triangle; and the line S as auxiliary line. The line S determined by 2i and X2 is given by =: 0, and will be written SiXi -f ^3X2 + S^Xs = 0. Since this line is taken as auxiliary line ^1=^^2=^3=1; O-i -f (Ta + (Tg = 0, $i + $2 + $3 = 0, /g-j^x also $? Cj Gk — 5V S/fc (J i = C* 5"* — cij Sj , where i, /, h are each 1, 2, 3 successively. The equations of the Pappus lines are the corresponding minors as re- presented thus, P\ P^ Pb Pz Pj Pfi ^1 <^2 <^3 il $2 $3 Xi X2 2^3 Xi X^ Xg Their respective intersections, corresponding with the nine points P^j of § 32, have coordinates : ai: (1,0,0), a^: (0, 1,0), a,: (0,0, 1), A: (o'lSi, o'i$2, <^3$i), ft: (<^i52> 0'2$2j <^2 53), ft: ((^sSi, <^3 $3)' a-i JC2 Xs O"! <^2 ^3 $1 52 $3 meet lines p. (I ti p. 11 n p 38 IV. Perspective Triangles. — Complete Pappus Hexagon. where the intersections are named as indicated here : Lines P3 P^ P^ P2 P4 resp. in points ai 02 ag Pi Pe '' " '' ft A ft (33) - A A " " " 73 73 n From coordinates (32) we write the coordinates of the joins: a^asi (1, 0, 0). agtti: (0, 1, 0), ajag: (0, 0, l). ftft: (t^aSa, 0-353, '^iCs, 0). Evidently the three lines of each column here also lie on a point, for the columns of determinants vanish identically on addition. The three points are called Ttg, 7ti, Ttg resp, and have coordinates, ^i' (<'8f2» ''iCs, ^^aCi), TTi: (r 'e T\ I \ I \ aifty J ^^' -^IJ -^2- ^^1 \P\^ 0'3j <^2), Ul, $3, $2)- \ Tti, Si, Sg. A, ($2; $3, $1), (<^3, <^1, ^%i' fk, ^2, Si. A, { fo, $2, $1), (<^3, <^2; ^^l)- 40 IV. Perspective Triangles. — Complete Pappus Hexagon. The twelve axes other than 2>i and Dg are seen to pass through the point with coordinates (1, 1, 1), which is thus the auxiliary point e, the pole of aS' as to the reference triangle. Since S has the same pole, e, as to all point- h has the same polar, E, as to all line- cubics consisting of three joins of the cubics consisting of three joins of the six lines P^, two and two, six points 7t(, two and two, (Salmon: H. P. C, §166, pp.143, 144), e is the pole of S as to every and any E is the polar of h as to every and any of the six triangles on all six lines P^. of the six triangles on all six points Ttj. The 18 lines have already been shown, (34) and (36), to lie by threes on the six points n^ which are on the lines D^ and Z>g three and three. The lines A and D^ as diagonals or false sides of the complete QUADRILATERAL OF THE HeSSIAN PAIRS. 36. The following linear relation exists as to the three lines P^ on 2i, Cl ((Tg iCg — (T3 X^) -f (Tg (Cg X^ — (Ti ag) + (Tg ((Ti Xg —<^2^l)=^' The Hessian covariant of the binary cubic is given by the sum of the squares of these terms separately, and the imaginary Hessian lines are (Ti ((Tg a^g — (Tg Xg) + 6) CTg ((Tg X^ — (Tj X^ + id^ (Tg ((Ti X^, — (Tg Xj) = 0, and ffi ((Tg OTg — (Tg arg) + 6)^ (Tg ((Tg x^ — (Tj iCg) + 6) (Tg {a^ x^ — (Tg Xi) = 0. These two and the analogous two on 2g may be written in reduced form respectively (Tg(Tga:i + 6)(Tg(Tia:g + (o^(TiCrgiCg=0, ]l) (^zO^Xi -f G)^(Tgcria:2 -f cj(Ti(TgiCg = 0, { 2[ Sggga-i + cjSggiiCg-f (j2$i$2a;3 = 0, j3f $g5'3iBi + a)2$g$iiCg + G)$i$2a:3 = 0, J4[ These intersect as follows : |1} and 1 3} in imaginary point J: (^^<'if3], '^sCsL'^iCz-^^aCi]). The line 7/ is the real line D^ given by (35). The line -£?^ is the real line D^ given by (37). Therefore, D^ and Dg are the diagonals of the imaginary quadrilateral of the Hessian pairs of the line-triads on 2i and Sg. Hexagon of three equispaced lines on each of two points. Theorem VI. Hexagon of three points on each of two lines. Theorem I. 42 IV. Perspective Triangles. — Complete Pappus Hexagon. Theorem III. follows without further proof. 37. As may be seen from Fig. 3, for the reverse or dual process the three line-triads of a set are mutually in triple perspective thus : Triang^ les. 1 6 8 2 4 9 1 6 8 7 3 5 2 4 9 5 7 3 10 14 18 16 11 15 11 15 16 17 12 13 10 14 18 12 13 17 Centers. Axes of Perspective. -1, «, h. p., A, A- c, 2i, d. A, A, A- e, f, 2i. A, A, A- 9, ^2, h. . A, A, A- ^) ^2) I. A, A, A- ^2, m, n. . A, A, A- The points a, h, c, .... w, ?i are the twelve points on E. 38. Without further proof because they follow analytically from data already given and may be tested in Fig. 3, we present several theorems: ,IV. A triangle of one set of three is in two-fold perspective with any one of the opposite set of three triangles, but for all such perspectives — there are only nine axes each taken thereareonly nine centers each taken twice and situated on the covariant twice and situated on the covariant point e. The centers in each case are line E . The axes in each case are The point- and line-triads are between themselves in single perspective. The center of perspective in each case is 1.^ if the two triads are both of the first or both of the second set of three as herein classified, and Sg is center if they are of oppositely named sets. IV. Perspective Triangles. — Complete Pappus Hexagon. 43 39. The special forms or arrangements for the three lines on each of two points to which attention is called are, (1) Two sets of equispaced triads; i. e., lines at angles of-^. (a) The points being the two equiangular points of the triangles of one set of three. {h) The points being on the circumcircle of equiangular triads. (2) The points taken at infinity, (a) arbitrarily, giving two sets of three parallel lines each, at an angle 3^ with each other. (6) at the circular imaginary points of the plane, giving two sets of perpendicular lines. These various forms are handled analytically best by using special coordinate systems for the several cases, and furnish nice work in devising expeditious methods. I omit as irrelevant the various methods that were employed. 40. From these special forms we have the theorems, VI. For a triad of equispaced lines on each of two points, one set of the three point-triads consists of equiangular triangles with sides respectively parallel (Fig. 4). Thus one of the lines D is at infinity and the other is perpendicular bisector of the line S between Sj and ^2. Further, the circumcircles of the three equilateral triads pass through Si and Sg, and those of the other set of three triangles intersect in e, the pole of S as to any of the six triangles. VII. If to the conditions of the previous theorem we add that one of the set of scalene triangles is also equiangular, then the vertices of the other two triangles of its set are inverse points as to its circumcircle, and all the circumcenters of the set of three equilateral triangles are on the finite Hessian diagonal D. Theproof of this last theorem is especially neat by use of circular coordinates Finis. VITA. Charles Clayton Grove was born December 19th, 1876, the son of Lewis and S. Elizabeth Grove, at Hanover, Pa. His early education was received in the private school of Miss Martha E. Grove and in the public schools of Hanover. In 1896, he was graduated from the Millersville (Pa.) State Normal School, where he prepared for college. After two years of teaching, he entered in September, 1898, upon the classical course of Pennsylvania College, Gettysburg, receiving the degree of Bachelor of Arts in 1900. The next winter he was Supervising Principal of the schools of Hummelstown, Pa. In 1903, the degree of A. M. was conferred at Gettysburg. In October, 1901, he entered upon graduate study at the Johns Hopkins University with Mathematics, Physics and Italian as subjects. At its very close, this course was interrupted by his taking an instructorship in Mathematics in Pennsylvania State College. During the winter of 1905-6, he was an instructor at the Baltimore Polytechnic Institute, and completed the work for his doctorate at the Johns Hopkins University. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RCTURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO S1.00 ON THE SEVENTH DAY OVERDUE.