f f-'^m^mmcM^'^t^. -v-:: -> -.r 1 twV*. QA UC-NRLF $C lb3 411 EXCHANGE EXCHANGE JUL 891913 Quadratic Involutions on the Plane Rational Quartic DISSERTATION Submitted to the Board of University Studies of the Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy BY THOMAS BRYCE ASHCRAFT 1911 V ,r»R I Of tHf \ rAU'r^^'*''*' -^ .«"' Press of the Mail Publishing Company Waterville, Maine Quadratic Involutions on the Plane Rational Quartic DISSERTATION Submitted to the Board of University Studies of the Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy BY THOMAS BRYCE ASHCRAFT 1911 V Press of the Mail Publishing Company Waterville, Maine A ^'^■(^1 ,< Quadratic Involutions on the Plane Rational Quartic. By T. B. Ashcraft. § I . The General Theory of Involution Curves of a Plane Rational Curve of Order n. Let R" denote a plane rational curve of order n, and let it be given by the equation Xo = Oo<" + a,^""' + ajr' + — - + a„ (i) x^^bjf+b.t"-^ +bjr-'+—- +Z>„ x^ = c^ +c,^-' +cjr' +— - +c. If we join the parameters t^ and /j by a line, where /, and /j are in an involu- tion of the form (2) A t^, + 5(/, + /j) + C = o, we shall show that the locus of this line is a rational curve of class n-i which touches R' 2){n-2) times and meets it in 2(n-2){n-T,) other points. This class curve will be called an involution curve, and will be denoted by r*"'. Cut the curve R" by any line (3) (fx) =fo»o + fi»i +^2^2=0 and we have (4) («„$) r + (« .0 /- + + {a J) = o. For convenience suppose we choose the involution with o and 00 as double points. Then / and -t must satisfy the last equation, and we have for n even, say n = 2 m, 263424 2 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic (5) («oO t"' + («iO ^•»-' +--- + (a„f) = o, (6) («of ) f ""— (« .f ) /'"-' + -— + (««0 = o. Whence by addition and then subtraction we get (7) («oO i"' + («2f) t""-' + ----- + («„f) = o, (8) («.f) f^-' + .(«3f) ^^"-^ + — + («„-xf) = o. Since only even powers occur we can divide the exponent by 2 and write (9) («of) i" + («20 t"-' +--- + (««f) = o, (10) («.f) r-' + («30 /«-' + ----- + (a„-.e) = o. EHminating $„, f „ f 2 from equations (3), (9) and (10), we have the locus re- quired in determinant form (11) /o(r) , /.(/") , /,(r) Or written parametrically its equation is and since n = 2w we have as the representation of the involution curve (12) f, = F,(r'), which is a rational class curve of order n-i. Similar argument holds for n odd. The point to be emphasized is that the parameter may be replaced by a new one which reduces the degree by one half; that is a T replaces a quadratic in t. This new parameter may be chosen in a triple infinity of ways depend- ing on the ratios of a, ^, v, 5 in a transformation of the form ctt + ?> 1 = vi + S . If the double points of the involution are given by (a ty, then we choose any two quadratics apolar to (a t)', say (a/)* and (^ /)'; then any convenient mem- ber of the pencil (« t)' + X (p i)' will serve as a new parameter T. To find the number of contacts of the r"'' with the R", we shall consider first the R* and its involution cubic r^. The R"* is of class six so there are 18 common lines. There are three ways in which we may have common lines. A line meets the curve in four points, /„ ^2, t}, tt. Let ^, and /j come together so that line is a tangent. We have common lines when T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic 3 i) Points ty and t^ are a pair of the involution, 2) Points t^ and /< are a pair of the involution, 3) Points /, and t^ are a pair of the involution. Case i) can happen twice, viz. when the line cuts out either of the double points of the involution. This accounts for two common lines. In case 2) a tangent at t meets the curve again, say at t^. For a given t there are two /,'s, and for a given ^, there are 4 i's, since there are four tangents from a point on the curve, the curve being of class six. The relation connecting t and ti is /. (<*) t* + f, (t*) t, + f, (t^) = o. The condition that the roots /, be in an involution is of the fourth degree in t, which means four common lines for case 2). Case 3) must contain all the other common lines, that is twelve. This case happens when /j and tj are a pair of the involution, but ^2 and ^3 are as well a pair of the involution; therefore the twelve common lines are six repeated. In other words the R* has six contacts with the r^. This is easily extended to the general case of R". The R" is of class 2(«-i), so the R" and the r"'' have 2(«-i)^ common lines.- There will always be two of these accounted for in case i), corresponding to the double points of the involution. For case 2) the equation connecting a point of tangency / and a point of intersection of the tangent at /, is of degree 2W-4 in t and n-2 in /„ and is of the form / (/"-^ /i"-») =0. For two values of ti to be in a given involution is a condition of degree n-2, in the coefficients of ^, and hence of degree 2(w-2)(«-3) in t; this then is the number of common lines for case 2). Subtracting the common lines for case i) and case 2) from the total number we have for case 3) 2(«-i)» — 2(n-2)(n-3) — 2 = 6n-i2. But since these pair ofif we have in general 2>n-6 contacts of R" and r""'. The order of r""' is 2 « - 4, so the R" and its involution curve intersect in 2 n (n - 2) points. The contacts count for 6 n - 12 intersections, so there are 2(«-2)(«-3) remaining intersections. If the parameters of a node of R" are in the involution, then the node is a factor of the involution curve, and the remaining factor is an r"''. This part of the locus being a double point of the R" will count for two contacts; 4 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic hence the remaining part, r"'', will have only 3 « - 8 contacts. If n-T, nodes are made a part of the locus then we always get for the remaining part of the locus an r' with n contacts or full contact, that is a conic all of whose intersections are contacts. Since two sets of the involution determine the involution it is a condition on the R" for n-3 nodes to be in the involution for n greater than 5.* We shall now consider the R^ and find the involution conic. We found that there are in general 3W-6 contacts, so in this case we get three contacts and no extra intersections. Let the double points of the involution be o and 00 , and let the reference triangle be the tangents at the double points and their join. Then the equation of R^ may be (I) Xo = aat^+bat' Xi =Cit +di X, = b,t' +cd. If o and 00 are the double points of the involution it is of the form <2) ^,+^2=0, and the line whose locus is the involution conic will join t and -t of the R'. Such a line is given by = 0. ^0 > X, X, a,t^+h,t* » c^t +i, b,t' +c,t -a^^ + b^t' > -c,t + di b,t'- c,t This determinant is readily seen to reduce to the following: (4) Xq , Xi , Xi bot' , d, , b,t aj.' , Ci , Cj which is parametrically (5) $0= — b^Cit' -\-Cidi f ,= ajb^t* — b^^f ^2= (^0^1 — a4i) t'. *In a Desargues Configuration B there is a sextic with nodes at the ten points of the Configura- tion. Any three nodes on a line would be in an involution, hence we could get ten conies having full contact with the sextic. T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic 5 It is seen that only even powers of t occur so we replace the parameter t' by a new one, t' say. For convenience we drop the primes and write the equation in the form f = — biCit + Cidi (6) $i=a„b2t' — 60^2^ $2= {b^Ci — a^di) t This is the involution conic in line form but we want the point form. We have ^0 = ^0^2 ((^odi—boCi) t' (7) ^1 = Cidi {aod^—boC^) X2 = — aJ)2Cit' + 20062^2^^1^ — bgCldi. Now in order to get the intersections of this involution conic with the R' we must eliminate the parameter from the equation of the conic and thus get an equation of the second degree in x. If we then substitute for the x^'s their values in the equation of the R^ we obtain a sextic in t which will give the intersections of the two curves. Eliminating / from (7) w e get bi'Ci'Xg' + b^c^x' + {a^d' — 2a^gCyd^ + b^c^) x* (8) + 2 {aJbaCidi — bg'c^c^ XyXi + 2 {aab^c^d^ — 60^2^1*) ^0*2 + 2 ( bobiCiCi — 2 aobiCidi) ^o^i = O- If we now substitute for the x/s their values in equation (i) we get ao'b^'Ci't^ + 2 a^'b^'c^dfi + (ao'b^'di' — 2ag'b2C,C2di) t* (9) — (2 ao'b^Ctdi' — 2 aobobiCiCjdi ) t^ + {a^'c^'d' — 2 aobJbiC^d^') f + 2 ajy^f^d^t + b^c^d* = o. This sextic is seen to be the square of the cubic (10) aJb^Cit^ + ajb^dif — agCidit — 60^2^1 = o, which gives the parameters of the three points of contact of the R' and its involution conic. We shall now prove that the points of contact of an R^ and its involution conic are given by the Jacobian of the cubic giving the parameters of the three flexes of R^ and the quadratic which gives the double points of the involution. 6 T. B. A slier aft: Quadratic Involutions on the Plane Rational Quartic We shall consider the R^ given by equation (i), and the involution whose double points are o and oo . The cubic giving the flexes is the fundamental cubic, that is, the unique cubic apolar to each of the three binary cubics in (i). Calculating that cubic in the usual way we have (ii) a^biCtt^ + 2>ciob2dit' + scoCa^i^ + b^Cidi = o. The quadratic giving the double points under consideration is (12) t = o. The Jacobian of (11) and (12) is (13) agbiCit^ + ajb^dit"" — a^Cidit — 5o^2^i = o, and is just the same cubic as (10), and the theorem is proved. We shall now consider R'* and its involution cubic r^. The number of contacts we found to be in general 3«-6 which is just the number of flexes of an R". Since the Jacobian of the flex cubic and the quadratic of the double points of the involution gave the contacts of R^ and r^ it seems natural to look for some such relation in the case of R'*. The Jacobian of the flex equation in general and the quadratic giving the roots of the involution will always be of the right degree in t, 3W-6, to give the points of contact of R" and r""'. But in the case of R'* we find the degree in the coefficients not the same as those of the contact equation. We shall find by a symbolic method the degree of the contact equation in the coefficients of the fundamental involution, as well as in the coefficients of the quadratic of the involution. Suppose the fundamental involution of R* is given by (i) (aty + 1 i^ty = o. Let the double points of the involution be (Qty = o, and let one set of the involution /, and t^ be given by (at)'. Let the line on /i and ^2 meet the R* again at T, and Tj. Since every line section is apolar to the fundamental involution we have (2) \aa\ (aT,) (aTj) = o and (3) M (^TO (^T,) = o. Eliminating Tj from (2) and (3) we get T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic 7 (4) |«^||«aMM.'|(«T,) (PT.) =0. Since every set of the involution is apolar to (Qt)', we have, (5) \aQ\' = o. Again, since when /, or /^ is T,, there is a contact, we have (6) (aT,)»=o. Solving for the a's in (5) and (6) we find them to be of the first degree in the Q's and of the second degree in the T's. If these values of the a's are put in (4) we get an equation of the first degree in the determinants of the funda- mental involution, of the first degree in the Q's, and of the sixth degree in T. This is the contact equation. The flex sextic is the first transvectant of the fundamental involution and is of the first degree in the determinants of the fundamental involution. The Jacobian of the flex sextic F and the quadratic Q giving the roots of the in- volution is a sextic Ji which is of the first degree in the determinants of the fundamental involution, but only of the first degree in the Q's. Taking the Jacobian of J, and Q we get a sextic J 2 which is of the second degree in the Q's and the first degree in the determinants of the fundamental involution. Now we propose to show that K, the sextic giving the points of contact of R* and r^, can be built from F, Q, J 2, A, and q, where F, Q, J 2 have the meaning just given, and where A is the discriminant of Q, and q is the third transvectant of the two members of the fundamental involution. The possi- ble combinations that are of the same degree as K are easily seen. We shall show that K = X A ¥ +^]2+vgQ\ where X, iJi, v are constants to be determined. Let the R* be referred to two flex tangents and the line joining these flexes whose parameters are o and 00 . Its parametric equation will be Xq = at* + bt^ -It c1^ (l) x^ = bt^ + ct' + dt Xj = cf + dt + e 8 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic We shall choose the involution whose sets are t and -/, hence whose double points are given by (2) Q -2/ o. Since we are not interested in the equation of r^, we proceed to find the equation giving the points of contact with the R*. Calculating the funda- mental involution of R*, that is two quartics which are apolar to the three binary quartics in (i), we get (3) bet* — 6b ef — 4 c e / = o, and (4) ^act^+6adf — c d = o. The polarized form of (3) and (4), that is where S refers to f,, t^, t,, t^, is (5) be St — beSi — ceSi=o and (6) aeS3 + adS2 — ed=o, which is known to be the condition that four points lie on a line. Now let two of the t's, say t^ and /j be equal to /, and let a" refer to t^ and 1^ Then (5) and (6) become (7) (bet' — b e) 2 — (2 6e^+ce)^ = — 1/5, V- = 2/5. > = i/5- Or to avoid fractions we have finally (18) 5K =i.Q' +2j,-AF. 10 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic If two nodes are in the involution, these two nodes are a factor of the r' and the remaining factor is some other point. We shall show that the re- maining factor of r^ is the third node. Let the parameters of one node be given by /' + a, a second by /* + h, and the third by a general quadratic c^t' + Cit + Cj. The R* referred to its nodes has the equation x^ = {f + a) (co t" + c^t + Cj) (19) X, = [f + b) (co t' + c^t + cj «2 = (/' + a ) {f + b) The sets of the involution are, if two nodes are in it, t and -/. The equation of a line joining t and -/ is given by the determinant Xa X. X, = o. {f + a) {cJ." + c,^ + Cj) , (/' + b) (cot' +c,t+C2) , (f + a) (f + b) (t'+a) Ic^t'—c^t+c,) , it' +b) Icot'—c^t+c^) , (f +a) (f +b) If now we take the sum of the second and third rows for a new second row, and their difference for a new third row, and remove the factor Ci^ from the third row, we have (20) x„ (f + a) (co t' + c,) t' + a it' + b) (Co t' + c) ,' (f + a) It' +b) t' + b . o = 0. Replacing f by T and expressing this equation in terms of f ,'s we have after removing common factors f = - (T + 6) (21) e. = T + a f J = o which shows on the face of it the other node to be the rest of r». T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic II § 2. Involutions Determined by Two Double Lines of the Plane Rational Quartic. If lines are drawn on the meet of any two double lines of the rational quartic, we obtain a quadratic involution. That is to say, while such a line meets the curve in four points the parameters pair off, /j and t^ say. By choosing o and oo as the points of contact of one double tangent we may write the curve (i) x^ = {aif + 2 bit + c^y X2 = (oj r + 2 62 / + C2)' . Any line on the meet of Xg and Xi is of the form x^ — X' a;, = o, or (2) /^t' — X' {aj' + 2b it + CiY = o which breaks into factors (3) [ 2 / — X (a, f + 2 Z>, / + c,)] [ 2 / + X ( a, /' + 2bit + c,)] = o. If ti is a root of the first factor, then (4) X = a, /' + 2bit + Ci Substituting this value we have, after removing the factor t -/, , (5) -^(0,1) = ai^^2 — Ci = o, 12 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic and this is the quadratic involution from the meet of x^ and Xi. We shall denote the double lines by o, i, 2, 3, and /„;^-) will denote the quadratic involution obtained by drawing lines on the meet of any two double lines * and j. We shall show that the double points of I ,0.,) are given by the points of contact of the two remaining tangents from the meet of the double lines o and i. The double points of /(o,i) are given by (6) a, ^ — c, =0. It is easily seen that the Jacobian of the quadratics which give the points of contact of the double lines give the points of contact of the two remaining tangents from their meet; for the Jacobian of two squared quadratics is the product of the quadratics and their Jacobian. Forming the Jacobian of the double lines o and I we have (7) a,/' — c, = o , which gives precisely the roots of /(o.d- We shall next prove that the points of contact of the two tangents that may be drawn to the quartic from the meet of any two double lines are on a line through the meet of the other two double lines. Having proved that the roots of the involution I (o,,) are the points of contact of the other two tangents from the meet of o and i, we have only to show that the points of contact of tangents from the meet of the double lines 2 and 3 are in the involution 7(0.1), that is to say, that the roots of I^2,)) are in /(o,i)- Not having the equation of the double line 3, we make use of the well- known fact that the three catalectic sets of the fundamental involution give the three sets of two tangents from the meets of double lines, such as from o and I, and 2 and 3. The fundamental involution of the quartic given by (i) takes the form (8) |a, bi b^c.lt* + I a, 6, c,' \ t> + {b^c, c," | t + X ( I o, &, 0/ I /3 + I 62C2 a,' I ^ + I a, ft, 6jCj|) =0, where | a, Z), 63 C2 | denotes the determinant a, bi , a^ 62 6,c, , 62 C2 and I a, &, c*\ is a, bi , a^ 62 and so on. T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic 13 Writing down the g^ of (8) we have (9) a, 6,^2* I + X I a, 6,a/I, o , j&jCjC," 1 +>> IftzCi^i' I o ,|62C2C,»| + Xl&jCja.'l, X|a, 6,6,^2! = 0. This is a cubic in X whose roots are at once found to be c,' c,' {biC, — b,c,y a,* at' (o, &2 — a^biY Substituting X = in (8) we get, after removing the factor a' biia^c^ — a^Ci), (10) o," 6j t* + a, (a, Cj + ^2 c, ) /3 — c, (a, c^ + aj c, ) / — 62 c,' = o. This factors into (11) [a^f — Ci] [ fli 62 ^' + ( <*i ^2 + ^2 c, ) ^ + ^2 ^1 1 = o. the first factor giving the points of contact of tangents from (o, i) and the second factor giving the points of contact of the two tangents from (2, 3), where (o, i) means the meet of o and I, and so on. We have now to show that the roots of (12) a, &2 f + (a, Cj + O2 c, ) / + 62^1 =0 are in /,o_,). The only condition necessary is that the product of the roots shall be c '/a,, and this is obviously so in the case of (12). We get 7(2,3) by polarizing (12). Thus (13) -^(2,3) = 2 a, Z)j /i ^2 + ( ^1 C2 + a^ c^) (ti + t^) +2 bj c, = o. It is then also readily seen that the roots of /,o.i) are a set of 7,2,3). The fact that the double points of 7,0,,) are in 7,2,3, and also that the double points of 7(2.3, are a set of 7,o.i) says the two involutions are commutative, that is •^(0,1) -^(2,3) = -^(2.3) Ao.D- Thus there are a single infinity of four-points on the curve for which (o, 1) and (2,3) are diagonal points. If we allow this four- point to run around the curve, (o, i) and (2, 3) will be fixed diagonal points 14 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic and the third diagonal point will have a locus. There will be three such loci corresponding to the three ways in which we may pair off the double lines. This question will be considered in a subsequent paragraph of this paper. We have proved that the points of contact of the two remaining tangents from the meet of any two double lines lie on a line through the meet of the other two. There are then six such lines and we shall prove they are on four points. We shall denote by L,o,,) the line on the points of contact of tangents from (o, i), that is from the meet of the double lines o and i. The other five lines are similarly named. It is obvious that if three lines are on a point they must be such as L(o,i).. L,(o,2) L(o,3)- To get these three lines we need the double points of /(o,i), /(0.2) and /(o,3). We have found the roots of /(o,i)tobe (14) a^t" — c, = o. From symmetry the roots of 7, 0,2) are (15) a^f — c, = o. Now finding the roots of 7, ,,2) and again making use of the catalecttc sets we find the roots of 7(o,3) to be (16) (ai&2 — Ojft,)/' — {biC^ — b^c\) = o. The roots in these three involutions are given by equations with no middle term; hence the three lines wanted are in each case on parameters t and -t. From equation (i) we get the line joining / and -/ as the determinant (17) X n 4 X 1 4f,{a,t'+2bit + c,y , (aj f + 2 62 / + c^ )' d^f ,{aif — 2 Z), / + cy , (a^ t' — 2b^t + c^Y - o. Expanding and removing extraneous factors and placing /'=Ci/a, we have (18) L(oi,) = [ai'biCi' + a'b^c^ — la^b^cfii — 2a^a^iC' -\-2 a^aJbiC^Ci + 4 a, 61 62* c, — 20, 61' 62 Cj — 2 ^2 ^1' ^2 c^ Xq + [ 2 Z)2 («! C2 + a^ c,)] Xi — 4 a^ bi c^ X2 = o. If t' = c-i/a-i we have (19) L(o.2) - [ — a^b^c^ — a^b-fi^* ^2a.^b^c^c.,+2a^a^yC^—2a^a^.f^Ci — 4 02 ^1 *^2 C2 + 2 a, 6, 62* C2 + 2 aj &, b^ c, ] x^ + 4 aj &j C2 «! — [ 2 &, ( a, Cj + flj c, ) ] acj = o. T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic 15 lit' = (5i Cj — ^2 ^1 )/(«^i ^2 — 02^1) we have (20) L(o,3) = b^'Xi — bi'x2 = o. If L(o,i), L,o,2), L,o,3) are on a point the determinant of their coefficients must vanish. The determinant may be written bi{alcl+alcl) — 2ai&2Ci(aiC2 +^2^1) + 2^16 [^.^(ajCi— 6162) +2bfi2Ci{2afi2 — a^b^^a^c^ +a2Ci,2a,Ci -b 2{a\cl + alc'^ -V2a.})^C2{a^c.^+a2C^—2a.i)2Cy{ayC2 — bp^—2b})2C2{2a]3^ — a^^,2a2C2,a^C2-Va2C^ o , ij . *i This expanded is readily seen to vanish which proves the theorem that the six lines on the points of contact of tangents from the meets of any two double lines form a complete four-point. These six lines together with the four double lines form a Desargues Configuration B. That is we have two triangles perspective from a point and having homologous sides meeting in three collinear points. We shall now study the four-point more in detail. The one point ob- tained was that determined by the fines L(o,i), L(o,2) and L(o,3)- Thus the point is paired off with the double line o. In the same way each of the four points is paired with a double line. Now there is reason to believe from other considerations, that these points are in some way related to the Stahl conic N, which is the locus of the flex lines of cubic osculants of the rational quartic. We shall show that the four points are the polar points of the four double lines as to the conic N. If the quartic is written (21) X, = a,- 1* + ^ bi /3 + 6 Ci f + 4 ^,- f + e.- , [i = I, 2, 3], it is known that N takes the form (22) — 36( bcx) (cdx) +12 {ad x) (c d x) +12 {b e x)(b c x) +4 (b d x)' + {aexY + 4 {ab x) {dex) — 8 (adx) (b ex) =0, where (b c x) Cq Ci C2 and so on. bo b. b. Co Ci Ci Xq Xi X2 i6 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic. Taking the quartic as given by (i) N is (23 )xo'[ — 4|oiii(a2C2+2&/) I |62C2(a,c,+2fe,*) | +4|62C2(aiC, +25,") | {ai't^c^l + 4 I o, ft, (flj C2 + 2 b^') 1 1 a, 6, C2' I +4 |a, ft, ^2 C2 1' + I a,' ^2' 1' + 4 I a," ^2 ^2 II by CyCi' I — 8 I a," ^2^2 I I o, ft, C2' | ] + Xi' [i6a2b2'c2] + x' [ i6aift,'Ci] + x, ^2 [ — i6ft,ft2(a^iC2 + Oz^i)] + Xo Xj [ — 8 6, c, I a, 6, (02 C2 + 2 62') I + 8 a, 6, I ^2 C2 ( = o, or (10) [ (a O' + ^ ( M )M [ ( « O' — ^ ( M )M = o. This shows that the tangents all along the gi line pair oflF into two sets of three. In our notation the pencil of cubics is (11) 2bt^ + 2,ct'+2,ct + 2d + \{t'+t)=o, which is readily seen to be a set of apolar cubics. The equation of the gj liiie is x^ — X2 — o. Hence the parameters of the four points in which this line meets the curve are given by (12) bt* + 2bt^ — 2dt — i = o. At these points two of the six tangents to the curve come together. The pairing of the tangents must be this tangent at the point with two of the remaining four, and this same tangent with the other two; or, the tangent at T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic 21 the point taken twice with one of the four, and then the remaining three paired. The latter is the case, as is proved by taking the Jacobian of the two members of (11) and obtaining (12). We note too that (12) is a self-apolar quartic which tells us again that the pencil of cubics are apolar, since they have a self- apolar quartic for Jacobian. The pencil of cubics has a unique apolar quartic and its Hessian is the Jacobian of the cubics. Finding the quartic we have (13) Q ^ b't* + 4bdt^ + 6bdt' + ^.bdt + d' =0. The Hessian of this quartic is at once verified to be (12). At the four points in which the g2 line meets the curve two tangents come together and we have just seen that one other tangent is paired with this one taken twice. There are then four such tangents. They are given by the Steinerian of Q to which the system is apolar, for if the Hessian has a root that is a double root of the cubic, the other root of the cubic is a root of the Steinerian. The Steinerian of a quartic Q is known to be gj Q + X giH =0, when g2 and g, are the invariants of Q, and H is its Hessian. But g^ of our Q is zero, so the Steinerian is the quartic over again. We assert further that Q is the quartic of which the system of cubics are the first polars. Salmon tells us how to find the quartic when the cubics are given. It is 12 H (J) + g^ J, where J is the Jacobian of the cubics, H the Hessian of J, and ^2 the self apolarity condition of J. But (14) ]^b^^ + 2bt^ — 2dt — d=o, and g2 - o, and (15) H (J) ^ b't* + 4bdt3 + 6bdt^ + ^bdt + d' =0. This is just Q over again. Q is then the quartic to which the system of cubics is apolar, as well as the quartic of which they are the first polars, and besides it is its own Steinerian, and gives the points of contact of the four tangents which are paired with the tangents counted twice at the points where the gj ^ine meets the curve. Consider now the pencil of cubics. To any one of the pencil we have a definite corresponding point /, on the curve, viz. the point with regard to which the cubic is the polar of Q. But paired with that cubic there is a second cubic, 22 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic and we get a point /j on the curve from it in a similar way. We have then a quadratic involution set up on the curve. Since the two cubics come together at the g2 points, these points correspond to the double points of the involu- tion. We can find the quadratics giving the double points in the following manner. The coefficients of the polarized form of Q must be proportional to the coefficients of the pencil of cubics. We can thus determine X in terms of /,. Polarizing Q we have (i6) (b't^+bd) t^+3ibdt,+bd) t'-h2,(bdt^+bd) t+bdti+d' =o. The pencil of cubics is (17) 2bt^ + {sc+\)t'+{3c + \)t + 2d=o, hence (6bd — ^b c) ti+6b d — ^cd (18) X = b ti + d Since our base cubics are those at the gj points, the double points of the involution will be given for X = o, and X = 00 ; that is one is given by the numerator of (18) and the other by the denominator of (18). We now look for this pair of points on the curve. The conic on the flexes meets the curve again in two other points q. If the fundamental in- volution of the curve is (a t)* + X (^ /)* then (19) g ^ («^)3(aO(M) =0. Using the fundamental involution as given by (5) and (6) we get (20) q = {b — c)t' — ct + (d — c) =0. Now operating with q on the pencil of cubics we obtain (36c — 6 b d) ti + 3cd — 6 b d (21) X = ~ . bti+ d T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic. 23 It is to be noticed that X in (21) is just the negative of X in (18). This shows a sort of cross working of the quartic of which the cubics are the first polars, and the quadratic g; that is if C, is the polar cubic as to /„ and Cj is the polar cubic as to ti, then q operating on C, gives ^2 and q operating on Cj gives /j. The double points of the quadratic involution are the points with regard to which the polars of the quartic are taken to give the two base cubics of the pencil, that is the flex cubics. Now at these double points we have a tangent from some point on the 52 line; that is each of these belong to one of the cubics of the pencil. We seek the relation of this cubic to the flex cubic. The polar of (13) as to the double point b t + d gives the flex cubic (22) t" + t = o. We can find the cubic to which bt+d belongs by equating coefficients in the identity (23) 2 b t^ + {2,c+-k)t' + {2>c+'>^)t+2d^{bt+d) (a^t'+a.t+a,), and we get (24) {bt+d){t'+t + i) =0. The last factor gives the pair of tangents to which bt+d belongs, and it is seen to be the Hessian of (22). Now the cubic /* + / = o which we have considered is special only geo- metrically; it is one of the pencil and behaves as any other member of the pencil. So any two of the three tangents given by (24) are the Hessian of some cubic of the pencil. We look for the three cubics thus obtained from (24). The factored form of (24) is (25) (bt + d) (t — w) (t — w') =0. We assert that the three cubics, of which any two of the three tangents given by (25) are the Hessian, are just the cubics which are the polars of the quartic Q as to the three roots of (25). The polar of Q as to / = w is (26) {b'w + bd) t^ — T)bdw'f--2)bdw't + bdw + d' =0. 24 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quar tic The Hessian of (26) is (27) (bt ^- d) {t — w') = o, and our assertion is proved. That is we have a system of cubics which are the first polars of a quartic, such that if the roots of any one be, /,, /j, t,, then the cubic which is the polar of the quartic a^ to any one of the /'s, and is therefore one of the pencil, has the other two t's for its Hessian. T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic 25 § 4. The Case with Three Double Lines on a Point. First of all we shall find the parametric equation of a quartic curve with a syzygetic point, that is with three double lines on a point. We may choose three points of the curve, say o, 00 , i. Let o and 00 be the points of contact of one double tangent, say «, = t". Let «o be a double tangent with points of contact i and a ; then X, = (t—lYit—a)'. If there is another double tangent on the meet of x^ and Xi, it is of the form a;o + ^ ^1 = o, and we find that « = i, and X = 4. The fourth double tangent will be written generally, and the equation of the curve is x,=^{t — iy{t + iy (1) x,=t' Xt = (/*— 5,/ + s^y. By transformation we can write the curve in the better form x^ = t* + I (2) X. = 6 /' 2(5, '— i) X2 = ^fi + ^Sjt + m. [m = ] The three double lines on a point are X, = O, 3 ATo + », = o, 3 »o — », = o. The Hessian of the three double lines is 26 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic (3) 3 Xo' + X,' = o. The cubicovariant of the three double lines is (4) x^ {x^' — Xo') = o. If we cut (2) by any line (5) (^ x) = $^x + ^,Xi + eja;^ = o, we have a quartic in t (6) fo ^* +4 f 2 t^ + 6 ^ 1 1' + /^ ^ 2 Sz t + m $ , + $ = o. Now if we form the two invariants, gi and g,, of (6) we get a line conic and a line cubic. We shall prove that the lines to the gi conic from the syzygetic point are the Hessian pair of the three double lines on that point; and that the three lines to the gi cubic from the syzygetic point are the cubicovariant of the three double lines. Writing the gi of (6) we get (7) fo' + 3f."— 4^2f2" + Wf 0^2 = o. Put 62=0 and we get the tangents to g^ from the syzygetic point. Changing the result into points we have (8) 3 X,' + X." = o. Next writing the g^ of (6) we have (9) f,3+^f^3_f^»f^ + (5^^ + l) f„3f^=.+2 52f,f3»+mfoflf2 = o. Setting f 2 = o, and changing the result into points we have as the tangents from the syzygetic point (10) x^ ( x^" — x^') =- o, and our theorem is proved. T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic 27 The Stahl conic N was defined in section 2 of this paper. The cubic osculant at a point t of the quartic (2) is Xo = T /' + I (11) X, = 3i' + 3 ^ t X2 = t^ + 3 -: t' + 3Sjt + S2 ■: + m. Cut this cubic by two lines {u x) and (vx). Making them cut in sets of apolar points we get the equation (12) :«i (3^2 — 3 -f') + x^is^x' + rtfz — i) = o. . For a given t this a line, the line of flexes of the cubic osculant at the point T of the quartic. For a varying x we get the locus of this line, which locus is the Stahl conic N. Its parametric equation is ^0 = 3^2 — 3^' (13) $1 = S2 x' + m X — I. . f 2 = o Since fj = o all the flex lines of the cubic osculants pass through that point, which is the syzygetic point. That is N is the syzygetic point counted twice. The cubic osculants at the points of contact of double tangents have interesting properties. Consider the double tangent with points of contact I and — I. The cubic osculant at t = i is x^ = t^ + I (14) ^1 = 3 ^' + 3 ^ X2 = t^ + 3f + 3S2t -{■ S2 + m. This curve passes through N and of course has a flex there. The flex tangent is (15) Xa + x^ = o. • The cubic osculant at t = — i is Xo = — t^ + I (16) x, =^ St' + 3t X2 = t^ — 3 /* +3^2^ — ^2 + m 28 T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic It also passes through N, and has then the same flex tangent as (14). Hence the theorem: When three double lines are on a point the cubic osculants at the two points of contact of any one of these three double lines pass through the syzy- getic point, each having ajlex there, and both have the same flex tangent. The three such tangents are also the three tangents tog, from the syzygetic point. We have seen that a line on the points of contact of the two tangents from the meet of any two double tangents passes through the meet of the other two double tangents. In the syzygetic case the six such lines are on the syzygetic point, three of them being the double lines themselves. The equations of the other three may be found by use of the catalectic sets of the fundamental involution. Again the Jacobians of the factors of the fundamental involution give the nodes, and thus we can get the equation of the lines to the nodes from the syzygetic point. We have then four sets of three lines on the syzygetic point, viz.. The three double lines Xi = W I 2x0 + Xi = , (B) 3x0 X, = . (C) The lines to ^3 Xo = (A) II Xi — Xo = , (B) :», + :Vo = . (C) The lines to the nodes 6 S2X0 + {Si' + I ) :!C, = o , (A) III 3{s, — iyx, — [2s^' + (s, — iy]x,=o,(B) 3 (s, + i)' Xo — [2 5," — ( 52 + I )=■ ]x, =0. (C) Lines on the points of contact of tangents from the meets of the three double lines on the syzygetic point with the fourth double line es^Xa— (52* + I ) :JC, = O , (A) IV 2>s,'x, — [2{s^ — i,y+s,']x^=o, (B) 3Si'Xa—[2{S2 + iy — Si']Xt=0. (C) T. B. Ashcraft: Quadratic Involutions on the Plane Rational Quartic 29 We have already seen that the second set is the cubicovariant of the first set. We say furthermore that the lines marked (A) are harmonic, because it is readily seen that they are of the form {