IMPORTANT COVARIANT CURVES AND A COMPLETE SYSTEM OF INVARIANTS OF THE RATIONAL QUARTIC CURVE BY JOSEPH EUGENE ROWE 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 PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER PA 1911 IMPORTANT COVARIANT CURVES AND A COMPLETE STSTEM OF INVARIANTS OF THE RATIONAL QUARTIC CURVE* BY J. E. ROWE Introduction. It is well understood that different domains of rationality are useful in discussing different properties of curves. Two domains are employed in the following, to render possible the geometric interpretation of certain invariants and covariant loci of the rational plane quartic. Section 1 is divided into two parts: In the first part is given a straightforward proof of the covariance of curves derived from R]" by a certain translation scheme; in the second part SALMON'S work on the combinants of two binary quartics is applied to those covariant curves of the R4 which can be found as combinants. In Section 2 the most important invariants of the R4 are discussed, and four invariants are found in terms of which any other invariant relation on the R4 can be expressed algebraically; in this sense these four invariants constitute an algebraically complete system. Section 3 contains a treatment of the invariants of the f4 when the R4 is taken as the section of the Steiner Quartic Surface by a plane; in this scheme the invariants occur as symmetric functions of the coefficients of the cutting plane. ~ la. Certain Covariants of Rational Curves. Let the rational curve of order n, which may be called R", be written parametrically (1) X = ait +b =t1- + ct'-2 +.. (i= 0, 1, 2). If (1) is cut by two lines (2) ) (2) (ax) = ~0oX + t 1X1 + 22 + = 0, and (3) (rsl) =r0o + t X + ba + r2 = 0, the results of this operation are the two binary n-ics (4) (a.)tn + (b6)r-t + (c)t-2... = 0, * Presented to the Society, September 6, 1910. 295 296 J. E. ROWE: COVARIANTS AND INVARIANTS [July and (5) (ar)t ( + ()t- (Cb)t)t- -2..= 0, which yield the n parameters of the points in which the lines (2) and (3) cut the curve. The combinants of (4) and (5) are expressible as rational functions of the tworowed determinants of the matrix (at) (b ) (cf) (d4)... (6) (an)) (be) (cu) ()... But if in any two-rowed determinant of (6), the quantities x,, s1, x2 are substituted for the coordinates of the point in which (2) and (3) intersect, the result may be expressed as a three-rowed determinant. For instance, ( (0 )0 0 (7) a b a1 b, x =labxl; (a.) (b?7) a2 b2 2 and evidently the other determinants of (6) assume similar form. Consequently, by means of the above translation scheme * from the combinants of (4) and (5) certain loci are derived which are related to the R" in a special manner. Certain sets of points on the R" defined by some projective relation are connected in this way with a point of the plane, namely, the intersection of (2) and (3). This fact alone warrants the assertion that loci related to the R'" in such a manner are covariant curves f of the R". But it is important to give an actual algebraic proof of this fact. In order to prove formally that these curves are covariants, it is sufficient to show that they are unaltered by any linear transformation of the x's or t's. Let the transformation of the x's be (8) x = lI 0 + mi l + n, (i-o, 1, 2). After this transformation (1) assumes the form (9) x, = (ia, + m na)t + mal + + (1, b + m + n*2)tC-.. or (10) x, = (li a) t + ( ib) t-1 + (lc)t —2.., where ( l a) = i ao + m. a + ni Ga2, etc. * GRACE and YOUNG, Algebra of Invariants, pp. 314-318. t GRACE and YOUNG in the place already referred to have practically proved that such loci are covariant. A' r Pr 1911] OF THE RATIONAL PLANE QUARTIC L ' From what has been stated the combinant curves of (10) are expressible as sums of terms, each term being of the same degree in determinants of the type (l0a) (lob).(lox) (11) (1,a) (1,b) (1x). (12,a) (12b) (ix) But by the theorem for the multiplication of determinants (11) becomes (12) lmnn abx; also each three-rowed determinant of the type (11) formed from (10) is equal to the product of Ilnmnl by the corresponding determinant formed from (1). Hence, as l Imn Ifactors out of each term of the combinants of (10) to the same degree, these curves are unaltered by a transformation of the x's. This is a property which is possessed by any homogeneous function of the determinants of the type abxl; however, not every such function is a covariant. But if such a function is also unaltered by a linear fractional transformation of the parameter t, then such a function is a covariant. Therefore it remains to show that combinant curves possess also this property. Let the transformation of the t's (t = t1/t2) be (12) t, mt1 m2,t, t - 11t + 1t2. Transforming (1) by means of (12) we obtain (13) = a t" + t- + c -2.. (i=O,, 2). If we cut (13) by the two lines (2) and (3) we shall obtain two binary n-ics, (14) (a.')t- + (b')tn-1... =0, and (15) (a' )t + (b6)tn-1... =0. But (14) and (15) are exactly what we should have obtained by operating upon (4) and (5) with (12). However, the combinants of (4) and (5) are unaltered by any such transformation as (12); hence the combinant curves are unaltered by a transformation of the parameter t and are covariant curves of the Rn. All loci obtained in the manner indicated are covariants, but other covariants occur which cannot be derived as combinants. Salmon* has discussed the combinants of two binary quartics from an algebraic standpoint and we shall apply his result with some extensions to a study of those covariants of the fR4 which can be derived in this manner. * Higher Algebra, third edition, pp. 200-206. 298 J. E. ROWE: COVARIANTS AND INVARIANTS [July ~ lb. Covariant Curves of R4 as Combinants. Let the equation of the Rt be written parametrically (16) xi- aat4 + 4b(t3 + 6cit2 + 4dOt + e (i=O, 1, 2). If the R4 is cut by the two lines (2) and (3) we obtain the two binary quartics, (17) U= (a)t4 + 4(b6)t3 + 6 (c )t2 + 4(dZ)t + (et) 0, (18) V= (ag)t4 + 4(bv)t3 + 6(cr)t2 + 4(dy)t + (er) = 0, giving the parameters of the points in which (2) and (3), respectively, cut the curve. Considering the u and v above to be the same as those of Salmon * by reason of (5) and (7) we have a=(ab)=-\abxl, a'=(de)=\dex, Am=(ad')=jadxj, 3'=(be')=Ibex{, (19) y=(ae')= aex, 8=(bd')=Ibdxl, X==(ac')=lcacxl, '=(ce')=lccexl, u = (be')==bcxj, u'=(cd')=lccdxl. Consider the pencil u + Kv as a single binary quartic, substitute its coefficients in its two invariants g2 and g3, and let the resulting quadratic and cubic in K be written (20) S K2 + S'K+ = S 0 and ToK3 ~K2+ K2 K+ K+ = 0; the quadratic gives those two values of K for which u + Kv becomes a selfapolar quartic; and the cubic those values of K for which u + Kv becomes a catalectic quartic. Equating to zero the discriminant of the quadratic of (20) and making the substitutions (7), we have A =-I aex 12 + 16 ] bdx '2 + 12 1 acx I I cex I - 481 bex I I cdx I - 8 1 abx I I dex I (21) - 8 adxIlI bex = 0, which is the locus whose tangents cut R4 in self-apolar sets of points. If the g2 of (4) were formed it would be the envelope of lines which cut R4 in self-apolar quartics. Hence A is the point equation of the conic g2. A line which cuts R4 in three consecutive points is a line of A, for the g2 of a quartic having a triple root vanishes. Consequently the six inflexional tangents of R4 touch A, and indeed A is often spoken of as the conic on the six inflexional tangents of R4. The condition for u and v to have the same apolar cubic, or that each may be expressed linearly in the same three fourth powers, leads to a second combinant which with the substitutions (7) becomes *Higher Algebra, third edition, pp. 200-206. 1911] OF THE RATIONAL PLANE QUARTIC 299 B l acx II cex - I bcx lI cdx - l adx I cdx I (22) - Ibex 11bcx I + Ibdx I - _Iabx I ldex | = 0. This conic may be identified with the conic which Stahl * gives parametrically; it is the locus of the vertices of flex-triangles of all first osculants. The eliminant of u and v expresses the condition for a common root or the condition that they intersect on the curve. Hence by reason of (7) R becomes the point equation of the R4. The Jacobian of u and v is (23) at + 3Xt5 + (3/ + 6u )t4 + (y + 88)t3 + (33' + 6) t2 + 3X't+ a= 0. By a well known property of the Jacobian (23) gives the six tangents from a point to R4. The condition that these form a self-apolar set is (24) -402 t= A + 48B. Consequently the locus of points such that tangents from them to R4 form selfapolar sets is the conic (25) A + 48B = 0. Further we find that the following relation holds (26) 320 (rI2 -1004)= ( A-16B)2 - R, where 1, and 14 are the well known invariants of the sextic (23). But 2 _ 100l 4 = 0 for a sextic which has either a triple root, or a double root 2 p4 and the other four forming a self-apolar set. Consequently such points lie on the quartic curve (27) (A —16B )2- R =0. The points on R which behave in this manner are on the conic A-16B=0. Hence the six flexes and the two points qi (which have the property that tangents drawn from them to Rt form self-apolar sets) lie on the conic (28) A -16B=0. It is worthy of notice that the above not only gives an easy way to write down the equation of the conic on the flexes but is an independent analytical proof that they do lie on a conic with the two points qi. Salmon's combinant equation C = 0, expressing the condition that a member of the pencil u + Kv can have two squared factors, and from our standpoint the locus of points such that lines drawn from them cut the R4 in quartics which *W. STAHL, Journal fur diereineundangewandteMathematik, vol. 101 (1886). t 12 = 0 is the self-apolarity condition of (23). 300 J. E. ROWE: COVARIANTS AND INVARIANTS [July have two squared factors, can be nothing else than the point equation of the four double tangents of the R4. Since it may be written (29) 128C =(A - 48B)2 -R, it is evident that the conic on the eight points of contact of the four double tangents is (30) A - 48B = 0. Forming the Hessian of the cubic of (20) and requiring it to be apolar to the quadratic of (29) yields a combinant which may be expressed thus, (31) 1281= R - (A - 16B)(A + 48B). When I- 0, the R4 breaks up into two conics. If the two lines from a point which cut the R_ self-apolarly are apolar to the Hessian pair of the three lines which cut R4 in catalectic quartics, then I vanishes; therefore together with (7) it gives the locus of such points. Forming the third transvectant of u and v we get a quadratic in t whose discriminant vanishes on the locus (32) A- 12B = 0. This may be identified with Stahl's conic N. The first osculant of R4 at a point t is an R3 and therefore has three flexes on a line. The equation of this flexline is a quadratic in t' and the discriminant of this quadratic may be identified with (32). Or writing the R4 symbolically (33) (ao)(at)4= and (bo)(,t) = 0, then taking the third transvectant we have (34) Iab Iaa I 3(at) (3t)= 0, which is the conic N given parametrically. Of course the letters in (33), (34) are only symbols and must not be confused with the same letters already used with different meanings. The eliminant of the cubic and quadratic of (20) is the condition for such a K to exist that u + K1v = 0 has roots that are both self-apolar and harmonic pairs. But a quadratic for which g2 = g3 = 0 has a triple root, and therefore with (7) Salmon's D = 0 becomes the equation of the six flex tangents of the R?4. The combinant E, the discriminant of the cubic of (20), evidently yields the locus of point whose lines cut R1 in catalectic quartics. It is therefore the point equation of the g3 of equation (4). Suppose that the polar of the quadratic of (20) as to the cubic of (20) is 1,, and the polar of 01 as to the quadratic is 02, by using the cubicovariant of the 19111 OF THE RATIONAL PLANE QUARTIC 301 cubic instead of the cubic itself we obtain bI and (2 by the same operations. The condition for 0- = 0 and _0 (or b' and b2) to be the same is the combinant M, and by using (7) we have the locus of such points. Before leaving covariant curves we observe that from Salmon's identities (35) D -E= - 2B - 2(A- 16B), and (36) 1281 = R - (A + 48B)(A - 16B), it follows that (37) 64(D - E)= B[(A - 16B)2- ]. Consequently,* twelve of the intersections of the flex tangents of R4 and E lie on the conic B. Also from (26) and (37) we have (38) 320(D- E)= B [(A + 48B)2 -100. If 14- 0 (i. e. if tangents drawn from any point to R4 form a catalectic set), the intersections of D and E lie on two conies; the flex tangents touch E along A + 48 B = 0 and intersect E along B. This particular R4 is of the lemniscate type. ~ 2. Invariants of R4 as Combinants of the Fundamental Involution. If the R4 be referred to a special triangle of reference its equations are X = at + 4bt3, (1) x2= 4dt + e, 2= 4bt3 +6ct2 + 4dt. The condition for R4 to have a triple point,- or the condition for R4 to have a perspective point for (1) becomes (2) BI _ 36a2 c2e2 + 256ab2 de - 96a2cd2e - 96ab2 ce2 - 16a2 bde2 = 0. If the g2 of u of Section 1 is found for (1), its discriminant multiplied by eight may be written (3) A' - 6a2 c2e2 - 96b22 c2 8+ a2 bde2' + 48abc de - 32ab2d2e = 0. Since the six inflexional tangents are lines of the conic g2, A' vanishes when the R4 has three concurrent flex tangents. *See THOMSEN, American Journal of Mathematics, vol. 32 (1910), p. 222. tW. STAHL, Mathematisohe Annalen, vol. 38 (1891); GRoss, Mathematische Annalen, vol. 32 (1888). Trans. Am. Math. Soc. 21 302 J. E. ROWE': COVARIANTS AND INVARIANTS [July Calculating the conic on the flexes from the preceding section for (1) and taking its discriminant, we have 256 (a 3cd4e2)* - 64(a c2d4e2) + 32 (a c3de3) (4) - 192a3 b2 C2d2e3 - 512a2 b3 c2d3e2 + 128 (a3 bc3 d3e2) 32a3 bc4 de3- 4a4c4e4 = 0. Since it is known that if an RI has three flexes on a line it has a fourth, Cl'= 0 is the condition for four collinear flexes of the R4. These three invariants are independent, as may be shown in the following manner: A' and B' are evidently independent. Further if C' were expressible in terms of A' and B', such a relation as C' = XB'2 + EA'B' would have to hold because C' contains no term in b4 c4 d4, i. e., B' would have to be a factor of C', but this is' disproved by making b = 0 in (2), (3) and (4). Similarly D' is shown to be independent of A', B', and C'. The condition for A - 16B in lines to be apolar to A in points we shall call RI = 0, and we find that (5) 36R' ~ 108 C~ + B I2~10A1B~ + 16A2 = 0. If H= 0 is the apolarity condition of A in lines to A - 16B in points, then (6) 6H~ A B1 ~ 8A'2 = 0 The discriminants of A and B are known, and therefore, the discriminant of the whole pencil (7) X(A - 16B)~c+A = 0 is (8) C~~~hC3 I + RIX21k +HX/12 - A' 2 1o = o (8) 1 But a more convenient formula than (8) is the discriminant of (9) X'(cA + dB) +,W(aA + l6bB)= 0, which is j -(C ~ d)3A12 A d(c + d)2H+ d2(c ~ d)RI d C-dCJX'3 +[-3(a b)(c d)2A 2_(b(c d)22+ d(a + b)(c + d))H ( (d (a + b) ~ 2bd(c + d))R' - 3bd2Cjx'2p. (10), + [- 3(a + b)2( d)A2 (2(a + b)(c+ d)b +(a + b)2d]H + (2bd(a~+ b) + (c + d)h2) Ii - 3b2dC']X4L'v + [- (a + b)3 A 2- (a ~ b )2 bH'i (a + 6)b2R2 - Rb3CI3= 0. * The expressions in parentheses carry with them their conjugate expressions. 1911] OF THE RATIONAL PLANE QUARTIC 303 By substituting proper values in (9) and (10) any invariant of any member of the pencil of conics may be easily found. For instance, by a method to be given presently the discriminant of B* is found to differ only by a numerical factor from the undulation condition. Hence by substitution in (9) and (10) the undulation condition is (11) R2 - -— A2 - C _=0, or by reason of (5) and (6) (12) 144 C + (B' + 2A')2= 0. Also the invariant whose vanishing is the condition for three concurrent double tangents is the square root of the discriminant of the conic A - 12B, or at least it differs only by a factor. Hence the condition for A - 12B to degenerate is (13) -(B' + 6A )2 0, and for three concurrent double tangents, (14) B + 6A = 0. The conic A - 16B meets R4 in two points qi besides the six flexes. By substituting (1) in A - 16B and comparing with the sextic giving the flexes we find a quadratic giving the q's. Its discriminant is (15) B~ + 8A = = 0; hence (15) is the condition for the q's to unite. Having taken the R4 in a workable form to find relations among its invariants, we shall now give the most important invariants in their most general forms. It is well known that all line sections of the R4 are apolar to a pencil of quartics which constitute the Fundamental Involution. From the theorem of Grassmann t on the proportionality of determinants it follows that the combinants of the Fundamental Involution are invariants of the R4. An easy method to pass from the combinants of line sections of an R4 to combinants of the Fundamental Involution is to replace each determinant by its complementary, i. e., I acx I and [ bkx I must be replaced by | bde I and I ace |, respectively, etc. Of course, allowance must be made for binomial coefficients. For instance, if these changes are made in the combinant B of Section lb we have B' = I bde I abd - I ade I Iabe I - bee II abe I (16) - I acd I ade + ae I ce 2 - I cde abe l = 0, *This was proved in a different way by Dr. THOMSEN in his dissertation, Johns Hopkins University, 1909. t W. F. MEYER, Apolaritdt und Rationale Curven, ~ 11. 304 J. E. ROWE: COVARIANTS AND INVARIANTS [July which is the triple-point condition for R] written without binomial coefficients. Further, if these changes are made in A of Section lb the result is -768 times the discriminant of A for R4 written without binomial coefficients. We shall summarize the results of these two sections in a table. Let (R4) mean the R4 written parametrically with binomial coefficients and (R4) mean R4 written in the same way without them. Also let A' stand for the combinant A of the Fundamental Involution which may be derived from A as already explained. In the first column is given the name of the combinant of two line sections of (R4); in the second we have the locus resulting from the process explained in Section la; in the third is the same combinant of the Fundamental Involution giving an invariant of (R4) whose meaning appears in the fourth column. Also if A' means an invariant of (R4) we shall understand that A' stands for the same invariant of (R4), possibly differing by a factor. Evidently any such invariant as A' can be obtained from A' by substitution of binomial coefficients, i. e., cdel and Ibcdl of A' would be replaced by 24 1cdel and 96 1 bcd, respectively, to obtain a multiple of A,. Table of Related Invariants and Covariants. Combinaht ofComb oCombinant of C~omb inant~ of~ F. I. giving Condition for to have two line sec- Covariant Curve of (R4). an invait Condition for (t to have tions of (R4). irian A Locus whose lines cut (R4) A 3 concurrent flex-tanin self-apolar sets. gents. B Locus of vertices of flex A's ' Triple point. of 1st osculants. C Product of 4 double tan- C' 4 collinear flexes. gents. D Product of 6 flex tangents. D' Cusp. 2I Point equation of ( R4). R' Undulation. E Locus whose lines cut (R4) E' Tac-node. in catalectic sets. A -12B Locus of flex lines of 1st A - 12B' Its 2 q's unite. osculants. lM Remote meaning. M' Skew invariant. For the purpose of deriving these invariants a special form of the R4 has been used; and after they have been derived, to identify them with the proper combinant of the Fundamental Involution is a comparatively easy matter. This was the case with A' - 12B', which as a combinant of the Fundamental Involution one would hardly have expected to give the condition for three concurrent 1911] OF THE RATIONAL PLANE QUARTIC 305 double tangents. On the other hand, forming the cubic of (8), Section la, of the Fundamental Involution, we have three catalectic sets associated with the double points; if two of these are the same we have a tac-node, the condition for which is E' = 0, and this would probably be a difficult result to obtain directly from the equations of the curve. D' = 0 is the condition for k R4) to have a doubly perspective quintic, and therefore the condition for a cusp. Let the invariants A', B', C', and D' be called, I1, I], and I6, respectively. These four invariants constitute a complete system in the sense that any other invariant relation on the R4 can be excpressed in terms of them, i. e. any other invariant of the R4 is connected with these four by an algebraic relation. The proof of this fact depends upon a theorem due to Stroh,* which applied here says that every combinant of two binary quartics multiplied by the proper power of an invariant (which in this case is the undulation condition) is an invariant of a binary sextic. t In this case I('2) 2, I( R'2)2, L ( R'2)4, 6 (R2) 6, and Ik(R'2)7 (where k is the degree of an invariant) become J6, J,,, J, and J, of a binary sextic. But I, 1 I,, and I are independent, hence J0, J6, J,2, and J(s are independent. Suppose then that there is an invariant Ik of the R4 which gives rise to an I3 of the binary sextic. Then we should have five invariants of a binary sextic, four of which are independent, and these are always connected by an algebraic relation. Consequently any 1 of the 24 is algebraically expressible in terms of the four invariants I, I>, J4, and 16. ~ 3. 7he R4 as Plane Sections of the Steiner Quartic Surface. The invariants of the R4 may be treated very neatly by considering the R4 as plane sections of the Steiner Quartic Surface I which we shall call S4. The point equation of the 84 referred to its trope planes is (1) V/x = I/o + 1- /x + 1/x2 + /x, =; 0 we propose to cut this by the plane (2) ( ax)) = ar+x + alXz + a22~ + a3x = 0; in this manner any relation among the a's is of (2) yields a special R?, or the invariants of the R4 may be expressed in terms of the symmetric functions of the a's of equation (2). For instance, any plane through the point (1111) cuts out an R4 with a triple point and therefore the condition for an R4 with a triple point is that (2) be on *E. STROH, Mathematische Annalen, vol. 34 (1889), pp. 321-323. t W. STAHL, Crelle's Journal, vol 104 (1889), p. 302. T The Steiner Quartic Surface may be obtained as the polar of a plane as to a tetrahedron. (1) is the polar of the plane (1111) as to 401t23=0. It is of the third class, fourth order, with an enveloping cone of order six. Hence a plane section is a curve of order four and class six-an RI. 306 J. E. ROWE: COVARIANTS AND INVARIANTS [July the point (1111), which is a0 + al + a- + a = 0, and this as a symmetric function of the a's is (3) s1 =0. Any plane on a pinch point cuts out an R4 with a cusp; the six pinch points have the coordinates (1100),.., (0011), hence we have an R4 with a cusp when any one of the six factors of (4) (o + al)(aO + a2)(a, + a3)(a, + a2)(a, + a3)(2 + a3)= 0 vanishes. Expressed in terms of symmetric functions, (4) becomes (5) S1 S2S3 S2-S4 =o which is the general condition for a cusp. Similarly when any two factors of (4) vanish we have two cusps, and by taking the product of the factors of (4) five at a time we obtain the condition for a second cusp after (5) has vanished. I find this to be (6) 2S, + 2 + S 2 S3-S = 0. By the same argument, after (5) and (6) have vanished the condition for a third cusp is (7) S S3 + 32S4 + S2 + 2 S1S2 = 0. If (2) is a tangent plane of (1), the point of tangency is a double point in the curve of the section. Nodes also occur where (2) cuts the double lines of the S4. The condition for an extra node is that (2) be a plane of (1). The plane equation of (1) is (8) (1/^=) 1/o + 1/1 + 1/+ 2 + 1/ = 0. and hence the condition for an extra node, or for R4 to degenerate, is (9) 3 =0. The points where (2) cuts the trope conics are points of contact of the double tangents in the curve of section or the double tangent are the intersections of (2) and trope planes. Hence if (2) is on a vertex of the reference tetrahedron the corresponding R4 has three concurrent double tangents. As this can occur in four ways, the condition for three concurrent double tangents is (10) S4 =0. Evidently this could not occur twice unless (9) holds, in which case R4 degenerates. The line joining the two points where (2) cuts a trope conic is a double tan 1911] OF THE RATIONAL PLANE QUARTIC 307 gent; if these two points come together the intersection of (2) and a trope plane becomes a line of the corresponding trope conic and in the section this point is an undulation. Hence the condition for an undulation is that the intersection of (2) and a trope plane be a line of the corresponding trope conic. This may occur in four ways and the general undulation condition is the product 4 II ( I + al a2 + + 2)= 0, or (11) 02 S-S183S4 + 2 0. By the same argument as in case of cusps, the condition for a second undulation after (11) has vanished is found to be (12) 82 + 182 S3 S,4 + 2 7S4 0. Also for a third, after (11) and (12) have vanished we have (13) 2 + S 83+2S4= 0. By the use of a result of Richmond and Stuart * the quadric associated with the S4 containing all those points which in plane sections are flexes is found in our notation, (14) 8 (a3 + a + 3 2 + a( 3 - S3 (X2 -- 2Exzl) = 0. By its form the quadric in the second parenthesis touches the edges of the tetrahedron at their mid-points, and in sections gives the points of contact of the double tangents. Knowing the meaning of (14) and of the quadric just mentioned and recalling that they (or the corresponding conies in Section lb) are proportional to A- 16B and 48 B -A, respectively, we see that the quadric Ya3 X2 = 0 is the locus of points which in plane sections yields the conic which is the envelope of flex-lines of all first osculants, as it is proportional to A - 12B. It is desirable to form the discriminant of the pencil of conics as has been done n Section 2. In order to do this in a way that involves the a's symmetrically, we find the plane equation of (14) and substitute the a's for a's. Thus by using a result of Salmon t after dividing out by 83, the discriminant of (15) L xa3X - \8 ( 2 - 2Zxo ) = 0 is exhibited as follows, 2 3-( s2 - 51 83 + 82 ),2+4( 81 S32 —2 2- 4)2 (16) + 4S 2 ( S - 4S1)X3 0; *Proceedings of the London Mathematical Society, series 2, vol. 1 (1904), pp. 129-132. t Geometry of Three Dimensions; p. 50, i 67, p. 58, ~ 79. 308 J. E. ROWE: COVARIANTS AND INVARIANTS [July and (16) in the notation of Section lb is the discrininant of (17) P(A - 16B) + x(A - 48B)= 0. Equating (17) to A and snbstituting in (16) we find the discrininant of A to be (18) A - [2 (SIS3 48)2 0 therefore three flex tangents are concurrent if (19) S SS3 -44`4 =0. Or by equating (17) to B we find the discriminant of B (20) 12 ( 84 S1 3 S ) = 0, 4.36(S~SS-SSS) but (20) is only a multiple of (11), showing that B degenerates when I,' has an undulation. If we identify (17) with (21) vA + K(A - 16B)= 0 we find that the discriminant of (21) is 2~ ( S 83- 4 S4)] 2V3 + 4, ( S2 S2 - 20 X, S3S4+ 64S')v 'K1 (22) + -1 —(64K - 5S2S2 + 12S2 + 4 vK 8 1 4 1 3T ~/ 3 S2 84 +4T(8S 3S4+ 2 S2 - 42 S32S)K3=0. Let us write equation (22) briefly as (23) A2 V3 ~ ll2K~ R, v-7+ K2 + C'2 K3 = 0. The importance of (23) will appear later. If a point where a double line of S' cuts (2) is moved up to be a point of (14) we have brought a node and a flex together and have a fleenode; except that at the two pinch points which are on each double line we get cusps. The points where the double lines of S4 cut (2) are x0- x1i + a+31, a2 = X3= -(a~ + 0); (24) X0= X2=1a+,, X1a=9 x I =-( 3 0 o+ '2); - X3 a, a 29~xi = X2 -(a + ~a3). Substituting these in equation (14) and factoring the cusp condition twice out each time, I find that the conditions under which a flecuode can occur are three expressions of the type (25) (a a 2 (a + a a a )2 (ao + a,)= 0; OF THE RATIONAL PLANE QUARTIC 1911] 309 i. e., the vanishing of any one of the three expressions of the type of (25) insures a flecnode. By arguments similar to those already used the general condition for a flecnode is shown to be 32S 82 34S2 S2 8 3 + 2 328, 52 - 4. S - 643 3 _,- 5X A2 (26) +16SAS 2 4- S 54= 0 The condition for a second flecnode after (26) is satisfied is (27) S2 2 16AS S2,S3+ 48S] + S 8 -- 488 S = 0. Also, for a third, in addition to (26) and (27) we find (28) 1S2a-6 -3 = O. Obviously an R4 cannot have three flecnodes without one of them being a biflecnode. The Rf is cut in eight points by A - 16B = 0; six of these points are flexes and the other two are the points q. For a flecnode, a q and a flex must come together; this could occur only twice and a third flecnode would require two flexes of the R' to cross, thus forming a biflecnode. The matter of biflecnode conditions is somewhat unsatisfactory. To find them we introduce a conic of fundamental importance. The conic (29) X(X + x2 + x2) + 2fxlx, + 2gxx + hx = 0 )0 1 2. 2 + X + = may be transformed by xi = 1 /x into the R4 referred to its nodal triangle, and its equation is (30) (l\+x2+, f+ hx2), + x + X0X, =0. It may be shown that the invariants of the Rl4 calculated directly from equation (30) are connected with the invariants we have found from the S4 by the following relations 2X = a' + a, + a2 +a3 = S1, 2f = a0 + a1- a2-, (31) (2g= aO- a1 a+ a —a, 2h = a0 - a - +a3. The equation of a pair of nodal tangents may be written down from (30). If we impose upon each of these tangents the condition that it cut the R4 in three consecutive points the results are the conditions for a biflecnode. These operations yield three pairs of equations of the type (32) hl- gf +gy =,gf f - hx + g = 0, 310 J. E. ROWE; THE RATIONAL QUARTIC where X = 0 is the triple-point condition and = 0 is the cusp condition. Ruling out the vanishing of X or ry, one such pair as (32) will vanish if only any two of the quantities (f, g, h) vanish. A pair of conditions expressible in symmetric functions of the a's which are equivalent to these two conditions are 3 (33) 0f2g2=0 and j2g2=O, and these are expressible as (34) 3S4 16S S2+ 16S2 + 16S13 -644 =0, and (35) S6 8S1 S2 + 16.S S22 + 48S 16S3S - 64S1 S2S3 + 6482 =0. It may be easily verified that the conic on the flexes degenerates when (34) and (35) vanish simultaneously. VITA I, JosEPeI EUGENE ROWE, was born in Emmitsburg, Md., March 21, 1883. I received my early education in the schools of my native place, and was graduated in 1904 from Pennsylvania College, Gettysburg, Pa. The year 1904-5 was spent at the University of Virginia, Charlottesville, Va.; the following year I taught in Mercersburg Academy, Mercersburg, Pa., from which place I came to study Mathematics in the Johns Hopkins University and to teach in the Jefferson School for Boys. Having remained in this work three years I became University Scholar for the year 1908-9 and held the University Fellowshipin Mathematics during the year 1909-10.