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 On the In-and-Circumscribed 
 Triangles of the Plane 
 Rational Quartic Curve 
 
 By 
 Joseph Nelson Rice 
 
 A DISSERTATION 
 
 Submitted to the Faculty of Sciences of the Catholic University of 
 
 America in partial fulfilment of the requirements for 
 
 the degree of Doctor of Philosophy. 
 
 Washington, D. C. 
 June, 1917. 
 
On the In-and-Circumscribed 
 Triangles of the Plane 
 Rational Quartic Curve 
 
 By 
 Joseph Nelson Rice 
 
 A DISSERTATION 
 
 Submitted to the Faculty of Sciences of the Catholic University of 
 
 America in partial fulfilment of the requirements for 
 
 the degree of Doctor of Philosophy. 
 
 .• . . . :v:'\*;ri'. 
 
 Washington, D. C. 
 June, 1917. 
 
•- ; i^-" .' 
 
 NATIONAL CAPITAL PRESS, INC., WASHINGTON, 0. C. 
 
ON THE IN-AND-CIRCUMSCRIBED TRIAN- 
 GLES OF THE PLANE RATIONAL QUARTIC 
 CURVE 
 
 INTRODUCTIOX 
 
 The question of the existence of simultaneously inscribed and 
 circumscribed triangles has received considerable attention, most 
 of which, however, has been directed to the consideration of poristic 
 cases. 
 
 R. \. Roberts^ investigates the possibility of the existence of 
 an infinite number of closed polygons simultaneously inscribed 
 in, and circumscribed about, a unicursal quartic. In this connec- 
 tion he discusses only the case of the nodo-bicuspidal quartic, 
 and shows that, for the triangle, the results are irrelevant. 
 
 Morley," in his article entitled "The Poncelet Polygons of a 
 Limacon," shows that the results obtained are poristic, except 
 for the triangle, in which case they are irrelevant. 
 
 Cayley,^ in "On a Triangle In-and-circumscribed about a Quartic 
 Curve," shows that the binodal quartic 
 
 ('-;)■ 
 
 has four such triangles. These are such that two of the sides are 
 tangent to the inner loop and the third to the outer. 
 
 The same author, in "On the Problem of the In-and-circum- 
 scribed Triangle," considers the following problem: "required 
 the number of triangles the angles of which are situate in a given 
 curve or curves, and the sides of which touch a given curve or 
 curves."^ He discusses 52 cases of the problem, according as the 
 curves containing the angles or touching the sides are, or are not, 
 distinct curves. The simplest case is when all the curves are 
 
 1 Proceedings of the London Mathematical Society, Vol. xvi, p. 53. 
 
 2 Ibid., Vol. xxix, pp. 8,3-97. 
 
 3 Collected Mathematical Works, Vol. v, pp. 489-492. 
 
 Philosophical Transactions, t. XXX (1865), pp. 340-342. 
 
 4 Collected Mathematical Works, Vol. viii, pp. 212-258. 
 
 Philosophical Transactions, t. clxi (1871), pp. 369-412. 
 
 5 Collected Mathematical Works, Vol. viii, p. 567. 
 
 Reports of the British Associations for the Adrancement of Science, 1865-1873. 
 Report, 1870, pp. 9-10. 
 
 3 
 
 :Mj2L0i 
 
4 •Tiib-&iid-Circnnt8cribed Triangles of Quurtic Curve 
 
 distinct; the number of triangles is here equal to 2aceBDF, where 
 a, c, e are the orders of the curves containing the angles respec- 
 tively; and B, D, F are the classes of the curves touched by the 
 sides respectively. The last and most difficult case is when the 
 six curves are all of them one and the same carve. The number 
 of triangles is here equal to one-sixth of 
 
 -46(.43+a3)-420(.42a+.4a2) + 221(.42+a2)+704Ja 
 + 172(.4+a)-fa[-9(^2_^a2)-12.4a+135(.4+a)-600] 
 
 where a is the order, .4 the class of the curve; a is the number, 
 three times the class +number of cusps, or, what is the same, three 
 times the order + the number of inflexions. 
 
 It is to be noted that this formula gives the same number of 
 triangles as has been found by the method used later. For 
 example, in the case of the rational quartic, where a = 4, .4 = 6, 
 a = 18, the number of triangles is 8, which corresponds to that 
 found on page 18. For the cuspidal quartic, where a = 4, A — 5, 
 a =16, the number is two, which alsD corresponds to the number 
 found on page 2*2. 
 
 In this paper it is proposed to look into the existence and actual 
 number of such triangles for the following types of rational quartics : 
 I. Quartic with three double points. 
 II. Quartic w^ith one double point and a tacnode. 
 
 III. Quartic with a triple point. 
 
 IV. Quartic with two double points and a cusp. 
 
 This discussion was led up to by preliminary work on the three- 
 cusped rational quintic. Upon subjection to a quadratic trans- 
 formation this curve goes into a rational quartic, wdiich, it will 
 be show^n, has triangles of the kind here mentioned. Accordingly, 
 it will first be proved that the quintic can have certain conditions 
 imposed upon its coefficients so that it may acquire an additional 
 cusp or a tacnode without degenerating. It will also be shown 
 that it cannot have a triple point. 
 
THE THREE-CUSPED RATIONAL QUINTIC 
 
 This quintic may be expressed parametrically as follows: 
 
 that is, x,= (t-ai)(t-l32)Ht-l3s)' 
 
 This quintic has cusps at the vertices of the fundamental tri- 
 angle, the values of the parameters thereat being 
 
 t^^i, (i = l, 2, 3) 
 as is readily seen by considering the common intersections on 
 
 any two of the lines 
 
 .r, = 0, (I'-l, 2, 3) 
 
 Consider, for example, the intersections on a;i = and x^ = 0. 
 They have in common /Ss taken twice. These common points 
 must be at the intersection of the lines themselves, viz., at the 
 vertex three. 
 
 For the purpose of more ready computation, specialize this 
 quintic by letting the values of the parameters at the cusps be 
 0, 1, 00. The parametric equations of the curve may then be 
 written as : 
 
 i-oc 
 
 Xi = 
 
 r- 
 
 Xi = (t-a)if-iy 
 
 xz^t-y x,= {t-y){t-l)H^^ 
 
 The Plucker numbers of the curve are: 
 
 m = 5, 5 = 3, • K = 3 
 
 n = 5, T = 3, t = 3 
 
 It is proposed to derive the conditions, which must be imposed 
 upon the parameters a, (3, y, so that the quintic may have, 
 
 I. A fourth cusp, 
 II. A tacnode, 
 III. A triple point. 
 
 It will be seen that, excluding the case of the triple point, the 
 quintic can have such extra points in addition to the three cusps 
 
6 In-and-Circumscrihed Triangles of Qiiartic Curve 
 
 at the vertices of the fundamental triangle. As preliminary, 
 there will be derived : 
 
 ( 1) The cubic equation connecting the three parameters at the 
 three flexes; 
 
 (2) The sextic equation connecting the six parameters at the 
 three double points. 
 
 (1) The Cubic whose Three Roots are the Flexes. 
 Let Lxi-{-Mx2-{-Nx3 = be a flex-tangent. 
 
 Then Lfi{t) + J//2(/) -\-Nf3(t) = has a triple root, 
 
 and Z//(0 -\-Mfo\t)-\-Nf/(t) = has the same root doubled, 
 
 also Lfi'\i) -\-]\W(i) +yfs"{t) = has the same root once. 
 
 From these three equations eliminate L, M, N, then 
 
 Mt) m hit) 
 
 //(/) //(O Uit) 
 
 fl"(t) f2"(t) h"{t) 
 
 0, is the cubic of flexes. 
 
 For the functions of / and their derivatives substitute their corre- 
 sponding values and reduce, thus giving as the required equation: 
 
 + (7-/3-b2ai3-4a7-2^7)/ + 3a(i3-7 + 2^7) = 0. . . . (1) 
 
 If one of these flexes coincides with one of the three cusps, 
 that is, if t has any one of the values 0, 1, co, there is then at this 
 point a cusp of the second kind.* Hence, the condition that 
 there be a cusp of this character at : 
 
 /-O is /3-7 + !2/37 = 
 
 /=! . . 3a+7-2a7-2 = 
 /= ex. . . o:_^_|_2 = 
 
 (2) The Sextic connecting the Six Parameters at the Double 
 Points. 
 
 Let X and ix be the parameters at one of the double points. 
 
 ^-=^-(m-t)=^"-(X-7) _ 
 
 * Salmon: Higher Plane Curves (French Edition), Chap, ii, p. 70. 
 Cayley: "On the Cusp of the Second Kind or Nodecusp;" Collected Mathe- 
 matical WorJiK, Vol. V, pp. 2fi.5. 266. 
 
 Quarterly Journal of Pure and Applied Mathematics, Vol. vi (1864), pp. 74, 75. 
 
In-and-Circumscribed Triangles of Quartic Curve 7 
 
 These equations reduce to 
 
 (\-a)fx^-\-{\-a)(X-y)pL-a\(\-y) = (1) 
 
 (X-|8)M"'+(X-/3)(X-7-^2)M-(^X2-/37X-^2i3X + 2j87+i3-7) = (2) 
 
 Eliminate /j, from these equations. The eliminant is 
 
 I (X-«)(X-7) (X-/3)(X-7-2) I 
 I (X-a) (X-/3) r 
 
 I (X-a)(X-7) -aX(X-7) 1 
 
 I (\-l3){\-j-2) -(^X2-/37X-2/3X+2j87+/3-7) I 
 
 I -aX(X-7) -(^X^-/37X-2i3X+2/37+/3-7) |2_^ 
 "^1 (X-a) (X-^) I -" 
 
 On expanding this gives 
 
 (a-i8)(a-/3 + 2)X6-2(a-/3)(a-^ + 2)(l + 7)X^ 
 
 + {2(a-/3)(7-+a7-«i3 + 2a:+/3+37)-2(2^7 + i3-7) 
 + («7-/37-2/3)2| X-* • 
 
 -2!(a+^)(/37"'+«7-+a/3+2/37 + 3a7)-2(a+^+7) 
 
 (2^7+^-7)-(«7-/37-2/3)(2^7 + 2ai3+^-7)j X^ 
 
 -{ 2(2/37+^-7) (a2y_a/3-y,4-«2 + 2a7+i37)+2a/37(a-/3) (7+2) 
 -(2/37 + 2a^+/3-7)-iX- 
 
 + 2a(2/37+i3-7)(7-/5)(«+l)X+a^(^-7)(2/37+/3-7)=0. . . (3) 
 
 It may be noted here that, according to Cayley's analysis,* a 
 cusp of the second kind takes up a double point. If a — /S+2 = 0, 
 then the sextic has /= oo for a double root. But this is the condi- 
 tion that the quintic have a node-cusp at / = <» . 
 
 Equations (1) and (2) may be written as follows: 
 
 (Tia2 — cx(ai^ — a2)~'yo'2~\~c(y<T2 — (4) 
 
 cTia2-/3((7i'^-<T2)-(7+2)(ro+(/37 + 2/3)(Ti-(/3-7 + 2i37) = 0.. (5) 
 
 where ai=X+ju and o-2 = Xju. 
 
 Eliminating a^ from (4) and (o), the result is: 
 
 (a-i3)ai3-2(a-/3)(7+l)ai2+72(a-^) + 2a(|3+7) 
 
 -(/3-7 + 4/37)(ri-(a-7)(/3-7 + 2/37)=0 (6) 
 
 /. The Condition for a Fourth Cusp 
 
 It has been seen that this quintic has cusps at the three vertices 
 of the fundamental triangle, the values of the parameters at these 
 points being 0, 1, co. If a fourth cusp be possible, let the value 
 of the parameter thereat be t = T. 
 
 *Cayley: "On the Cusp of the Second Kind of Xode-Cusp;" Collected 
 Mathematical Worku, Vol. v, pp. 265, 266. 
 
In-and-Circumscribed Triangles of Quartic Curve 
 
 Join Ir. The equation giving the parameters on the join is 
 
 r- 
 
 T — a 
 
 ~^2 
 
 {t-iy 
 {r-iy 
 
 
 
 t — T 
 
 Throwing out the factor -. -r-„, 
 
 this becomes: 
 
 t-y 
 
 T — 7 
 
 
 
 
 putting t — T, and reducing, 
 
 2r2-(3i3+7)r + (/3-7 + 2i37) =0 (1) 
 
 Similarly, the equation giving the parameters on the join of 2t is. 
 
 t-a 
 
 T—a 
 
 
 
 {t-\Y 
 
 1 
 
 /-7 
 
 T — 7 
 
 
 = 
 
 which reduces to: 
 
 (2) 
 
 2 
 
 -3/3-7 
 
 
 
 2 
 
 2 
 
 -3a-7 
 
 
 
 2 
 
 = 
 
 2T2-(3a + 7)r + 2a7 = 
 
 Eliminate r from (1) and (2). The eliminant is, 
 
 i3-7 + 2/37 
 
 -3/3-7 /3-7 + 2^7 
 
 2a7 
 
 — 3a — 7 2a7 
 
 which reduces to: 
 
 272(a-/3)2-(57-9a)(«-i3)(i3-7)+2(i3-7)- = 
 
 which is the required condition. 
 
 A cusp may result by the coming together of two neighboring 
 inflexions as is readily seen by considering a penultimate case. 
 Hence the condition for a fourth cusp should be contained, as a 
 factor, in the discriminant of the equation of the flex cubic 
 (Equation I, p. 6). 
 
 This was not shown in general, as the work involved considerable 
 algebraic difficulty. In the special cases attempted, it was, how- 
 ever, proven to be true, as the following example will show. 
 
 As previously shown, the condition for a cusp of the second kind 
 at / = 00 , is : 
 
 a-i3+2=0 
 
 The discriminant of the flex-cubic is then the eliminant of 
 
 16(2a+ 1)«- (2+37 + 25a + 6a7-}- 12a") = 
 8(2a+ 1)^2 _ 2(2+37 + 25a 4- f)a7+12a2)/ + 9a(2+a + 37 + 2a7) = 
 
I)i-and-Circi(niscrihed Triangles of Quartic Curve 9 
 
 Therefore E= (2+3y-\-'25a-\-6ay-\-na^)^ 
 
 -96a('2a-\-l)(2-\-a-\-3y-\-2ay) 
 
 = lUa*-'i^0a^y-\-36a^y^+40Sa^-396a'-y+3Qay^ 
 - 193a2- lUay-\-9y'i-^ Uy + 4 = 
 
 The condition for a fourth cusp reduces to 
 
 ia''-6ay-\-7a-3y-2 = 
 This divides into E, giving for quotient, 
 
 36a--Qay + 39a-3y-2 = 
 
 II. The Condition for a Tacnode 
 
 A tacnode may arise by the coming together of two double 
 points. The joins of the three double points, two and two, 
 intersect the quintic in an additional fifth point each. If two of 
 the double points coincide, then two of these additional points 
 will also coincide. It is necessary to find the cubic equation 
 connecting these three points. 
 
 Let AiXi-\-A2X2-{-A3X3 
 
 =Ay(t-a)(t-l)-'+Ao(t-0)r+A3(t-y)(t-l)H-' 
 
 = Azt'-{2^y)A3t'+\As{l^2y)-hAo-^Ai\P-\A'y+Ao^ 
 
 + Ji(2+«)1 r~-\-Ai(l-{-2a)t-A,a = 0, 
 
 be the equation of the five parameters of the points of intersection 
 of any line with the quintic. 
 
 Then si==2-\-y is the sum of these five parameters, 
 
 and - = is the sum of their reciprocals. 
 
 From the sextic of the parameters of the double points, it is 
 seen that 5/= 2(1+7) is the sum of the six parameters thereat, 
 
 and --, = is the sum of their reciprocals. 
 
 Se a 
 
 It is readily seen that, if Su S2, S3 be the symmetric functions 
 of the three extra points of intersection with the curve of the 
 joins of the double points, two and two, then, 
 
 Si = 35i-25/ = 3(2+7)-4(l+7) = 2-7 
 
 '^2 0^4 ^Sr,' 3 + 6a 4a+4 2a- 1 
 
 and Tr = 3 2-7= = 
 
 03 s^ Sq a a a 
 
10 In-and-Circumscrihed Triangles of Quartic Curve 
 
 To determine S3: 
 
 Let the parameters at the double points be ti and <2, ^3 and ti, 
 t;, and te. Then the remaining parameter on the 
 
 join of (^1, t2) and (ts, U) is (2+7) — (/i+/2+i3+i4) 
 join of {h, h) and {h, U) is (2+7)-(^i+^2+^5+/6) 
 join of (^3, /4) and {h, U) is (2+7) - (^3+/4+/5+/6) 
 
 Si = 3(2+7)-22<i = 3(2+7) -4(1+7) =2-7. 
 
 Sz = { (2 + 7) - (^1 + ^2 + ^3 + ^4) } { (2 + 7) - (/l + /2 + /5 + ^6) } { (2 + 7) 
 
 -{h-\-ti+h-\-te)} 
 = (2+7)'-(2+7)'(22/0 + (2+7)[(S//)- + (/i+/2)(/3+/4) 
 
 + (/l + /2)(/5 + /6) + (/3 + /4)(/5 + /6)l-2/.i(/i + /2)(/3 + /4) 
 
 + (^l + ^2)(/5 + /6) + (/3 + ^4)(/5 + ^6)}+(/l + /2)(/3 + /4)(/5 + /6) 
 
 From equation (6), page 7, 
 
 s/.-= 2(7+1) 
 
 (^l + 4)(/3 + ^4J + (/l + /2)(<5 + ^6)+(/3 + ^4)(/5 + /6) 
 
 ^ 7^(«-/3) + 2a(^+7)-(^-7 + 4i37) 
 
 a — p 
 Substitute these values in Ss and reduce: 
 
 aW-y) 
 
 '.^3 = ^ 
 
 a — p 
 
 But | = ?^1 
 
 03 a 
 
 rj., f ^ (2a-l)(/3-7) 
 I hereiore 02 = 7; • 
 
 a — p 
 
 The cubic equation connecting the parameters of the three 
 required points of intersection is : 
 
 a—p a—p 
 
 (a-i3)X'^-(a-i3)(2-7)X- + (2a-l)(|3-7)X-a(^-7) = 
 
 If this cubic have a double root, it means that two of the double 
 roots coincide, thus giving a tacnode. The cubic will have a 
 double root if its discriminant is zero. The discriminant is the 
 eliminant of 
 
 S(a-^)\2 + 2(a-i8)(7-2)X + (2a-l)(i3-7)=0 
 (a-/3)f7-2)X2 + 2(2a-l)(/3-7)X-3a(,8-7)-0 
 
I)i-and-Circiiinsciibed Tr'uuigles of Quartic Curve 
 
 11 
 
 E = 
 
 3{a-^) 2(a-/3)(T-2) (^2a-l)(l3-y) 
 
 3(a-^) 2(a-/3)(7-2) (2a-l){^-y) 
 
 («-/3)(t-2) 2(2a-l)(^-T) -3a(^-7) 
 
 («-|3)(7-2) 2(2a-l)(/3-7) -3a(/3-7) 
 
 = 3(a-|3)(i3-7)(a-T)[4a^T-(i3-a)+4«(8a^2^a7--3a^7-/372 
 -5/3'-7) + (36a|37-61a2^+13a27+13ai32-a72+^72_^2^) 
 
 -\-^{8a--5ay-Sa(3-l3y-{-^-)-\-My-^)] = 
 
 a = (3, /3 = 7, 7=a are the conditions that a branch of the curve 
 passes through one of the cusps. Since the fifth intersection on 
 the sides of the fundamental triangle is given by t=a, 13, or y, if 
 two of these are equal, then the points of intersection must be at 
 points common to the sides of the triangle, viz., the vertices, at 
 which are the cusps. The remaining factor then must be the 
 condition that the quintic acquire a tacnode. 
 
 The Triple Point 
 
 If the quintic can have a triple point, let the value of the para- 
 meter thereat be / = r. 
 
 The join of Ir is 
 
 t-a 
 T — a 
 
 1 
 
 {t-\y 
 
 
 t-y 
 
 T — 7 
 
 
 = 
 
 which reduces to 
 
 The join of 2r is 
 
 (r-/3)/2 + (x-^)(r-7-2)^ + 
 ■i3r^>/3(7+2)r- (^-7 + 2/37) 1=0. 
 t-a 
 
 T — a. 
 
 (1) 
 
 
 
 {t-\r 
 
 r-/3 
 
 1 
 
 t-y 
 
 T — y 
 
 
 = 
 
 which reduces to 
 
 (T-a)^2^(r-7)(r-a)/-«r(r-7)=0 (2) 
 
 These two equations give the remaining points of intersection 
 of \t and 2r with the curve. For a triple point these must coincide. 
 Hence 
 
 (T-^):(r-a)(r-7-2):-/3r2+^(7+2)r-(/3-7 + 2/37) 
 
 = (r— a:):(r — q:)(t — 7): ^Q:r(T — 7) 
 
12 I it-a )ul-Ci rcinnscrihed Triangles of Q tun-tic Curve 
 
 Hence 
 
 (r-/3):-/3T2+/3(7+2)r-(|8-7 + 2/37) = (r-a):-ar(r-7) 
 and 
 
 (r-^)(r-7-2):-/3T2+^(7 + 2)r-(^-7 + 2/37) 
 
 = (r — a)(r — 7) :— 0:7(7 — 7) 
 
 That is, 
 
 (a-/3)r3 + (/37-a7 + 2/3)T2-(2a^ + 2/37+iS-7)r 
 
 +«(/3-7 + 2i37)=0...:... (3) 
 and 
 
 (a-/3)r3 + (a-/5)(7+2)r--(/3-7 + 2^7)r+«(/3-7 + 2^7) = 0...(4) 
 Eliminate r from (3) and (4) 
 
 £=(/3-l)2(/3-7)(a-|3)=0 
 
 As previously shown, ^ — y, a = ^, are the conditions that a 
 branch of the curve passes through one of the cusps. If /3=1, 
 then the quintic degenerates, that is, there can be no triple 
 point. 
 
 A second proof of the non-existence of the triple point will now 
 be given. 
 
 Subject the quintic to the quadratic transformation 1/^ = —. 
 
 The curve becomes a quartic with the following parametric 
 equations : 
 
 y.^ = (t-l)%t-a)(t-y)=t'-{a-\-y-^2)t^+\ay-\-2(a-\-y)-\-l\r- 
 
 — {a-{-y-\-2ay)t-\-ay 
 
 From the five column matrix formed b}^ the coefficients of the 
 above three equations, the following ten determinants are derived: 
 
 2Moi2=- (a -18+ 2) 
 16Aoi3=(« + /3)(a-^ + 2) 
 
 4Aoi4 = ai3(«-/3 + !2) 
 24Ao23 =-{a + ^) {uy -^y + 2a) - ( /3 - 7 + 2^7) 
 
 6A024 =^ «[/3(a7 - /37 + 2a) + (/3 - 7 + 2/37) ] 
 
 4Ao34=-«-(/3-7 + 2i37) 
 J)6Ai23 = 7-(«+/3)(a-/3+2) + (^ + 7)[2a- + (2a+l)(^-7)] 
 24Ai24--aiS7'(«-^ + !2)-a(/3+7)(/3-7 + '2ai3) 
 16Ai34 = a-(/3 + 7)^/3-7 + '2«/3) 
 24A234=-a''^7(/3-7 + 2i37) 
 
 If ao-«i^ + 02/'-«3/^+a4^''==0 
 
 and bo-bit-\-bof'-b,f-'-\-b,f' = 
 
Oo 
 
 ai 
 
 Oo 
 
 03 
 
 60 
 
 h 
 
 62 
 
 63 
 
 «1 
 
 (12 
 
 as 
 
 04 
 
 61 
 
 62 
 
 63 
 
 64 
 
 Jn-aii<]-('irciniisciibed TriaiKjlefi of QiKtrtic Ciirre 1>\ 
 
 be two quartics apolar to the three above quartics, then the 
 determinants formed from the matrix of the coefficients of these 
 two latter equations are proportional to the determinants in the 
 first matrix; so that 
 
 There is no loss in generality in placing p= 1. 
 
 The condition that the quartic have a triple point is 
 
 = ot 
 
 that is, 
 
 Aoi . A34- A02 . A24 + A03 . A23 + A12 . Ai4-Ai3"+A23 . Al2 = 
 
 24A2.34 . 24Aoi2- I6A134 . 16Aoi3+'24Ai24 • 4Aoi4+4Ao34 • 24Ao23 
 
 -(6Ao24)-+4Aoi4.4Ao34 = 
 
 Substituting the values of the AijkS, this becomes 
 
 aH0y(a-l3-\-2) (l3-y-\-2(3y) - (a-]-^)((3-\-y) (a-/3 + 2) {^-y-\-2^y) 
 + /3(a-/3 + 2)!^7-(a-/3 + 2) + (/3+7)(/3-7 + 2a^)! 
 + (/3-7 + 2/37)|(a + i3) (a7-i87 + ^2a)+ (/3-7 + 2^7)l 
 -\l3(ay-0y-\-ia + ){i3-y + Wy)\'-\-ct^(a-l^-\-'2)W-y-\-^20y)] = O 
 
 Which reduces to 
 
 2«^'(/3-l)-(a-/3)(/3-7)(«-7) = 
 
 But it has been seen already that none of the conditions here 
 obtained can be the necessary condition for a triple point. 
 
 A third proof of the non-existence of a triple point may be stated 
 thus: 
 
 If possible, let the three cusped rational quintic have a triple 
 point. 
 
 Subject the quintic to a quadratic transformation. 
 
 Let the vertices of its singular triangle be two of the cusps and 
 the triple point, the cusi)s being at the vertices 1 and 2, and the 
 triple point at 3. 
 
 The resulting curve is of order 
 
 10-3-2-2 = 3,t 
 
 * W. F. Meyer: "Apolaritat und Rationale Curven," Chap, i, p. 3. 
 t Ibid., Chap, ii, p. 184. 
 
 JR. Sturm: "Die Lehre von dem Geometrischen Verwandtschaften," 
 Vol. iv, p. 44. 
 
14 In-(nid-Circiimsc)ihed Triangles of Qitartic Curve 
 
 having points of tangency on the sides 1'3' and 2'3' of the singular 
 triangle in the transformed plane; also a branch of the curve goes 
 through the vertex 3', since there is one extra intersection on the 
 side 12 of the original triangle. 
 
 The original quintic had a third cusp, which in the transforma- 
 tion remains a cusp. Hence the new curve is a cuspidal cubic, 
 and therefore of the third class. That is, from a point of the 
 curve, but one tangent, excluding the one at the point itself, 
 may be drawn. 
 
 But this curve would be on the vertex 3' and have as tangents the 
 sides 1'3' and 2'3', which is clearly an impossibility. i\ccordingly, 
 the three cusped rational quintic cannot have a triple point. 
 
 The Quadratic Transformation 
 t-ai 
 
 If the quintic xi = ^^_ ' (i = 1, 2, 3) be subjected to the 
 
 quadratic transformation Xi = - (i— 1, 2, 3), the resulting curve is 
 
 t—ai 
 or yi = {t-^i)Ht-a2)(t-as) 
 
 y2 = {t-^2)-{t-a^){t-a{) 
 yz = {t-^^y{t-a,){t-a.^. 
 
 This is a curve of the fourth order. It passes through the 
 vertices of the fundamental triangle of the transformed plane, 
 and is at the same time tangent to the sides. That is, the funda- 
 mental triangle is simultaneously in-and-circumscribed to the 
 quartic, its vertices being t=ai, and its points of tangency t = ^i. 
 
 It has been seen that the original quintic has three double 
 points; and, further, that conditions can be imposed upon it so 
 that it can acquire a fourth cusp or tacnode without degenerating. 
 By the quadratic transformation, quintics of each of these types 
 will go into quartics having three double points, a cusp or a tacnode 
 res])ectively. 
 
 The existence, therefore, of one triangle in-and-circumscribed 
 to the quartic led to the investigation of the number of such 
 triangles in each of these cases. 
 
 It was shown that the rational quintic with three cusps could 
 not have a triple point. Accordingly, the rational quartic with 
 a triple point cannot have triangles in-and-circumscribed; for, 
 
hi-and-Circumscrihed Triangles of Quartic Curve 15 
 if it were possible, by subjecting the quartic to the quadratic 
 transformation i/i = - (with such a triangle as reference triangle), 
 
 it would go into a quint ic with a triple point. This has been shown 
 impossible. 
 
 The Rational Quartic 
 
 Consider now the.in-and-circumscribed triangles of the rational 
 quartic. 
 
 Let its parametric equations be : 
 
 Xi = aot'^-\-ait^-\-a2t~-{-ast-\-ai 
 X2= biP+bor^+bst 
 
 X,= CitS-^c2t'-\.c,t 
 
 thus making the vertex (1, 0, 0) a double point with parametric 
 values thereat / = and t= co. 
 
 Let (as) = and {(Ss) = be the conditions upon a set of four 
 parameters that they lie on a line. 
 
 Then aoSo-{-aiSi-{-a2S2-\-oc3Ss-}-aiSi = 
 
 i80'^0 + l8l5l+|8252+|8353+/S454 = 0, 
 
 where {a^)ik is proportional to the determinant Aimn formed 
 from the matrix of the coefficients of the parametric equations of 
 the curve.* 
 
 That is, 1 q:/5 | oi — '2M2,34 = «4 | be \ 23 = fcX, say, 
 \ a^ \ 02- lQAui = ai\ be \n = kY 
 I a:/3 1 03 — 24Ai24 = rt4 \ be \ 12 = kZ 
 |«/3 |o4^96Ai23= \abc\viz = P 
 1 a/3 1 14 - '24^023 = Qo I be \ 23 = X 
 la/3 |24-16Aoi3 = ao I 6<? I i3=F 
 I a/3 I 34 — '2Moi2 = ao 1 be \ n = Z 
 I a^ I 12 = 1 a/3 I 13 = I a/3 1 23 = 4Ao34 = 6A024 = 4Aoi4 = 
 
 where h = — -. 
 aa 
 
 In (a5) = and (^.'^)=.0 take si as the symmetric functions of 
 Xi, X2, X, X, so that, 
 
 50=1, 5i = Xi+X2+^2X, .y2=XiX2+2X(Xi+X2)+X2 
 
 53 = 2XiX2X+(Xi,+X2)X2, 6'4-XlX2X2 
 
 * W. F. Meyer: "Apolaritat und Rationale Curven," Chap, i, p. 33. 
 
E = 
 -4 
 
 16 Iii-und-Circumscribed Triangles of Qiiartic Curve 
 
 Substituting these values for s„ 
 
 (as) = [ao+ai(Xi+X2) +0:2X1X2] + 2[ai+a2(Xi+X2)+a3XiX2]X 
 
 + [a2+a3(Xi+X2)+a4XiX2]X2 = 
 (^5) = [i3o+^i(Xi+X2) +/32XiX2] + 2[/3i+/32(Xi+X2) +/33XiX2]X 
 
 + [/32 + /33(Xl+X2) +/34XiX2]X2 = 
 
 Eliminate X from these two equations : 
 
 0:2 +«3(Xi+X2) +0:4X1X2 ao+o!i(Xi+X2) +02X1X2 1 2 
 
 i32 + ^3(Xl+X2) +^4X1X2 i8o + ^l(Xl+X2) +^2X1X2 1 
 
 ao+a3(Xi+X2) +0:4X1X2 ai+a;2(Xi+X2)+03XiX2 I w 
 
 /32 + ^3(Xl+X2) +134X1X2 /3i+/32(Xl+X2)+/33XiX2 I ^ 
 
 I Q:i+02(Xi+X2) +0:3X1X2 0:0 +o;i(Xi+X2) +02X1X2] _q 
 I i3i +/32(Xi +X2) +133X1X2 ^o+i3i (Xi +X2) +/32X1X2 1 
 
 Expanding the eliminant, and substituting therein for the 
 (aB)i/s their respective values, there results an equation in Xi 
 and X2, which is the condition that the join of Xi and X2 be a tangent 
 to the curve. By cyclically permuting, the conditions that the 
 joins of Xo and X3, Xi and X3 be tangents to the curve can be written. 
 These three equations are contained in the following schema, 
 viz., equations (1), (2), (3), page 17. 
 
 From these equations there are derived three equations in 
 Si, S2, S3, the symmetric functions of X], X2, X3. They are equations 
 (4), (5), (6) of the schema, page 17. 
 
 It may be well to indicate here how these equations have been 
 obtained. Multiply equation (1) by X3^ (2) by Xl^ (3) by X2*. 
 Subtract these two by two, and throw out the factors X3— Xi, 
 Xi— X2, X2— X3. This gives three new equations. Subtracting 
 any two of these, and throwing out one of the above factors, 
 equation (4) comes out. In an analogous way by multiplying the 
 original equations by Xr'^ and by Xl^ equations (5) and (6) are 
 derived. 
 
 Equations (5) and (6) are linear in 81 and S2. From these the 
 following values of Si and So in terms of S3 are derived: 
 
 -YiSs'-\-^VHPX-3kYZ)S,'-\-2k(3PY'-\-iXn^'--^2PXyZ 
 
 -j-u-x7J-8ky'Z'-x'Z)s,'+u-H,Pxz-'-ky/j-'ipy'z 
 -5X-'rz+nxy)S3''-\-k'{kZ'-^X'-z-'-2Prz-'-\-ioxy'Z 
 
 +4 ¥*) S3 + ^Ik'Y'Z 
 
 '^' ~ -X^'~Sz'-{-U-\XY(XZ-^2V')S,^-\-k-'{ IGXY'Z - UXY^Z 
 
 -SX^Z'-UY')Ss^-\-U-n'Z{XZ-^2Y-)S3-k'Y^Z'- 
 
 <iXY'S,'+{X'^U-Y'-U-X-'Z'^-2PX^'-\-10kXY^Z)S3' 
 -\-2k{PX~Z-2PXY-'-Xn'-5kXYZJ + UkY'Z)Ss' 
 -\-2k''i4kY'-Z''-\-fiPY'-4PXYZ-\-iX'Z-SX'-Y^-kXZ')Ss"- 
 -\-2kU'^{PZ-3XY)S-i-k*Y' 
 
 S2 = 
 
 -X^Y\S,'-^4k''XY{XZ-'2Y'')S3'+kHl(^XY-Z-4kXY'-Z 
 -3X^-Z^-UY')Ss'-^U-n^Z{XZ-2Y~)Ss-k'Y'~Z'- 
 
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IS In-and-Circumscribed Triangles of Qiiartic Curve 
 
 These values substituted in equation (4) give an equation of 
 the eleventh degree in S3. The constant term and the coefficients 
 of Ss^^ and 83 vanish identically, thus reducing the equation, 
 after dividing through by 83^, to the eighth degree. 
 
 The coefficients of this octavic are such that when F=0, the 
 coefficients of 83^, S3'', and 83, as well as the constant term, become 
 identically zero, and the equation then reduces to a quartic. This 
 condition, viz., F = 0, is the condition, as will be seen later, that 
 the quartic have a tacnode. 
 
 The equation has eight possible solutions, each of which gives 
 a single value for 81 and 82. There are then eight sets of values 
 for 81, 82 and 83, each set leading to an in-and-circumscribed 
 triangle, the vertices of which are found from the equation 
 
 X3-8iX2+82X-83 = 0. 
 
 It is to be noted that the number of triangles given here is the 
 same as that found by Cayley's formula (page 4) for this case. 
 
The Quartic With a Tacnode 
 
 Let the parametric equations of the quartic bs 
 
 a:i = ao/^+ai/^+«2^~+a3^+rt4 
 X2= biP:i-b2t^-^bd 
 
 X3= C2t'^ 
 
 thus making the vertex (1, 0, 0) a tacnodal point with parametric 
 values thereat / = and t— <^, and at the same time making .i'3 = 
 a tangent to the curve at the same point. 
 
 Let {as) = and {^s) = be the "Schnittpunktformen," that is, 
 the conditions that a set of four points he on a hne. Then using 
 the notation of the previous case, 
 
 I a/S 1 01 — "2^234= —aib3C2 = kX, say, 
 
 I «/3 I 03 — 2Ml24 = a4&lC2 = ^'-^ 
 
 |ai8 I 04-96Ai23=-C2|a6| n = P 
 I a/3 I 14 — 24Ao23= —aobsC2 = X 
 1 a,S 1 34 - 24/^012 = (lobiCo = Z 
 
 I ajS| 12= \al3 I 13= |a|S|23 = 4Ao34 = 6Ao24 = -l-Aoi4 = 
 
 |a^|o2=l«/3|24=r = 
 
 Substituting }^ = in equations (4), (5), (6), page 17, the 
 following relations between Si, 82, *S3 result: 
 
 X^S3^S2-{P''-GkXZ)S3-'-2kPZS2S,-\-4kXhSiSs-k'~Z'~S2- = 0. . I. 
 
 -X^Sz'~Si-^2PXS3'^-^2kXZS2Ss-4kX\S3-\-k-'ZhSs = 0. . . .IL 
 
 {X''-^kZ'')S3-2kXZSiS3-2kPZS3-k-'Z'-S2 = III. 
 
 From II and III, linear in Si and <S2, it follows that : 
 
 2Xi^kZ''-X'-)Ss'^-\-2kPXZSs - kZjU-X'- -k'-Z') jy 
 '^'~ -SkX-'ZSs '" .' 
 
 _ -X{^kZ-^-X'-)S,'+2kPXZS,+2kZ{^kX-'-k-Z-) 
 ^'~ -3k-KXZ-' 
 
 Substituting these values for .Si and .S2 in I, there results the 
 following equation in »S'3: 
 
 4Z2(4^'Z2 - X^) (X2 _ A-Z2) .S3^ - U'PX-'Z(X''+ 2kZ'')S3' 
 -k'~XZ(lGX'-32k^X^Z-'-\-P-'XZ-\- l6k''Z')Ss- 
 - ^k'PXZ'~{2X--\-kZ'-)S3 - 4/.-''Z2(4Z2 - kZ-') (X'' - kZ-') = 0. 
 
 19 
 
2(1 fn-<iii«J-('ir(iiiiiscribe(l TruuKjlcs of Qiitirtic Curve 
 
 This equation factors into : 
 
 [2Z(.Y+2V A-Z)(X+ ^JlcZ)S■i~-\-kPXZSs-\-2k'~Z{2X-{- ^J kZ) 
 (X+VA'Z)] 
 [2X(Z-2V kZ){X- V A-Z)83-+A-PA'Z83+2A-2Z(^2A'- V kZ) 
 {X-^kZ)]^0 VI. 
 
 There are four possible solutions for iS's. Each value gives a 
 single value for 81 and So. Hence, there are four sets of values 
 for iS,, each set determining an in-and-circuinscribed triangle. 
 
 The fact that equation VI. factors into two c^uadratic factors 
 must be especially noticed. This indicates that the triangles 
 fall into two sets of two each, and not into one irreducible set of 
 four triangles. 
 
 The Quartic with a Triple Point 
 
 Let (as) = and (/3s) = be the "Schnittpunktformen" apolar 
 to the three equations representing parametrically the rational 
 quartic, viz., 
 
 Then the condition that the quartic have a triple point is 
 
 ao tti 0:2 "3 
 
 /3o ^1 1^2 ^Z 
 
 ai ai a3 0:4 
 
 ^1 /32 /33 i34 
 
 = 
 
 which reduces to | aj8 1 01 • | ccjS I 34 — I a/S [02 • | a^S 1 24 = 0. 
 
 That is, I a/S 1 01 : 1 a^ I 02 = j a/3 | 24 :| «/? I 34^ which in the notation 
 previously used is kX:kY = Y : Z, or A'Z= Y~. 
 
 It is possible, therefore, to take A'= 1, Y — m, Z — m'-, and 
 also k=\. 
 
 Substituting these values in equations (4), (5), (6), page 17, 
 they reduce to : 
 
 S>mSz^-{P^-^m^)S3''+Si''S2+{Qm-^Pm''-)S2S3+{4>-^mP)SiSz 
 -m%2_|_(53_5^^S2)2m3+?n2(S2-Sr)=0 1. 
 
 m2S3'-(2P-rm^)S3'-SiS3- + (w^+!2?"P-4)S3+2m2S283 
 
 + 27^i3S3+m2Si = II. 
 
 m 2S2S3 - + 2w.SuS3 - + ( 1 + 2mP - 4w ') S, 2 + 2/» ''SiSs 
 
 -(2P/?i2-6m)^'3 + w^.S2 + w-' = III. 
 
In-and-Circuin.scribed Triangles of Qiiartic Curve 21 
 From equations II and III, linear in Si and 82, it follows that: 
 
 So 
 
 (Sz + m)-' 
 ■2mhS3^- (Gm^-'iPmA-DSs'-- (MPm''-^m)S3-{-m^ 
 
 m^(S3-\-m)^ 
 Substituting these values in I, this equation becomes: 
 
 ( - m^-{-'2Pm^-^2m'-P-^m^-\-2Pm - 1)83^(83 + ^) ^ = 0. 
 Neither one of the values ..S3 = 0, — m gives proper triangles. 
 
 The Quarlic with a Cusp 
 Let the parametric equations be : 
 
 thus placing the cusp at the vertex (1, 0, 0) with / = thereat, and 
 also making the sides of the triangle of reference tangents to the 
 curve. The sides 13 and I'-Z have points of tangency at t = b 
 and f — c respectively; while 23 is a double tangent with t= ^a 
 at the points of tangency. 
 
 Let {as) = and {0s)^O be the conditions that a set of four 
 points lie on a line, then, 
 
 \a^\ 3io2iAoi2 = 2{2a'-bc){c-b) 
 I a/3 1 23 <> -UoM=-2aXc-6) 
 |a^|i3^ 6Ao24 = a*(c2-fe2) 
 I aj8 I 03 - 2Mi24 = - 2a ^bc{c - b) 
 
 I a/3 |oi= I a/3 |o2= 1 a/3 [04= I a/3 | — 12= I a/3 1 14= | a/3 [24 = 
 In {as) = 0, (j3s) = let st be the symmetric functions of Xi, Xo, 
 X, X. Substitute for Si their respective values. From these 
 ecjuations eliminate X. The result is the condition that the join 
 of Xi and X2 be a tangent to the curve. Upon substituting for 
 a^ik their respective values the eliminant reduces to, after 
 throwing out the linear factor, 2X1X2— (6+^)(Xi+X2) + -^<"» and 
 thus making it a (3 — 3) correspondence : 
 
 aH6+c)(Xi+X2)='+6a^(Xi+X2)-XiX2-2a^6c(Xi+X2)- 
 
 -8(2a2-6c)Xi%'^-4a^(6+c)(Xi+X2)XiX2-8a%%2=0 
 
 The linear factor thrown out is the condition that the join of 
 Xi and X2 pass through the cusp. The condition imposed upon 
 
22 , In-and-Ch'Cwmscribed Triangles of Quartic Curve 
 
 this join is that the remaining two parameters shall be equal, a 
 condition which is satisfied if the line passes through the cusp. 
 
 Two similar equations, giving the conditions that the joins of 
 Xi and X2, X2 and X3 be tangents to the curve, can be derived. 
 From these three, then, the following equations in Si, S2, and S3, 
 the symmetric functions of Xi, X2, and X3 can be derived. They are: 
 
 8{2a^-bc)S2Ss-a'ib-\-c){So-S,'-)-\-2a\Ss-2a*bcSi = 0. . . I. 
 
 (^+^)(S2--SiS3)+6.SVS3+26cS3 = 11. 
 
 S3-+(6+c)S.>S3+fecS2- = III. 
 
 From III, S3 = — 6S2 or — 082 
 
 c u ,; f ^ AC • TT • ^- (5b-c)S 2-\-2b'~c 
 
 bubstitutmg »S3= — 002 in 11, gives Oi = 7-^—, — ^ , 
 
 b[b-\-c) 
 
 and substituting these values in I, there results 
 
 a'b'^l (b-{-c)(3b^c) -2c(5b-c)] 
 
 S2 = 0, or 
 
 a*{5b-c)''-8b\2a--bc){b-\-c) 
 
 Similarly using 83= —082, gives 
 
 (5c -6) 82 + 26c 2 
 
 Si = - 
 
 cib+c) 
 
 „ a'cmb-\-c)(b-\-3c)-2h(5c-b)] 
 
 and 02 = — i7"z rv^i — o -i/ -» ■> — 1 wl , — {■> or U. 
 
 a*{oc — b)- — 8c^i'-2a- — bc){b-\-c) 
 
 The value S2 = evidently does not lead to a solution. Hence, 
 there are two possible solutions, one for each value of S3. 
 
 It is to be noted that the number of triangles given here is the 
 same as that found by Cayley's formula (page 4) for this case. 
 
 The (3 — 3) Correspondence 
 
 The in-and-circumscribed triangles of the cuspidal quartic 
 were found by means of a (3 — 3) correspondence. This corre- 
 spondence is not, however, of the most general kind. The most 
 general correspondence of this type is that set up by means of the 
 lines drawn from a point of a conic to a line-cubic. 
 
 The question then arises, what are the conditions which the 
 line-cubic must satisfy, either in itself or in its relations to the 
 conic, in order that the general (3 — 3) correspondence reduce 
 to the type set up by the method here employed? 
 
In-and-Circmnscrihed Triangles of Quartic Curve 23 
 Take as the conic the norm-conic: 
 
 .-To = 1 
 
 Let the hne-cubic be (a^) ^ = 0. 
 
 The condition that the join of two points Xi, Xo, of the conic be a 
 hne of the cubic is : 
 
 1 2Xi Xi 
 1 2X2 Xr^ 
 
 ._ . 2 
 
 ' ■ ~\2 A2 
 
 
 where cri, a^ are the symmetric functions of Xi and X2. 
 
 Substitute these vahies in (a^)^ = 0, thus giving the equation 
 of the (3 — 3) correspondence between Xi and X2 : 
 
 o O"!^ I o 0'lO'2" I Q ■> I Q 0"i-0-2 (Ti^ 
 
 Uooocr-i cim-^ +0222 — «3aooi"^ -roaoo20'2'+ocion 7 ''^'^112-7- 
 
 3 
 
 3ao220'2 — 7Qi220'i — 3aoi2Ci(r2 = 
 
 These coefficients being all independent, this is the most general 
 (3 — 3) correspondence. The equation of the (3 — 3) corre- 
 spondence set up by the rational cuspidal quartic has but six 
 terms, lacking the constant term and those in 0-1(72-, a2, ai. In 
 order that the general correspondence become of the same type 
 as the special one, then 
 
 OOOI = O022 = ai22 = fl222 = 
 
 The equation of the line cubic then reduces to : 
 
 aooo^o^+3aoo2^o"^2+3aoii^o?i-+6aoi2?o|i^2+3aii2^r^2+aiii^i^ = 
 
 This cubic lacks the terms in ^2^ and t2~, thus indicating that the 
 line 0-2 = is a double tangent of the line-cubic, and is at the same 
 time a line of the conic. 
 
 The results obtained indicate some of the conditions which 
 must be imposed upon the line cubic so that the general (3 — 3) 
 correspondence may reduce to the special type here under con- 
 sideration. It is to be noted, at the same time, that these condi- 
 tions are necessary but may not be sufficient. 
 
 R. A. Roberts, in a paper entitled, "On Polygons Circumscribed 
 about a Conic and Inscribed in a Cubic," proposes "to consider 
 . . . the general problem of finding conies and cubics related 
 to each other in such a manner that it may be possible to circum- 
 
24 I n-and-Circii inscribed Triangles of Quartic Curve 
 
 scribe about the conic an infinite number of polygons which are 
 inscribed in the cubic."* 
 
 That is, he takes the hnes of a conic and the points of a cubic, 
 which is the dual of what has been used above, and finds various 
 relations between the two curves so that the results obtained are 
 always poristic. 
 
 It has been shown here that, if a tangent to a point conic is a 
 double line of a line cubic, the solution cannot be poristic. Dually, 
 this would say that if a point cubic have a double point and this 
 double point be on the line conic, the solution cannot be poristic. 
 
 The (4 — 4) Correspondence 
 
 The correspondence set up in the case of the quartic with 
 three double points is a (4 — 4) correspondence. As in the pre- 
 ceding case, it is, however, not of the most general type. The 
 most general (4 — 4) correspondence is set up by means of a 
 point-conic and a line-quartic. As in the preceding instance, 
 what are the conditions which the quartic curve must satisfy, 
 either in itself or in its relations to the conic, so that the general 
 (4 — 4) correspondence reduce to the type set up by the method 
 employed here.^ 
 
 Let the conic be the norm conic : .ro = 1 ; Xi = 2X; X2 =X-. 
 
 And let the line quartic be (a^) ^ = 0. 
 
 Then the conditions that the joins of two points Xi and X2 of the 
 conic be a line of the quartic is: 
 
 11 2X1 \r i 
 
 ^'~1 1 2X2 Xs^l 
 
 where au ao are the symmetric functions of X] and X2. 
 
 Substitute these values in (a^)^ = 0, thus giving the equation 
 of the (4 — 4) correspondence between Xi and X2 : 
 
 «oooo0'2 — 4aoooi~^ +4aooo2a'2 +t>flooii — | I<i0ooi2 2 i-oaoo220-2 
 
 — 4aoiii—^ +1200112-^7— — 12aoi22~T- +-4ao2220-2-|-flniiy^ — -l«iii2-^ 
 
 2 
 + 601122— r 4ai222^^ +02222 = 
 
 These coefficients are all independent, and so this is the most 
 general type of (4 — 4) correspondence. 
 
 * Proceedings of the London Mathematical Society, Vol. xvii, p. 158. 
 
 0^2 : — T : 1 
 
In-and-Circumscrihed Triangles of Qitartic Curve 25 
 
 The equation of the (4 — 4) correspondence set up by the rational 
 quartic has but twelve terms, lacking those in ai^, ai^a2, ci^. 
 (See Eqn. 1, p. 17). In order that the general correspondence 
 become of the same type as this special one, then, 
 
 fliiu = Qoiii = f'lll2 = 0. 
 
 The equation of the line-quartic then reduces to : 
 
 aoooo^o^+4aoooi^o''^i+4aono2^o^^2+6aooii?o^^i^+ 12a 0012^0 -^i^2+6aoo22^o-^2- 
 
 + 12ao]12^0^l"^2+12aoi22^o|l|2"+4ao222^0^2''' + 6ail22^1-^2- + 4ai222^1^2'^ 
 + 02222^2'* =0 ^ 
 
 This quartic lacks the terms in ^i^ and ^i^, thus indicating that 
 the line .ri = is a double line of the quartic. This line is the join 
 of the points of the norm-conic whose parameters are X = 0, X = oo , 
 which are also the parameters at one of the double points of the 
 rational quartic. Consequently, the lines, which are the joins of 
 the points whose parameters are the same as the parameters 
 at the other two double points, are also double lines of the line- 
 quartic. These conditions are some of those which the quartic 
 must satisfy, but are not necessarily all. 
 
 In the case of the quartic with a tacnode, the (4 — 4) corre- 
 spondence has only eight terms (as is readily seen by placing 
 }'=0 in Eqn. 1, p. 17). Then the general correspondence, in 
 order to be of the same type, must have, in addition to 
 
 fliiii = Ooui = aiii2 = 0, 
 
 also CtoOOO = QOOOI = ai222 = fl2222 = 0. 
 
 The line quartic then becomes, 
 
 4aooo2^o'?2+6aooii^o-^r+12aooi2^o"^ie2+6aoo22?o'?2- + 12aoii2^o^i-^2 
 
 + 12aoi22^0^1^2-+4ao222^0?2^ + 6ail22^r^2^=0. 
 
 The absence of the terms in ^i^ and ^i'^ indicates that the line 
 x^ = is a double line of the quartic; while the absence of the terms 
 ^o"* and ^2^* indicates that the lines .ro = and .r2 = are also lines of 
 the quartic. The absence of the terms in ^o^^i and ^1^2^ means 
 that the quartic passes through the vertices and 2, and since 
 the lines .Vq = and .r2 = are lines of the quartic these vertices 
 must be points of tangency. 
 
 It was shown, in the case of the quartic with three double- 
 points, that the lines joining the points of the conic, whose para- 
 meters are the same as the parameters at the double-points, are 
 lines of the quartic, so in the case of the tacnodal quartic, where 
 
26 In-tuid-Circumscribed Triangles of Qiiartic Curve 
 
 two of the three double-points come together at the taenode, the 
 line joining the points of the conic whose parameters are the 
 same as those of the remaining double-point, is a line of the 
 quartic. 
 
 Reality of Solutions 
 
 A solution is to be regarded as real if the cubic giving the 
 
 vertices, viz., 
 
 t'-S,P+S2t-Ss = 0, 
 
 has real coefficients. In all the cases examined, the ec^uations 
 obtained enable Sy and »S'2 to be expressed rationally in terms of S3. 
 Every real value of S3 will, therefore, lead to real values of .Si and 
 S2, and, accordingly, to a real solution. The number of real 
 solutions will therefore be simply the number of real roots of the 
 equation in S3. It may be noted that the above cubic will not 
 always lead to a triangle with three real vertices. In the tacnodal 
 case, the special values given (p. 30) yield four real solutions. Of 
 these four real values of S3, only one leads to a triangle with three 
 real vertices. 
 
Special Cases 
 
 The following section is added in order to include a few cases 
 showing in-and-circuniscribed triangles with three real vertices. 
 It is to be noted that in the various cases it may not be possible 
 to draw the maximum number of real triangles. For the rational 
 quartic, a case was found where three of the possible eight tri- 
 angles could be drawn. For the quartic with a tacnode only one 
 could be found. However, in the case of the cuspidal quartic, 
 it was possible to find two triangles with three real vertices each. 
 
 (a) The Rational Quartic 
 
 It is proposed, here, to find a rational quartic symmetrical as 
 to the fundamental triangle of reference. 
 
 Let the three double points be at the vertices of the triangle, 
 
 771 " ~|~ 771 "4~ 1 
 
 and have as parameters thereat (0, oo), (l, ), 
 
 771 
 
 ( — 771, 771 --\- 771 -{-!). These values are determined as follows: 
 If the six parameters taken in order of continuity along the path 
 of the curve be 0, 1, a, oo, b, — w, then a and b can be determined 
 in terms of m by means of the relations, 
 
 (0, 1/a, oo) = (a, co/b, -m) = {b, -m/O, 1), 
 whence a = m--f/n + l and b= , 
 
 771 
 
 Then ., = A,^(/-l)(/+ ^^^^+"^ + ^ ) 
 
 771 
 
 .r2 = ht(t-j-77l)(t-[77l'-\-77l-\-l]) 
 
 .r3=(/-l)(/+m)(^+ ^^'+J" + ^ (/-b7z^ + /» + l]) 
 
 It is necessary to determine A-i, ko so that the curve have tri- 
 angular symmetry. This will be the case if the tangents at the 
 double points be oppositely inclined. 
 
 The tangent at is, 771-'- V" =0 
 
 The tangent at 00 is,^ 'j^ =0. 
 
 A"! A*2 
 
 These are oppositely inclined if ki- = 77i~k2-. Take ki = 77ik; 
 k2=-k. 
 
 27 
 
28 In-and-Circumscrihed Triangles of Quartic Curve 
 
 Substituting these values in the 'above equations, the equations 
 
 f)2 ~ -4- fYi -X- \ 
 
 of the tangents at 1 and — are found to be : 
 
 m 
 
 xi ( 1 + m) - + A-.r3 = and .Ti ( 1 + m) ^ + km ^-x^ = 0. 
 
 Theseareoppositely inchnedif A-- = — . Take 1c — — . 
 
 m~ m 
 
 The parametric equations of the curve then become 
 
 Xi = 77i{l-\-m)-t{t-l){t-\ — — ) 
 
 m 
 
 It may readily be shown that the tangents at the remaining 
 double point are oppositely inclined, and that, consequently, the 
 curve has triangular symmetry. 
 
 The line .ri = .r2 intersects the curve, aside from / = 0, <», in the 
 parameter-pair 
 
 /2-(w2 + ;7? + l)=0 
 
 It is required to find what value m must have so that the point 
 
 f=V/w^+m+l may be a vertex of an in-and-circumscribed tri- 
 angle. On account of the triangular symmetry, it is clear that 
 .the other two vertices will be the remaining intersections of the 
 
 tangents at t= —^lm--\-}n-{-l with the curve, and that there will 
 be three such triangles. 
 
 The equation of the tangent at this point can be derived. 
 From this the symmetric functions of the parameters at the three 
 vertices, in terms of m, can be written down. Substituting these 
 in equation 5, p. 17, this reduces to 
 
 69m>2_|_3i2mii-f-50877iio+422w9-256-lw«-6238//i7-8736OT'' 
 
 -7160wS-3551//i^-304w3+668w2+396w+64 = 0, 
 
 a positive root of which is m = '-2.15. 
 
 The parametric equations of the curve become 
 
 Xi = 2.15(S.15)H(t-i)(t-^3.615) 
 
 X2=-{3.l5)H{t-\-2.15){t -7.77^5) 
 
 .T3 = 2.15(/-l)(/+2.15)(^+3.615)(^-7.7725). ' 
 
 The vertices of one triangle, given by t='^m~-\-m-\-l and equation 
 of parameters giving the intersections of the tangent at 
 
 / = -^Jm^+m+l with the curve, are f = 2.788, l.OJ), and 6.898. 
 
In-and-Circumscribed Triatujlcs of Qiiartic Carve 
 
 29 
 
 On account of the svmnietrv of the curve, the vertices of the 
 other two triangles may be found in an analogous way. For the 
 construction of these triangles see Figure 1. 
 
 (6) The Quarlic with a Tacnode 
 Let the parametric equations be 
 
 .r, = /(«-4)(/-|) 
 thus making the vertex (1, 0, 0) a tacnodal point with parametric 
 
30 In-and-Circiiniscribed Triaiifflest of Qiiartic Curve 
 
 values thereat / = 0, oo, and at the same time making Xs — a 
 
 tangent to the curve at the same point, and also making Xi = a 
 
 4 
 
 double tangent with parametric values / = 3 and / = „ at the points 
 
 of tangency. 
 
 From the matrix of the coefficients, 
 
 A = — -, Z^5, F = — -, A- =10. 
 
 Figure 2. 
 
 Substituting these values in equation VI, p. 20, there follows: 
 
 [S32-26S3+I2O] [-35S3--26S3 + 2XI 1X3X28] = 0, 
 
 whence 83 = 6, 20, 6.9, or -7.65. 
 
 (83 = 6 is the only value which leads to triangle with three real 
 vertices. Substituting this value in equations IV and V (p. 19), 
 
 then 
 
 37 
 »Si = -r and ».S2=11. 
 
 

 f. • , . I. 
 
 In-and-Circintiscribed Triangles of Qiiartic Curve 31 
 
 The vertices of the triangle are then given by the roots of the 
 equation : 
 
 6 
 
 (i- 1.5)(<-3.535)(/- 1.132) = 
 
 For the construction of this triangle, see Figure 2. 
 
 (c) The Quartic with a Cusp 
 The following values for a, b, c, lead to two solutions, both of 
 which give real triangles, as is seen from the accompanying table: 
 
 a 
 
 b 
 
 c 
 
 Si 
 
 So 
 
 S3 
 
 ^1 
 
 t2 
 
 ^3 
 
 1 
 
 -4 
 
 2 
 
 8.487 
 
 -.177 
 
 -.708 
 
 .283 
 
 8.496 
 
 -.292 
 
 
 7 
 
 .293 
 
 -.586 
 
 6.94 
 
 .319 
 
 -.264 
 
 For the construction of these triangles, see Figure 3. 
 1 
 
 Figure 3. 
 
BIOGRAPHICAL SKETCH 
 
 Joseph Nelson Rice was born at Weymouth, Nova Scotia, on 
 the 26th of December, 1890. He received his elementary and 
 high school education at the public school of this town. In the 
 fall of 1906 he entered St. Francis Xavier's College, Antigonish, 
 N. S., and w^as graduated therefrom with the degree of Bachelor 
 of Arts in 1910. In 1912 he received the degree of Master of 
 Arts. During the years 1910 to 1913 he was an instructor in the 
 department of jNIathematics at this same college. In the fall of 
 19 13, he entered the Catholic University of America as a graduate 
 student in the department of Mathematics. He has followed 
 courses under Dr. Landry, Professor of Mathematics; Dr. Shea, 
 Professor of Physics; and Mr. Crook, Instructor in Mechanics. 
 
 He desires to take this opportunity of expres.sing his thanks to 
 Professor Landry for many valuable suggestions offered during 
 the preparation of this thesis. 
 
liiiiliii 
 
UNIVERSITY OF CALIFORNIA UBRARY 
 
I