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 ^XCHANGIS 
 MAR 8 J917 
 
 COMPLEX CONICS AND THEIR 
 
 REAL REPRESENTATION 
 
 BY 
 
 BENJAMIN ERNEST MITCHELL ^<.- 
 
 DISSERTATION 
 
 Submitted in Partial Fulfillment of the Requirements 
 
 FOR THE Degree of Doctor of Philosophy, 
 
 in the Faculty of Pure Science, 
 
 Columbia University 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 
 UAJMCASTER, PA. 
 
 1917 
 
1) 
 
COMPLEX CONICS AND THEIR 
 REAL REPRESENTATION 
 
 BY 
 
 BENJAMIN ERNEST MITCHELL 
 
 DISSERTATION 
 
 Submitted in Partial Fulfillment of the Requirements 
 
 FOR THE Degree of Doctor of Philosophy, 
 
 IN THE Faculty of Pure Science, 
 
 Columbia University 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 
 LANCASTER, PA. 
 
 1917 
 
TABLE OF CONTENTS. 
 
 Chapter 
 
 Introduction 1 
 
 1. Historical 1 
 
 2. The Laguerre-Study Representation of the Imag- 
 
 inary quantity 2 
 
 I. The Real Conies 6 
 
 3. Some Simple Cases 6 
 
 4. The Real Ellipse 7 
 
 5. The Real Hyperbola 13 
 
 6. The Real Parabola 15 
 
 II. The Complex Conic, Preliminary Considerations 18 
 
 7. The Complex Line and Its Conjugate 18 
 
 8. Canonical Equation of the Complex Line 19 
 
 9. Real Representation of the Complex Line 21 
 
 10. Canonical Equation of the Complex Circle 21 
 
 11. The Complex Circle and Its Conjugate 22 
 
 12. Reflection with Respect to the Complex Circle. . 23 
 III. The Complex Conic, Reduction of Equation 27 
 
 13. The Complex Conic and Its Conjugate 27 
 
 14. Canonical Equation of the Complex Conic 29 
 
 15. Localizing the Complex Conic 32 
 
 IV. The Complex Conic, Its Real Representation 33 
 
 16. The General Case, Z 4= 33 
 
 17. The Special Case, Z = 42 
 
 18. The Special Case, 6 = 44 
 
 o4:.>i^'J 
 
COMPLEX CONICS AND THEIR REAL 
 REPRESENTATION. 
 
 INTRODUCTION. 
 
 1. Historical. — The introduction of the imaginary quantity, 
 or the complex quantity comprehending both the real and the 
 imaginary, into analysis had the effect not only of extension 
 and generalization but also in many cases of simplification. 
 Such results in the realm of geometry have not yet been fully 
 realized. In the preface to his "Einfiihrung in die analytische 
 Geometric," Kowalewski says: "Eine grosse Schwierigkeit in 
 der analytischen Geometric ist die exakte Behandlung des 
 Imaginaren." 
 
 But the incorporation of the imaginary in geometry does not 
 require any more of reconstruction and readjustment than it did 
 in the case of analysis.^ "A satisfactory theory of imaginary 
 quantities of the ordinary algebra . . . with difficulty obtained 
 recognition in the first third of this century . . . , it . . . was 
 not sought for or invented — it forced itself, unbidden, upon the 
 attention of mathematicians, and with its rules already formed."'" 
 
 No sooner had the imaginary won its rightful place in analysis 
 at the hand of Gauss and Cauchy than it began to knock at the 
 door of geometry. Indeed before its full recognition in analysis 
 there had appeared the geometric method of representing the 
 imaginary quantity due to Argand and Wessel. But the imag- 
 inary in geometry must play the role of element of structure 
 comparable to that of number in analysis.^ The history of its 
 development is of intense interest. 
 
 1 Convention. Toute expression ayant un sens geometrique quand les 
 elements dont elle depend sont reels conservera, par definition, le meme nom 
 quand quelques-uns de ces elements deviendront imaginaries. Niewenglowski , 
 Cours de Geometric Analytique, p. 114, Old Edition. 
 
 2 Gibbs: "On Multiple Algebra," Proc. Am. Asso. Adv. of Sci., 1886. 
 
 3 Cf. C. A. Scott: "On Von Staudt's Geometric der Lage," Math. Gazette, 
 Vol. 1, p. 307. 
 
 1 
 
•.^.'.-" COMRLEX CGNICS AND THEIR REAL REPRESENTATION. 
 
 Out of the school of Monge came a class of mathematicians, 
 beginning with Poncelet and culminatmg in Von Staudt, who 
 were somewhat exclusive in their methods. On the principle, 
 geometry for geometers and geometry all-sufficient and self- 
 sufficient, they set about to build up a body of doctrine wholly 
 independent of analysis. The principals in this program were 
 Poncelet, Chasles, Steiner and Von Staudt. The imaginary 
 enters through the so-called Principle of Continuity and makes 
 its first appearance in Chasles' "Traite de Geometric Superieure" 
 (1852). Regarding the second of the three advantages which 
 he claims for his geometry he says: " Je veux parler de la gener- 
 alite dont sont empreints tons les resultats de la geometric 
 analytique, oii Ton ne fait acception ni des differences de positions 
 relatives des diverses parties d'une figure, ni des circonstances de 
 realite ou d'imaginarite des parties, qui, dans la construction 
 generale de la figure, peuvent etre indifferemment reelles ou 
 imaginaries. Ce caractere specifique de I'Analyse se trouve dans 
 notre Geometric." 
 
 But the investigations of Poncelet and Chasles had their 
 origins in analysis and in their completed forms were not free 
 from analytical considerations. To George Karl Christain Von 
 Staudt belongs the honor of constructing independently of 
 analysis a geometry involving imaginary elements. Thus ac- 
 cording to Von Staudt: Two conjugate imaginary points may 
 always be considered as the double points of an (elliptic) involu- 
 tion on a real line; and as (in analysis) we pass from an imaginary 
 number to its conjugate by changing i to — i, so (in geometry) 
 we may distinguish the two imaginary points by associating 
 them respectively with the two senses of the line. 
 
 Now the most essential or characteristic ideal of geometry 
 is to render all configurations visualizable, intuitive: the de- 
 sideratum is, to use the German, "Anschaulichkeit." Whilst 
 Von Staudt's purely projective methods were theoretically suf- 
 ficient, yet they were found to be in use cumbrous and compli- 
 cate; accordingly we find the diverging lines of analysis and 
 geometry beginning to change direction and to come together. 
 "From this moment a brilliant period opens for geometrical 
 research of every kind. Analysts interpret all their results and 
 
INTRODUCTION. 3 
 
 set to work to translate them by constructions. Geometers 
 endeavor to discover in every question some general principle — 
 in most cases impossible to prove without the aid of analysis."^ 
 This was but reflecting the spirit of the great Monge who "has 
 shown from the outset . . . that the alliance between analysis 
 and geometry was useful and fruitful and that perhaps their 
 alliance was a condition of the success for both of these branches 
 of mathematics."^ For example, in the conclusion of his lecture 
 "On the Real Shape of Algebraic Curves and Surfaces/' as 
 interpreted by geometric models and Riemann surfaces, Klein 
 says: "These methods give us the actual mental image of the 
 configuration under consideration, and this I consider the most 
 essential in all true geometry."^ 
 
 This ideal of geometry is entirely consistent with the ideal of 
 mathematics as presented by Von Staudt himself: "Indem die 
 Mathematik darnach strebt, Ausnahmen Von Regeln zu be- 
 seitigen und verschiedene Satze aus einem Gesichtspunkte auf- 
 zufassen, wird sie haufig genothigt, Begriffe zu erweitern oder 
 neue Begriffe aufzustellen, was beinahe immer einen Fortschritt 
 in der Wissenschaft bezeichnet."^ 
 
 In these ideals we have the spirit and aim of the great program 
 proposed by Professor Study in his lectures and elsewhere,^ a 
 program in accordance with which on the one hand we are not 
 to be hampered by assumptions regarding reality or non-reality 
 and on the other hand all configurations whether real or imaginary 
 are to receive intuitive representation. 
 
 In his work "Vorlesungen iiber ausgewahlte Gegenstiinde der 
 Geometric, erstes Heft : Ebene Analytische Kurven und zu ihnen 
 gehorige Abbildungen" Study has blazed a path through the 
 great domain contemplated by his program. My aim in this 
 
 iDarboux: "A Study of the Development of Geometric Methods," Con- 
 gress of Arts and Science, St. Louis, 1904. 
 
 2 Darboux, supra. 
 
 3 The Evanston Colloquium. Lectures on Mathematics, Lecture IV. 
 ^ "Beitriige zur Geometrie der Lage," Vorwort. 
 
 6 " Vorlesungen liber ausgewahlte Gegenstande der Geometrie" (1911), 
 "Zur Differential-geometrie der analytischen Curven," and "Die naturlichen 
 Gleichungen der analytischen Curven im Euclidischen Raume," Trans. Am. 
 Math. Soc, vols. 10, 11. 
 
4 COMPLEX CONICS AXD THEIR REAL REPRESENTATION. 
 
 paper, as indicated by its title, is to apply his general method 
 to an interesting and important detail. 
 
 2. The Laguerre-Study Representation of the Imaginary. — There 
 is a great variety of ways of representing the imaginary element 
 by a real figure.^ The most efficient for purposes of analysis is 
 that due to Laguerre- extended and developed by Study .^ 
 
 The two families of minimal lines, right- and left-sided, to 
 use Study's term, have for equations 
 
 ^ + ^77 = const, and ^ — i-q — const. 
 
 Where ^ and t) are the rectangular cartesian coordinates of the 
 00* finite complex points in a projective plane. These lines 
 through a point (^', -q') of the plane have for equations: 
 
 s + 'i'n = ^' + W and ^ — i-q = ^' — ir]'. 
 
 On each of these lines there is one and only one real point. 
 Taking {x' , y') for the coordinates of the real point on the second 
 and {u', v') those for the real point on the first, we have: 
 
 t' + irj' = m' + iv' and ^' - irj' = .r' - iy'. 
 
 Considering ?/ + iv' and x' + iy' as geometric pictures of two 
 gaussian numbers w' and z' we write 
 
 (I) ^' + iv' = «'' and ^' - iv' = z' , 
 
 where z' = x' — iy' , the conjugate of %' . It is agreed to take 
 the two real points s' and ?r' of the two gaussian planes as the 
 Real Representation of the complex point (^', 77') of the cartesian 
 plane. Study symbolizes this representation by z — > w, and 
 calls it the First Picture (das erste Bild) of the imaginary point. 
 Again we have for the conjugate point (|', rj') of (^', 77') 
 
 . _ ,-, =1'. i-^', I consequently | ,, _ ._, ^ _,_ 
 
 Thus the picture of (4', 77') is w' 
 
 r.' 
 
 1 Encyklopadie der Mathematischen Wissenschaften, III, AD, 4o, 13-16. 
 
 2 Oeuvrcs, Tome II, pp. 89-98. 
 ' "Vorlesungen," p. 9. 
 
INTRODUCTION. 5 
 
 Hence by virtue of relation (I) a perfect correspondence is set 
 up between the totality of finite complex points of the plane and 
 the totality of finite real point-pairs: to any complex point 
 corresponds uniquely a real point-pair, and vice versa.^ 
 
 The cartesian plane (^, rj) and the picture planes (s) and {w) 
 may be considered superposed or not; in either case they are to 
 be considered distinct. The oo^ real points of the plane have 
 IV = z; that is, if the planes are considered coincident, these 
 points are their own pictures. 
 
 1 "Vorlesungen," p. 10. 
 
CHAPTER I. 
 REAL CONICS. 
 3. Some Simple Cases. — If we have given an equation, 
 
 <pa, V) = 0, 
 
 ^, 7] being rectangular cartesian coordinates, there is simul- 
 taneously given, by virtue of the relations 
 
 ^ + iv = w, 
 
 
 ^ - irj = z 
 
 (1) or |=" + ^ 
 
 
 w — z 
 
 ^ - 2i ' 
 
 a corresponding equation 
 
 
 
 fiio, 
 
 z) 
 
 = 0, 
 
 setting up a correspondence between the two picture planes. 
 This latter equation, being a relation between two complex 
 variables, expresses a conformal transformation between the 
 two planes. Since one of the variables appears in its conjugate 
 form, angles under the transformation, while preserved in 
 magnitude, are reversed in sense. Such a transformation Study 
 calls an improper (uneigentlich) conformal transformation. We 
 shall refer to such a transformation as reverse conformal. 
 
 If the coefficients of (p are real the curve is called real, although 
 it may contain no real points. Compare § 4 of this chapter, 
 and Study, p. 46. 
 
 Accordingly, for the sake of completeness and reference, we 
 state some simple cases without giving the proof, and develop 
 others. 
 
 (A) The picture of the oo^ pQi^^s on a real line, a^ + 6r; + c = 0, 
 a, h, c being real,^ is the ensemble of point pairs symmetric (the 
 
 1 We shall observe the following notation throughout this paper: Complex 
 quantities shall be represented by Greek characters, a, X, |, etc., and real 
 quantities by Latin characters a, I, x, etc., with the single exception w and z; 
 these beiBg complex and equivalent to u + iv and x -{- iy respectively. 
 
REAL CONICS. 7 
 
 one the reflection of the other) with respect to the real branch 
 of the line. (Study, p. 25.) 
 
 (B) The picture of the oo 2 points on a real circle is the ensemble 
 of point-pairs symmetric (geometrically inverse) with respect to 
 the circle. (Study, p. 36.) 
 
 Study gives the real ellipse as an example in his lectures on 
 this subject. I shall, however, consider this case along with the 
 real hyperbola and real parabola, the method used differing 
 from Study's in no essential respect. 
 
 4. The Real Ellipse. — We have two cases, (1) where the ellipse 
 has a real branch or arc, (2) where the ellipse has no real branch 
 or arc.^ 
 
 The equations corresponding to these two cases are of course 
 
 £2 2 
 
 (1) §+^=±1, a>b. 
 
 If we write e = 1 or i then e^ = ± 1 and we have 
 
 fc2 „2 
 
 (2) h+l-^'- 
 
 Form the pencil of lines 
 
 (3) r;=-r(^ + ea), t = s -\- it. 
 
 This line has one fixed intersection with the ellipse, (— ea, 0), 
 and one free intersection. The range of r through the domain 
 of real numbers, s arbitrary, t = 0, gives the 00^ real points on 
 the ellipse e = 1. The same range of r gives 00^ complex points 
 of the ellipse e = i. The complete variation of r, s and t assum- 
 ing all possible real values, gives the <x>- points of the conic. 
 
 Expressing ^ and 77 in terms of the parameter r, using equa- 
 tions (2) and (3), 
 
 1 - T- 2ea 2ebT 
 
 (4) ^ = €a^-^py., ^ + ^a = Y-^2-^ ^=r+72- 
 
 1 In investigations in this field, where a curve is considered as consisting of 
 00 2 points, a configuration consisting of 0=^ points is called by Segre filo, which 
 Study translates, Faden, that is thread. The difference, then, in these two 
 ellipses is that one has a real thread and the other has not. 
 
8 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 Then for 2c and z in terms of the parameter we have: 
 
 . . ^ o + ibr 
 
 IV = ^ + 17] = - ea + 2e -^r^— ., , 
 
 (4') _^ .,_ 
 
 To simplify the expressions on the right hand of these equations 
 and at the same time to bring them to well-known forms we 
 change our parameter bj' means of a linear transformation, 
 
 .t' - 1 
 Effecting this change in (4') we have 
 
 € 
 
 w = 2 
 (5) 
 
 if, ,,_, a — 61 
 
 The function |[f + (l/i")] is one of the most common in func- 
 tion theory. The Riemann surface belonging to it is two- 
 sheeted, the two sheets being connected along the branch-cut 
 running from — 1 to + 1. To concentric circles about the 
 origin in the ^-plane correspond confocal ellipses on the surface. 
 To the pencil of straight lines through the origin correspond 
 hyperbolas confocal with the above-mentioned ellipses. A single 
 ellipse lies entirely in one sheet, but the branches of a single 
 hyperbola lie half in one sheet and half in the other. The 
 Riemann surfaces of ?r and z defined by (5) differ from this 
 surface in no essential respect.^ 
 
 However, for some further simplification, and for uniformity 
 with cases to be discussed later, we make a second change of 
 parameter. We may write 
 
 , , , , a + 6 , , ^ , or — IP' c^ 
 
 (a - by + — r- = (a - by + , ^^ = Tx + -, 
 
 Ti = (a — b)T'. 
 I See Lewent, Konforme Abbildung, p. 63. 
 
REAL CONICS. 
 
 Where ^ = c? — Ir, c being the focal distance. Thus we have 
 
 (0) ,„ = l[,, + f], 
 
 where 
 
 Ti = (a - h)T'; 
 and 
 
 z = 
 where 
 
 5[^'+S]' 
 
 T2 = (a + 6)r'. 
 
 Certainly the simplest complex configuration is the real straight 
 line, and the simplest position of the real straight line is in 
 coincidence wdth the ^-axis, 77 = 0. The picture of the line in 
 this position is given by the ensemble of point-pairs connected 
 by the relation w = z by virtue of 77 = (w — z)/2i. It is 
 known in Conformal Geometry^ that any real analytic curve is 
 conformally transformable into a straight line, in particular, 
 into the ^-axis (small regions about a point of each curve being 
 taken). Conversely the ^-axis may be transformed conformally 
 to any analytic curve. The transformation effecting this, 
 operating on iv — z gives w = f(z) or more generally 
 
 F(z, iv) = 0. 
 
 For a small region about a point ^' on the ^-axis the points 
 symmetric (in the Schwartzian sense) to each other with respect 
 to the ^-axis go over into points symmetric with respect to the 
 arc of the analytic curve through the transform of ^'. We shall 
 give this principle prominent place in this paper. And, since our 
 transformations are of the simplest sort, consisting only of trans- 
 lations, rotations and inversions, the principle just stated holds 
 in the large, that is, throughout the finite portion of the plane. 
 
 The parameter r is complex and we ascertain first the corre- 
 spondence between the r-plane and the r'-plane, the two vari- 
 ables being connected by the relation: 
 
 .r'- 1 
 
 ^Kasner, "Conformal Geometry," International Congress of Mathe- 
 maticians, Cambridge, 1912. 
 
10 
 
 COMPLEX CONICS AND THEIR REAL REPRESEXTATIOX. 
 
 Solving for t' and calling for the moment t = x -^ iy and 
 t' = re' + iy' we have, following the usual method of Holz- 
 miiller/ 
 
 . , X -\-i(y -\- 1) 
 
 x' -h ly' = - 
 
 and 
 
 X — ly = — 
 
 + / = 
 
 X + i{y - 1) ' 
 
 a- - i{y + 1) 
 X - i{y - 1) ' 
 
 x''+(y + ly 
 x' +{y- ly 
 
 
 Where jJ and q are radii vectores from the points — i and + i. 
 The variation of j) and 7 gives the pencil of circles with ± i as 
 limiting points. The real axis of r is given hy p = q and since 
 r = piq = ij we see that to the real axis of r corresponds the 
 unit circle of r'. Further, to circles of the pencil in the upper 
 half of the r-plane, p > q and hence r = p/q > 1 correspond 
 concentric circles about tlie origin of the r'-plane outside the 
 unit circle, that is, the upper half of the r-plane maps into the 
 outside of the imit circle, and consequently the lower half of the 
 r-plane maps into the region inside the unit circle of the r'-plane. 
 It is eas}^ also to show that the imaginary axis of the r-plane goes 
 over into the real axis of the r'-plane. 
 
 Returning now to equations (6) let us write 
 
 (7) 
 
 w = ecTi 
 and s = 6(72 
 
 where 
 
 1 / c" \ 
 
 0-1 = - I ri + — j , Ti = (a - 6)r'; 
 
 0-2 = :^ I To + — 1 , r2 = (a + o)T . 
 
 Let (Tk = Sk' + isf:" and n- = tk + itk"; U>^ = 1, 2). Separating 
 the last equations of (7) into their real and pure imaginary parts 
 we have 
 
 (8) ..' "' "' 
 
 ''=l[r;5TI^^+i] 
 
 and 
 
 * "Theorie der Isogonalen Verwandschaften." 
 
REAL CONICS. H 
 
 
 Sk' = 
 
 To circles about the origin in tlie Xft-plane, tk'^ + th""^ = r*, 
 correspond ellipses whose equations are 
 
 (7-) (7-'J 
 
 To lines through the origin of the r^t-plane, tk" = mtk, correspond 
 hyperbolas whose equations are 
 
 (10) -4^ - -4^ = 1. 
 
 1 + m^ 1 + m2 
 These ellipses and hyperbolas are confocal for 
 
 K7-T-i(f-0^= 
 
 and 
 
 1 + m2 ^ 1 + w2 
 
 c being the focal distance. 
 
 If equations (9) and (10) be expressed in terms of a — 6 and 
 a + 6, that is, if cri and at be expressed as functions of r' the re- 
 sults are, writing a' for a + 6 and h' for a — b, 
 
 4*1'' 4si' 
 
 (90 
 
 (^+-) (^--y 
 
 =1, 
 
 (7 + »'rJ (7 -'*''•)' 
 
 /2 //2 
 
 1 + m^ I -{- VI- 
 
 These equations show that to any circle of radius r in the t'- 
 plane in general correspond different ellipses in the (Ty and 
 
12 
 
 COMPLEX COXICS AND THEIR REAL REPRESENTATION. 
 
 (r2-surfaces. There is one exception, namely, r = 1. In this 
 case each of the equations (9') becomes 
 
 (11) 
 
 .V' s" 
 
 To any line in the r'-plane through the origin correspond the 
 same hyperbolas, but we arc not to understand that as r' de- 
 
 scribes any given line in its plane <ti and o-o describe the branches 
 of the corresponding hyperbola together. A little examination 
 shows that they never move in the same direction on the hyper- 
 bola but always in opposite directions, passing each other on 
 the common ellipse (11) of the two planes. We do not carry 
 this investigation beyond this point. Further details are 
 mentioned by Study, pp. 98-103. 
 
REAL CONICS. 13 
 
 If our ellipse has a real branch, that is, if e = 1 the ic- and 
 2-surfaces are identical with the cti- and o'2-surfaces. If 
 € = i, the iv- and s-surfaces are got by rotating the ai- and 
 or2-surfaces through 90° and — 90° respectively. 
 
 We have shown in this section that the real axis of the para- 
 meter plane (r) maps into the real branch of the ellipse € = 1, 
 while for the ellipse e = i the real axis of r maps into the real 
 branch of the ellipse e = 1 turned through 90°. The picture- 
 points z and w are united in position for the ellipse e = 1, and 
 are diametrically opposite for the ellipse e = i. The elliptic 
 pencil of circles with limiting points ± i have been shown to 
 map into confocal ellipses, circles symmetric with respect to 
 the real axis in the r-plane go into ellipses symmetric with 
 respect to the real branch of the ellipse e = 1, and for the case 
 € = i symmetric with respect to the ellipse e = I turned through 
 90°. It may be shown that the hyperbolic pencil orthogonal to 
 the elliptic pencil maps into the confocal hyperbolas.^ Corre- 
 sponding curves and points are shown in the figures. 
 
 5. The Real Hyperbola. — 
 
 
 2 1.2 — ^> a > b. 
 
 Introducing the parameter in the same way as in the case of 
 the ellipse we have: 
 
 1 + T^ , 2« 
 
 26r 
 
 V = 
 
 Hence 
 
 1 - T^' 
 
 w = — a -\- 2 
 
 a -f- ibr 
 
 a + ibr 
 z = - a + 2- =;r. 
 
 1 — T 
 
 Changing the parameter by means of the linear transformation: 
 
 r' - 1 
 "■ ~ r' + 1 ' 
 * Holzmiiller, p. 63. 
 
14 COMPLEX COXICS AXD THEIR REAL REPRESENTATION. 
 
 we have, 
 
 w 
 
 (1) 
 
 
 Or, as in the case of the ellipse, if we put 
 
 (2) T^ = {a + ih)T' and to = (a - ih)r', and ar-\-h'^ = &, 
 
 c being the focal distance, we have. 
 
 (3) 
 
 W^+t]' 
 
 ^t' = 2 I 'Ti + T 
 
 \b4^- 
 
 When the equations of (3) are resolved into their real and pure 
 imaginary parts we have, as in the preceding case: 
 
 (4) 
 
 Since ti = (a + ih)r' and r-i = (a — f6)r', the configuration of 
 the r'-plane consisting of concentric circles and radial lines, is 
 not altered, lines and circles merely going over into lines and 
 circles. In particular the real axis of r' goes into the line with 
 slope hja in the ri-plane, and the line with slope — {bja) in the 
 To-plane. Since the expansion | a + i6 1 is the same for both 
 Ti and T2, a circle of r' goes into equal circles in the ti- and T2- 
 planes. To concentric circles and radial lines of the ti- and 
 To-planes there correspond ellipses and hyperbolas in the W' 
 
REAL CONICS. 15 
 
 and s-surfaces. Their equations are: 
 
 (7-) (7-0^ 
 
 and 
 
 ■"" /^2 \2 ^» ^2 „2^ 2 ~ Aj 
 
 (7-) (7-^J 
 
 W2 
 
 ^2 
 
 C2 ~ 
 
 ^m\- 
 
 a;2 
 
 1 +mi2 
 2/^ 
 
 c2 
 
 (?m<^ 
 
 1 + W2^ 1 + W2^ 
 
 where 
 
 tan~^ m\ = tan~^ m + tan~^ - : tan~^ iiio = tan~^ m — tan~^ - , 
 
 a a 
 
 m being the slope of a line through the origin in the r'-plane. 
 The ellipses correspond, but the hyperbolas are, in general, 
 different, the exception being where m\ = b/a and m2 = — b/a. 
 For this case the above equations reduce to: 
 
 a^ 0- a^ b^ 
 
 Now the transformation t = (t' — 1)/(t' + 1) converts the real 
 axis of r into the real axis of r', and as we noted the lines of 
 slope b/a and — b/a in the ri- and T2-planes correspond to the 
 real axis of t' and hence to the real axis of r. So the real axis 
 of r maps into the real branch of the hyperbola. 
 
 It can be shown^ that to concentric circles about the origin 
 in (r') correspond the pencil of circles having ± 1 for their 
 limiting points, and to the radial lines in the r'-plane correspond 
 the pencil of circles with vertices ±1. So we have the case 
 corresponding to that of the ellipse: the hyperbolic pencil of 
 circles in the r-plane with the common points ± 1 maps into 
 confocal hyperbolas in the w- and s-surfaces; in particular, the 
 radical axis of the pencil going into the real branch of the hyper- 
 bola. The orthogonal pencil with ± 1 as limiting points goes 
 into the confocal ellipses. 
 
 6. The Real Parabola. — We take for the equation of the para- 
 bola 
 (1) V' = 4p(^ + P), 
 
 ^ See Holzmiiller, p. 63. 
 
IG COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 with the finite focus at the origin. Form the pencil, 
 
 77 = 2Ta + p). 
 
 Then, 
 and, 
 
 1 — T- p 2p 
 
 , 1 + 2ir , 1 + 2ir 
 
 As in the previous cases we simplify these expressions by a linear 
 transformation of the parameter 
 
 (2) r = . for w, and t = , — ; for s. 
 
 Ti — ^ T2 -{- I 
 
 Thus we get 
 
 (3) W = pTi^, S = 2^7^2". 
 
 The Riemann surface belonging to pr- consists of two sheets 
 joined along the positive real axis. To lines parallel to the axis 
 of reals h" = const, correspond confocal parabolas, 
 
 To lines parallel to the imaginary axis, ti = const., correspond 
 the orthogonal trajectories of the preceding set, namely, 
 
 1)2 = - 4pt,'^(u - ph'-). 
 
 These two families of parabolas are confocal, having the 
 origin for their finite focus. 
 We get similar equations for s, 
 
 if = ^ph"\x + ph"^), 
 if = - 4pt/\x - ptof). 
 
 Solving (2) for ti and t-z we may write: 
 
 n = ^-^, a- = 1,2) 
 the upper sign going with ri and the lower with T2. Separating 
 
REAL CONICS. 17 
 
 this equation into its real and pure imaginary parts we have 
 
 To Hnes parallel to the real axis of Xk, U" = b, correspond a 
 parabolic pencil of circles through the origin with centers on the 
 imaginary axis, (6 T l){t'" + t"") ± t" = 0. In particular to 
 the lines 4" = ± 1 corresponds t" = 0, the real axis of r. Again, 
 to lines parallel to the imaginary axis, 4' = a, corresponds a 
 second parabolic pencil of circles through the origin with centers 
 on the real axis, the orthogonal trajectories of the preceding 
 pencil. The equation of this last pencil is a{t'^ + t"^) — i' = 0. 
 
 To the real axis of r then corresponds in the w- and s-surfaces 
 the curves t'^ = Ayi^u + p) and y- = ^y{x + p), that is, the real 
 branch of our parabola. As r describes a circle of radius r, in 
 its upper half-plane, say, r describes a congruent circle in the 
 lower half-plane. Correspondingly n describes a straight line 
 t\" = (l/2r) + 1 in its plane and r-i describes a straight line 
 h" = iXl^r) — 1, and in turn w and z describe the confocal 
 parabolas : 
 
 »— 4p(i+iy[«+p(^-+iy], 
 
 As T describes a circle of radius r' of the orthogonal pencil, t 
 describes the same circle and w and 2; describe the same parabola 
 
 ^2 ^ _ P 
 r 
 
 ''V 4r'V' 
 
 Both families of parabolas are double decked, that is, both 
 sheets of the Riemann surface are filled with parabolas. Those 
 extending infinitely to the right have both branches in the same 
 sheet. Those extending infinitely to the left change sheets over 
 the branch-cut running from the origin to infinity positively. 
 
CHAPTER 11. 
 
 THE COMPLEX CONIC, PRELIMINARY CONSIDERATIONS. 
 
 While the purpose of this paper is the reduction of the equation 
 and the real representation of the imaginary conic, yet we shall 
 find it advantageous to spend some time with the more elementary 
 configurations. As a result should come a proper orientation 
 in the field of "geometry in the domain of the complex"; also 
 suggestions as to methods of procedure in the question proper. 
 Accordingly we give our attention first to 
 
 The Complex Line. 
 
 7. Relation of the Pictures of a Complex Line and Its Conjugate, 
 or the Group Property of the Transformations Belonging to the 
 Complex Line and Its Conjugate. — The general equation of the 
 complex line is 
 
 A : a^ + /377 + 7 = 0, 
 
 where ^, t], a, 13, y have the form wi + ino and the ratios a : fi : y 
 are not all real. This geometric configuration has oo- complex 
 points.^ There are no conjugate pairs of points on the line, with 
 one exception, namely, the real point of the line which is its own 
 conjugate. 
 
 This fact leads us to consider in connection with A its conju- 
 gate, 
 
 A : a^ + /377 + 7 = 0. 
 
 A is the "locus" of the conjugate of the points of A, and con- 
 versely. The two loci intersect in their common, self-conjugate 
 real point. 
 
 The corresponding reverse conformal transformations picturing 
 the imaginary points of A and A are, by virtue of the relations 
 §2,(1) 
 
 A : a^ + /3r; + 7 = ; T : niv -}- vz -\- 2y = 0, 
 
 (!)-_-_ _ _ _ 
 
 A : o;^ + iSt? + 7 = 0; T' : vw + fjiz + 2y = 0, 
 
 where jx = a — ijS and p = a -{- zjS. 
 
 ' Hereafter the term point will mean complex point. 
 
 18 
 
THE COMPLEX CONIC. 19 
 
 Let us now for the moment regard the planes (w) and (z) 
 as coincident. Applying T to a point z' we have 
 
 (2) fjiiv' + vz' + 27 = 0, 
 then applying T' to to' 
 
 (3) vw" + '^iw' + 27 = 0. 
 
 On elimination of w' between (3) and the conjugate of (2), 
 
 'iiw' + vz' + 27 = 0, 
 there results 
 
 w" = z'. 
 
 The result is the same if we reverse the order of application. 
 Hence the theorem: 
 
 The reverse conformal transformations belonging to a line and 
 
 its conjugate applied consecutively leave the points of the plane in 
 
 place : 
 
 T' = 'r~^ TT' = T' T z= ^ 
 
 In case the line is real a = a, j8 = 6, 7 = c, n = a — ih, 
 
 V = a + ib =]! then T = T' : fxiv -\- Jlz -^ 2c = 0, that is, the 
 
 real line is its own conjugate and further TT' = T^ = 1, the 
 transformation belonging to it is involutorial. 
 
 8. Reduction of the Equatioti of the Complex Line to Canonical 
 Form. — Putting A and A in the form R -\- iT, R, T being linear 
 functions in $. 77 with real coefficients, we have 
 
 A, A : ii ± iL2 : ai^ + birj + Ci ± i(a2^ + b-zv + C2) = 0. 
 
 Thus A and A are identified as members of a complex pencil 
 
 n : Zi + KL2 = 0, 
 
 where k = k' -\- ik" . Belonging to this pencil there is of 
 course a single infinity of real lines, k' arbitrary, k" zero. On 
 each value of k', k/ say, there is built up a single infinity of 
 imaginary lines, k/' arbitrary. We may thus distribute the 
 double infinity of complex lines into a single infinity of sub- 
 pencils each with real bases and each containing two real lines, 
 the bases, k' = k/, and 00, and a single infinity of imaginary 
 lines, k" arbitrary. According to this classification A and A 
 belong to the sub-pencil k' = with bases Li and L2. 
 
20 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 We proceed now to simplify the equations of A and A by 
 referring them to other bases, namely a certain rectangular 
 pair. Assuming ai&2 — azbi 4= and aia2 + &1&2 4= ^ A and A 
 have their common real point finitely located. We translate the 
 origin to this point and at the same time write A in the so-called 
 normal form of elementary geometry, 
 
 a B 
 
 ^ ^ ^ + , V = 
 
 or 
 
 ^ cos d -\- 7} sin 6 = 
 where 
 
 cos 6 = cos (s + it) = cosh t cos s — i sinh t sin s, 
 and 
 
 sin 6 = sin (s + it) — cosh i sin s + i sinh t cos s. 
 
 Thus the normal forms of A and A are 
 
 (1) ^ cos 5 -}- 77 sin s ± i tanh t{^ sin s — rj cos s) = 0. 
 
 Functions of the angle 6, or of its component parts are furnished 
 
 by the relations 
 
 bi + 62 tan q Oi tan q — a^ 
 
 a\ + 0,2 tan 9 61 tan q — h^. 
 
 and 
 
 «i — ^2 cot g' 61 + &2 cot q 
 
 tanh ^ = 
 
 61 — 62 tan g ai + 02 tan g' ' 
 
 where q = ^ arc (a^ + /3^). The sign of Va^ + |3- is that required 
 when a and /3 are real. 
 
 Equation (1) shows A and A referred to new bases, two per- 
 pendicular lines belonging to the pencil 11. These new bases 
 are connected by the relation:^ 
 
 n : L' cosh t + XL" sinh i = (1 + \){Li + kU), 
 where 
 
 L' = ^ cos 5 + 77 sin s, 
 
 L" = ^ sin s — T] cos s. 
 
 Taking U and L" as coordinate axes our equation of the com- 
 
 1 If 0102 + ^162 = 0, Li and L^ are perpendicular to each other and the second 
 transformation worked out above is not necessary. 
 
 - Newenglowski, "Cours do Geometric Analytique," sec. 446, old ed. 
 
THE COMPLEX CONIC, 21 
 
 plex line reduces to the form •} 
 
 A, A : 77 = zhi tanh t ^. 
 
 9. Real Represcntatioji of the Complex Line. — We have thus 
 reduced the equation of the complex line containing two effective 
 complex coefficients to one in which there is but a single coef- 
 ficient and it is pure imaginary in form. Writing A and A 
 separately with their accompanying transformations we have 
 
 A : 7] = i tanh t^; T : iv = e~-'2, 
 A : 77 = - i tanht^; T' : lo = e^^z. 
 
 The limits of t and tanh ^are — co<^< + oo,— 1< tanh t 
 < + 1. If we allow t to become infinite and assume its upper 
 and lower bounds then we have the minimal pair of the pencil, 
 
 ^ + iv = 0, 
 ^ - *r7 = 0. 
 
 Since e-' and e~-' are real the picture-pairs z and w are observed 
 to lie on rays symmetric with respect to the first bisector of 
 A and A. 
 
 If we introduce with Study, page 52, a parameter p measuring 
 the distance from the vertex of the pencil to the points on A 
 we have 
 
 ^ = p cos it = p cosh t, w = —, p, 
 (2) H , ^ty> 
 
 77 = p sin it = ip sinh t, z = e'p- 
 
 Since wz =\p\^, for any p, to and z are seen to be inverse points 
 with respect to the circle of radius \p\, tv having the same 
 argument as p, and z its negative. 
 
 The Complex Circle. 
 
 In a manner similar to that used in the case of the line let us 
 take up the case of the circle. 
 
 10. Group Property of Inversion loith Respect to a Complex 
 Circle and its Conjugate. — The equation of a complex circle and 
 
 1 Cf. Study, pp. 29, 52. 
 
22 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 its conjugate with their accompanying transformations are 
 
 K:e + r + 2«^ + 2^77 + 7 = 0, 
 T : wz -\- fxw ■{■ vz + y = 0, 
 
 K : ^2 _^ 7j2 + 2a^ + 2^77 + 7 = 0, 
 T' : icz + ^zo + /xl + 7 = 0, 
 
 where ^, t], a, • • • are complex quantities and n = a — i^, 
 
 V = a -\- i0. Neither T nor T' is involutoric, but, as before, 
 applying T to a point z' of (z) 
 
 (1) w'z' + ixw' + ^r + 7 = 0; 
 and applying T' to iv' of (iv) 
 
 (2) iv"w' + pw" + Jiw' + 7 = 0, 
 then subtracting from (2) the conjugate of (1) we get 
 
 (f/ + v)iiv" - z') = 0. 
 
 Obviously iv' cannot equal — v for all values of z', hence w" = 2' 
 and r = T-\ TT' = 1. 
 
 In this case and in the preceding we might have expressed w 
 explicitly in terms of z and arrived at the same result by sub- 
 stitution. This relation is also evident by a mere examination 
 of the two expressions T and J", considering in the one z as the 
 independent variable and in the other iv as the independent 
 variable. When a = a, ^ = b and 7 = c that is, when the 
 circle is real (in the sense of Segre) we have n = a — ib and 
 
 V = a -\- ib = jjL, so T ^ T' : wz -{- fxw -\- fxz -\- c = 0. 
 
 11. Reduction of Equations of K and K to Canonical Form. — 
 Breaking up K and K into their real and pure imaginary parts 
 we have 
 
 K, K : ^2 + 77- + 2a,^ + 26177 + Ci ± i{2a2^ + 26077 + Co) = 0. 
 
 They are thus seen to be members of a complex pencil 
 
 n : Ci + kR = 0, 
 
 where Ci = ^^ + r?- + 2ai^ + 26177 + Ci = is a real circle of 
 the pencil, R = 2a2^ + 26377 + Co = is the radical axis of the 
 pencil (also real), and k = /c' + ik". 
 
THE COMPLEX CONIC. 23 
 
 Clearly we may simplify our configuration by a change of 
 axes. This we do making the axis of centers the ^-axis and the 
 radical axis the 77-axis. As a result we have 
 
 where 
 
 and 
 
 Il'.e + v'-h 2a'^ + c' + 2X^ = 0, 
 , 2(aia2 + 6162) - ea , ^ ,^ " 
 
 X = Vas^ + h.^K; (to, 770), 
 
 coordinates of the new origin. 
 
 Among the real circles of 11 there is one with its center at the 
 (new) origin and is given by X = — 2a'. Taking this circle 
 with the radical axis as bases we have 
 
 n : ^2 _^ 772 + 2ix^ + c' = 0, 
 
 where ^t = X + 2a'. The character of the pencil — whether 
 hyperbolic, elliptic or parabolic — depends on whether c' is 
 greater than, less than, or equal zero. 
 
 Our original circles, K and K referred to the new bases are 
 seen to be given by the values ju = a' ± i Vao^ + h<f, that is^ 
 
 / , • // / 2(aia2 + &1&2) - g2 „ J ..JO 
 
 2Va22 + fe2' 
 
 We have thus succeeded in reducing the equation of the complex 
 circle 
 
 K:e-i-T-\-2a^ + 2^rj + y = 0, 
 
 containing three complex coefficients to a canonical form 
 
 K : ^^ + 7,2 + 2m^ + c = 0, 
 
 containing but a single complex coefficient. 
 
 12. The transformations belonging to K and K expressed in the 
 canonical form are 
 
 K = 0; T :wz + ix(iv + 2) + c = 0, 
 
 (3) 
 
 K = 0; r : icz + )u(w + 2) + c = 0. 
 
 These transformations of course are not involutoric any more 
 
24 COMPLEX CONICS AXD THEIR REAL REPRESENTATION. 
 
 than the original ones were. So we cannot consider (2) and 
 (u') coincident. To study the corresponding movements in 
 (z) and (w) we introduce a parameter plane defined in the 
 following manner: Setting 
 
 77 = 0, • • v, w = z in (3), 
 then 
 
 and 
 
 r, r' = - M ± Vm- -c = w', ic" or z', z". 
 
 Form the pencil of lines 7? = t(^ — ^"), t = s -\- it. Hence 
 t t" 2p „ 2p 
 
 and 
 
 = ^^^ _ // _ ^P 
 
 where p = V/x- — c = ri + ii'2, the radius of the complex circle. 
 Setting w' = IV — iv", z' = z — z" and dropping the primes, 
 
 (A\ 2p 2p 
 (4) w = z r- and z = :, — . 
 
 1 — IT 1 — IT 
 
 If r describe its axis of reals, t = 0, we have, on elimination of s, 
 
 (u — TiY -{- (v — r^y = r^ 
 and 
 
 {x - rif + {y - TiY = ;•-, 
 
 where r- =\p\" = rr + r^^. w and z thus are seen to describe 
 congruent circles in their respective planes. 
 
 If T describe its axis of pure imaginaries, s = 0, there result 
 on the elimination of t 
 
 V = —II, and V = — —x. 
 
 Our picture planes are thus seen to be divided into four regions 
 corresponding to the four quadrants of the parameter plane. 
 The paired points of the two congruent circles in (s) and {w) 
 are the points which in the real case are united in position pictur- 
 ing the 00 1 real points of the circle. We superpose the planes 
 
THE COMPLEX CONIC. 
 
 25 
 
 now and notice a sort of unfolding process of the picture planes 
 due to the entrance of the complex quantity. Let us examine 
 the parabolic case. Here we have: 
 
 ^' =0, t" = _ 9 
 
 e-\-T + 2m? = 0, 
 
 Zfj., p = n = n -{- ir2. 
 
 Considering the planes superposed, the origins coincident and 
 the axes of reals of (to) and (z) coinciding with the ?-axis of the 
 cartesian plane we have: 
 
 2ju 2ifXT 
 
 IV = — 2^ + 
 
 1 — IT I — ir' 
 
 z = - 2fx-\- 
 
 2tx 
 
 If 
 
 If 
 
 2^/^r 
 ir I — vr' 
 
 T = - 00, - 1, 0, +1, + <x>, 
 
 W = 10^, W-i, 0, w+i, w«, 
 
 3 = Soo, Z—i, 0, Z-^-i, Zof. 
 
 T = - CO, - ^, - i^, 0, + ^i, +i, + co, 
 
 00 , Wi, 0, - f /z, - n, w^, 
 
 w = w^, 
 
 Z = Zgoy 
 
 21 on. 
 
 Thus having in mind that t -> w is a direct transformation and 
 T — > 2 is a reverse transformation, the angles being preserved 
 in sense in the one and reversed in the other, we shall see just 
 what regions of (z) and (w) correspond. They are so indi- 
 cated in the figure. The situation is apparent if we consider 
 // as a sort of parameter varying in its pure imaginary part 
 
26 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 only. Let /x = Wi + ^^2. ??Z2 = gives a real circle of the 
 pencil with its center (— mi, 0). The variation of irii from is 
 seen to cause an iinfolding of the pseudo-real pair from the real 
 circle, the centers of these circles being {—mi, ± W2). For m^ 
 negative merely interchanges the figures z and w. For mi 
 negative, the figures are reflected over the 77-axis. It is inter- 
 esting to trace the path-curves of z and w due to the variation of 
 W2, holding T fixed. They are straight lines as is evident when 
 we write 
 
 2ir 2iT 
 10 = r-(??2i + ^2) and z = , ^(mi — imo). 
 
 I — IT 1 — IT 
 
 In particular the points z, w, whose united positions represent 
 the 00 1 real points in the real case trace the lines 
 
 y = s{x + 2wi) and v = s{u — 2mi). 
 
 The hyperbolic and elliptic cases do not introduce any essential 
 differences. 
 
 Study has given the elementary transformations making up 
 reflection with respect to the complex circle.^ In the light of 
 the foregoing considerations we may state the following set, 
 which is equivalent to that given by Study. Calling the con- 
 gruent circles in (2) and (w) picturing the real axis of the para- 
 meter, the Congruent-Pair, and the lines picturing the axis of 
 pure imaginaries, the Basic-Lines, the transformations in question 
 are as follows: 
 
 The transformation picturing the complex points on an imaginary 
 circle consists of three elementary reflections, namely, a reflection 
 over the center axis determined by the circle and its conjugate, a 
 reflection over one of the Basic-Lines, and a reflection over one of 
 circles of the Congruent-Pair . These may he taken in any order 
 hut the Basic-Line and the circle of the Congruent-Pair must not 
 helo7ig to the same planes. 
 
 1 Page 32. 
 
CHAPTER III. 
 
 THE COMPLEX CONIC, REDUCTION OF EQUATION. 
 
 With the processes and results of the preliminary considerations 
 in mind we give our attention now to the subject proper. 
 
 13. The General Equation and Its Corresponding Transforma- 
 tion. — The general equation of a complex conic and its conjugate 
 are, 
 
 T :ae + 2I3^V + 7v' + 25^ + 2er7 + r = 0, 
 
 (I) - _ - _ - 
 
 T :ae + 2l3^ri + yr + 25$ + 2ir; + f = 0, 
 
 where all the quantities entering are complex and the equations 
 are irreducible. 
 
 Each configuration contains a double infinity of points and 
 the conjugates of the points of the one lie on the other. They 
 have four points in common. These may all be real or all 
 imaginary, or any of the intermediate cases. It is not necessary 
 to make a separate discussion for any particular case. 
 
 The transformations picturing the complex points on these 
 "curves" are given through the relation § 2, (1). 
 
 Thus we have 
 
 r = 0; 
 
 T : axw" + 2/3iZf2 + 712= + 25iz^ + 2€i2 + 4^ = 0, 
 (II) 
 
 r = 0; 
 
 T' : yiw"" + 2^iivz + ai2- + 2iiw + 25i2 + 4^ = 0, 
 where 
 
 ai = a - 7 - 2il3, 5i = 2(5 - it), 
 
 y^ = a-y + 2i^, ^^ " " + ^' ei = 2(5 + ie). 
 
 Each transformation is (2, 2)-determined and hence requires 
 a two-sheeted Riemann surface for unique determination. 
 
 We note that when /3i = the variables are rationally separable, 
 
 (III) aiio" + 2hw = - (712' + 2eiz + 4^). 
 
 27 
 
28 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 This requires that a -\- y = which corresponds to the real 
 case for an equilateral hyperbola. We give this case special 
 consideration, § 18. 
 
 If we take the conjugate of T and consider z the dependent 
 variable we observe that it is identical with T'; hence, as in the 
 previous cases, the transformations belonging to a complex conic 
 and its conjugate are connected by the relation that the one is the 
 inverse of the other: T' = T~^. 
 
 If the coefficients of V are real T and T' become identical and 
 reduce to 
 
 C = 0; 
 
 (IV) 
 
 T : aiw"- + 2biicz + a^z^ + 25iW + 2hz + ^h = 0. 
 
 The transformation z -^ w picturing the complex points on a real 
 conic is an equation of the second degree in z and w\ the coefficients 
 of the square terms and the terms of first degree being conjugate 
 complex in pairs; the coefficients of the product term and the constant 
 term being real} 
 
 If further we consider only those curves admitting oo^ real 
 points, we may put 
 
 ^ + ir} = z and ^ — iri = z. 
 
 Our equation then becomes 
 
 (V) T : axz^ + 2byzz + a^z" + 2hz + 25i2 + 4/i = 0.^ 
 
 Let us compare the expressions for the invariants and the 
 conditions for the different species of conies in terms of the 
 coefficients of (I) considered as real, a = a, ^ = x, • • • and 
 those of (V). We have 
 
 r = 0, 
 
 (D 
 
 C = • • 
 
 ' (V), 
 
 ) 
 
 a -\- c ■ • 
 
 •bu 
 
 
 ¥- ac ' 
 
 .•6i- 
 
 \<Xl 
 
 Hence in terms of the coeflBcients of (V) we have the following 
 
 ^ Perna, "Le Equazione delle curve in coordinate complesse coniugate," 
 Palermo Rendiconti, 17 (1905), p. 65. 
 
 2 Cesaro, "Sur la determination des foyers des coniques," Nouv. Ann., 60 
 (1901), p. 1. 
 
THE COMPLEX CONIC, REDUCTION OF EQUATION. 29 
 
 conditions for the different species of proper conies and their 
 special cases: 
 
 ai = 0, Circle, 
 
 61 = 0, Equilateral hyperbola, 
 Iai| < 61, Ellipse, 
 |ai| > 61, Hyperbola, 
 I ail = ^ij Parabola.^ 
 
 14. Reduction of the General Equation to Canonical Form.—li 
 we write (I) in the form: 
 
 r, f : Ci ± iC2 = ai^ + 261^77 + c^i' + 2di^ + 2^177 
 
 + /i ± i{a2e + 2b2^r] + C2r + 2d.^ + 2e.ri + /a) = 0, 
 
 we identify F and F as members of a complex pencil 
 
 n : Ci + kCo = 0, K = k' + ik", 
 
 F and F being given by /c = i and — i respectively. 
 
 Among the real conies, k" = 0, of the pencil there is one and 
 only one equilateral hyperbola,^ for we have : 
 
 (ai + k'a^)^ + 2(bi + k%)^rj + (ci + k'c^W + ' • • = 0, 
 
 and the value of k' which renders this conic an equilateral 
 hyperbola is 
 
 , , _ «! + Cl 
 
 ^2 + C2 
 
 If tti + Ci = 0, Ci is the required hyperbola and A;' = is the 
 value of the parameter giving it. If a2 + C2 = 0, C2 is the 
 required curve and k' = qo . If both ai + Ci = and 02 + c-i = 
 both Ci and C2 are equilateral hyperbolas and k' is indeterminate. 
 In this case all the conies of the pencil are equilateral hyperbolas. 
 (See § 18.) 
 
 Supposing the pencil to have only one equilateral hyperbola* 
 we have 
 
 H:C^ + k'C2 = 0, k' = -«-^\ 
 
 a2 + C2 
 
 Taking H and Ci as bases we have 
 
 n : /^ + XCi = (1 + X)(Ci + kCo), 
 
 ^ Cesaro, loc. cit. 
 
 2 Niewengloski, "Cours de Geometric Analytique," sec. 460. 
 
30 COMPLEX COMCS AND THEIR REAL REPRESENTATION. 
 
 where 
 
 X = 
 
 k' -_K , _ fll + Cj J 
 
 , K — 
 
 02 + C2 
 
 We now take the asymptotes of the equilateral hyperbola as 
 our coordinate axes. We have 
 
 (1) // : a'e + 26'^77 - aV + 2(^'^ + 2e'r] +/' = 0, 
 
 where 
 
 a' = ai + k'a2, h' = hx + !:%, 
 or 
 
 k' = - 
 
 gi + Ci 
 
 <l2 4" C2 ' 
 
 n = 
 
 ni 
 
 712 
 
 + 
 
 Wl 
 
 W2 
 
 |ai 
 
 <Z2 
 
 Ci 
 
 C2 
 
 L n = a, h, c, •••,/. 
 
 The transformations effecting this change of coordinate axes 
 are, 
 
 ^ = /r - mv' + U 
 
 7] = m^' + W + r/o, 
 where 
 
 D', E', F' being the algebraic complements of d', e', f in the 
 discriminant of //, 
 
 I = yl ^ , m = y\ ^ — , 
 
 ^Si — 52 ^*i — 52 
 
 Si and 52 being the asymptotic directions of H furnished by 
 
 a'e + 2h'^r) - a'jf = 0, 
 
 5l, S2 = -, . 
 
 As a result of this transformation the equation of H becomes 
 
 ■^(sO, ^o) r,, ,2 , ,,2 
 
 (2) 
 
 H' : 2^v + 
 
 ^IF' ' 
 
 F' = a'- + h'% 
 
 and Ci becomes 
 
 (3) C : ae + 26^^ + cr?- + 2d^ + 2er/ + / = 0, 
 
 where 
 
 Ci^i + 26i — ai52 
 a = , 
 
 51-52 
 
 ^ Niewenglowski, loc. cit., 446. 
 
THE COMPLEX CONIC, REDUCTION OF EQUATION. 31 
 
 b = 
 
 c = 
 
 Sl — S2 
 
 aiSi — 26i — C1S2 
 
 Si — So 
 
 0^0 ^Si — S2 arjo \5i — 52 
 ^so ^Si — So driQ y& 
 
 S\ —So 
 
 ^0, Vo being the coordinates of the new origin. 
 
 There is one real conic C of the pencil IT whose axes are parallel 
 to the asymptotes of H'. This conic is given by X = — 1/6, 
 taking H' and Ci in their reduced forms (2) and (3). Thus 
 
 C ^C -hH' = 0. 
 Taking C and H' as bases we have 
 
 where 
 
 - 1/6 + X 
 M = ^^ . 
 
 Thus we have finally 
 
 n : a^2 _|_ 2jLi^7; + erf + 2d^ + 2ey] + f + ixh = 0, 
 where 
 
 h = p= — . 
 
 The values of the various parameters giving our original 
 conies r and T referred to the different bases and the different 
 configurations of reference are assembled as follows : 
 
 (I) Hi : Ci + kCs; r, f : Ci + k"C2 = 0, /c" = ± i, 
 Uo.H + XCi; r, r : ^ + V'Ci = 0, 
 
 ± I 02 + <?2 
 
 After change of origin T,f :H' - X" VFC = 
 
 Ha : c + M^'; r, f : C + fx"ir = 0, 
 (III) ,, _ - 1/6 + V^VF _ _ . , 1 
 
32 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 We have thus reduced the equation of our conic 
 
 (I) ae + 2/3^^ + yr + 25^ + 2e77 + f = 
 
 containing five effective constants to an equivalent one 
 
 (VII) ae + 2/x^77 + cr?2 + 2d^ + 2er] + f + fxh = 
 
 containing only two complex constants which are linearly and 
 integrally related. 
 
 15. "Localizing" the Conic in the Complex Pencil. — We have 
 identified our conic as a member of a complex pencil of conies 
 having two real conies (proper or degenerate) as bases. We 
 now proceed to "localize" it among the double infinity of conies 
 of the pencil. The locus of centers of 11 is given by 
 
 ^ = M^ + C77 + e = 0. 
 
 Which on the elimination of ix yields 
 
 Cz : a^- — err + ^^s — ct) = 0, 
 
 an ellipse or an hyperbola according as C is an hyperbola or an 
 ellipse. 
 
 The double infinity of points on Cz are the centers of the 
 double infinity of conies of 11. The single infinity of real points 
 on Cz are the centers of the single infinity of real conies of 11. 
 The double infinity of imaginary conies have their centers 
 pictured by double infinity of point-pairs z-* w associated by a 
 Schwarzian reflection over the real branch of C3. 
 
 Thus at this stage of the investigation we are able to " localize" 
 our conic to the extent of determining the picture of the center. 
 
 In general: The point pairs Zo — » Wq, zvq — > Zq picturing the center 
 of an imaginary conic and its conjugate are symmetric ivith respect 
 to a real central conic, namely the locus of centers of the real conies 
 of a pencil determined by the real and pure imaginary component 
 parts of the conic. 
 
 Such localization of the foci is not so simple since they are 
 known to lie on a bicircular sextic. 
 
CHAPTER IV. 
 
 THE REAL REPRESENTATION OF THE COMPLEX CONIC. 
 
 In this chapter we consider the reverse conformal trans- 
 formation z-^iv of the form §13, II which pictures the oo^ 
 points whose coordinates satisfy an equation of the type § 1'], I 
 or § 14, VII. Or, stated more exphcitly, given a point (^', tj') 
 satisfying an equation of the above mentioned type, there is 
 simultaneously given, by virtue of the relations ^' + irj' = iv' 
 and ^' + it)' = z', a pair of associated points of the two picture 
 planes (w) and (z). This point-pair z' -* iv' we call the real 
 picture of the point (^', r;') and the ensemble of such pairs 
 picturing the double infinity of points on the conic we call the 
 Real Representation or Real Picture of the complex conic. We 
 shall find that these associated point-pairs may be grouped in 
 their respective planes on two orthogonal families of curves. 
 The two orthogonal nets thus formed in the two picture planes 
 are more or less similar depending on the complexity of the 
 case, being in the case of real conies identical. 
 
 We might take for the equation of our conic § 13, I and seek 
 its real representation through the corresponding transformation 
 § 13, II. The method here developed would be found suf- 
 ficient. But we shall find it somewhat simpler and more inter- 
 esting to take the so-called canonical form § 13, VII — simpler 
 because we have only one complex quantity among the coef- 
 ficients, and with the vanishing of its pure imaginary part we 
 have at once the real case — interesting because we shall be able 
 at various points of the development to observe just how the 
 entering of the imaginary affects the configuration. 
 
 Accordingly we take for the equation of our conic: 
 
 (1) ae + 2^^n + C7,2 + 2(1^ + 2e-n + ^ = 0, T = / + ^f'^ 
 
 writing /3 = 6i + ibi in the place of m in § 13, VII for uniformity 
 of notation. The corresponding equation in z and iv given by 
 
 33 
 
34 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 the relation 
 
 (2) 
 
 is 
 
 where 
 
 w = ^ + iri, 
 
 or ? = 2 (2 + w'), 
 
 = t 
 
 irj; 
 
 7; = 2 (2 - w), 
 
 (3) 
 
 aw- + 2biuz + 72- + 2div + 2€2 + ^' = 0, 
 a = a — c — 2zj3, 5 = 2(d — ie), b = a -\- c, 
 
 y = a- c + 2ifi, e = 2((f + ie) - 6, r' = 4i'. 
 
 We shall use the following notation for the discriminants of 
 (1) and (1') and the complementary minors of their elements; 
 the symbols in the first column in each table being for the 
 complex case, 62 4= 0, and those in the second column for the 
 real case, 60 = 0: 
 
 (1) ar + 2i3$77 + cyf + 2d^ + 2e7? + s" = 0; 
 
 (1') aw^ + 2hwz + 7|2 + 2bw + 262 + f ' = 0, 
 
 a 
 
 /3 d 
 
 13 
 
 c e 
 
 d 
 
 e f 
 
 ^e, H 
 
 c^ — e^ = A, A 
 (4) de-^t^B, B 
 
 a^ - d^=r, C 
 Pe - cd = A, D 
 ^8 - ae ^E, E 
 ac - fi^ ^Z, F 
 
 a b 8 
 
 b y e ^ X, T, 
 8 e r' 
 yr - e - 21, 31', 
 (4') 8e - 6r - 33, 33', 
 
 ar - 5^ - e, i', 
 
 be - y8 = ©, !D', 
 65 - ae = e, 2)', 
 «7 - &^ = 3, /^'. 
 
 We have the following relations between the quantities of the 
 two tables. 
 
 X= - 160, 3 = - 4Z, !D = - 4[A + z'E], 
 
 (5) 21 = - 4[A - r + 2iB], 33 = - 4[A + T], 
 
 S = - 4[A - r - 2iB], C? = - 4[A - iE], 
 
 Relation (2) may be looked upon as a transformation of 
 
THE REAL REPRESENTATION OF THE COMPLEX CONIC. 35 
 
 coordinates, in fact the quantities z and w are called Isotropic 
 Coordinates. 
 
 Considering iv and z as rectangular coordinates equation (2) 
 expresses an imaginary projective transformation. The deter- 
 minant of this transformation is 
 
 J = 
 
 = — 9i 
 
 ,1 - i 
 We rewrite (5) 
 
 (5') 51 = J-[A - r + 2iB], 33 = riA + T], 
 
 e = J2[A - r - 2iB], (g = J2[A - lE]. 
 
 Thus it appears that and Z are invariant under this imag- 
 inary projective transformation just as in the real case. 
 
 If Z 4= 0, we have for the coordinates of the center and its 
 corresponding picture 
 
 ^0 = 
 
 Vo = 
 
 j3e — cd _ A 
 
 ac - iS2 " Z 
 
 I3d-ae _ E 
 
 ac - iS-' ~ Z J 
 
 Zo-^Wo 
 
 Wo = lo + ivo = 
 
 20 
 
 — ^0 — ^770 = 
 
 z ~ S' 
 
 A-iE _ Q 
 
 z ~ S- 
 
 Here again we observe the similarity of notation. 
 
 We proceed now to a detailed examination of equation (T). 
 It is an implicit relation between two complex variables. Solving 
 for w we have 
 
 ty = 1 [_ (62 + 5) ± V(62 - ay)z' + 2(65 - ae)z -^ 8'' - a'H, 
 a 
 
 or using the notation of (4) and (4') 
 
 w = -[- (bz + 5) ± V- Sz' + 2(gl - d], 
 a 
 
 = -[- (bz + d) ±2 VZ22 - 2{A- lEJz + A-r - 2iB], 
 
 then 
 
 1 
 
 w = - [- (62 + 5) ± 2 VZ V2' - 2002 + (A, 
 
36 COMPLEX CONICS AND THEIK REAL REPRESENTATION. 
 
 and 
 
 iv = -[- (62 + 6) ± vi- n z' = ^., z = o. 
 
 a. z\X 
 
 If Z =1= 0, we have for the roots of the radical 
 
 z', 2 = 2 ± 2 = 20 ± a'^'/, / = I V2 ^ 3. 
 
 The corresponding values of w are 
 
 IV , w = ^ T 7^ = Wr) =F ba '■''I. 
 
 If the conic is real, it is known^ that the points corresponding 
 to the roots of the radical are the two real foci of the conic in 
 the case Z 4= 0, and the real finite focus in the case Z = 0. 
 In the case of imaginary conies then z' -^ 10' and z" — » 10" are 
 the pictures of the foci, now imaginar}', corresponding to the 
 real foci in the real case. 
 
 Equation (!') is of the second degree in either of the variables, 
 hence a two-sheeted Reimann surface is required for complete 
 depiction in case either is expressed as a function of the other. 
 The obvious disadvantage of proceeding in this way is that an 
 irrationality is introduced. We avoid this by introducing a 
 parameter as in the real case. This we effect in the following 
 manner : 
 
 The slope of (1) is given by 
 
 dt] a^ -\- ^Tf] -\- d 
 
 dk " ~ ^^-\-cv -\-~e' 
 The points where the tangent is perpendicular to the ^-axis is 
 given by intersection of (1) and ^^ + crj + e = 0. The ab- 
 scissas of these points are the roots of 
 
 (6) Ze - 2A^ 4- A = 
 
 or 
 
 (0 _ ^ 
 
 7 = Ve -^ Z = V- 2 4- 3. 
 
 The corresponding ordinates are 
 
 (8) VI, V2 = r7oT^j8c-l'2/. 
 
 1 Goursat, Nouvclles Annalles Mathematique (1887). Cesaro, Nouvelles 
 Annalles Mathematique (1901). 
 
THE REAL REPRESENTATION OF THE COMPLEX CONIC. 
 
 The pictures of these points are 
 
 37 
 
 i^uVi), Zi'-^wi] 
 
 Z\, Z2 = Zo ± I — Yir~ -'j 
 
 Wi, W2 = Wo ± Z 
 
 a/2 
 
 Forming the pencil of lines through (^1, 771) we have 
 
 (9) V - VI- r{^- ^1), T =s + it. 
 
 With the variation of r through the domain of complex numbers 
 the movable intersection of the line and the conic describes the 
 conic. Thus we introduce the parameter and expressing ^, 77, 
 tv and z in terms of it we have 
 
 2^- 
 
 (10) 
 
 ^ = ^l- 
 
 7; = 771 - 
 
 ce 
 
 2V-c0r 
 
 IV 
 
 IV I — 
 
 Z = Z\ — 
 
 2 V-ce(i+tV) 
 
 c{a + 2|8r + cr^) ' 
 
 2^l-cQ{\-ir) 
 cia^- 2i3r + ct-) ' 
 
 c{a + 2i3r + cr^) 
 
 Thus we have expressed iv and 2 rationally in terms of a para- 
 meter and we proceed at once to a detailed study of the functions 
 thus obtained. We shall find it simpler however to pass through 
 a series of linear transformations of the parameter and thereby 
 transform the functions iv and z into forms more frequently met 
 with in function theory. To this end we notice that the common 
 denominator of the above expressions may be wTitten {ct + 18)'' 
 + Z, so our first change of parameter is by the transformation 
 
 t' =CT+ ^, 
 
 and we get 
 
 ^ =b 
 
 V = Vi 
 
 2a/-c9 
 
 r'^ + Z 
 
 2<-cQ{t' -13) 
 c{t'' + Z) 
 
 IV = Wi — 
 
 2l - 
 
 2V-c9(c - i^ + It') 
 c{r'' + Z) 
 
 2 V^^(c + ^ig - ir') 
 cir'' + Z) 
 
 Again changing the parameter by the transformation 
 
 = iVz 
 
 1 -t" 
 1 + t" 
 
 we have after some reductions 
 ci/2/ / 1 \ 
 
38 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 77 = 770 - ^^-[(^ + iAfZ)r" +^-^], 
 
 Finally: 
 
 (11) 
 
 w 
 
 
 c - i^ + Vz „ 
 
 "^1 = 7u72 '" ' 
 
 ic^ 
 
 Zo + 
 
 T2 
 
 + ?'/3-Vz ,, 
 
 2C 
 
 1/2 
 
 W 
 
 = u'o 4- /o-i, cri = 9 ( '^1 + ~ ) ' 
 
 Z = Zo -\- I(T2, 
 
 For a detailed study of the Riemann surfaces belonging to w 
 and z let us assemble the series of transformations by which we 
 have changed our parameter and examine each turn. We have, 
 first of all, r the variable slope of the lines of the pencil through 
 
 (^b Vi), 
 
 77 — 771 = t(^ - ^1), T = 5 + it, 
 
 whose intersection with the conic gives the 00 ^ complex points 
 of the conic. 
 
 The quantities ^, 77, w and z expressed in terms of r are 
 
 2V-ce 
 
 ^ = .^1 - 
 
 V = Vi 
 
 2V-cer 
 
 2V = Wi — 
 
 2Al-ce(l+^>) 
 
 Z = Z\ 
 
 c{a + 2)3r + cj-) 
 Then by a series of transformations, 
 
 r' = CT-\- ^, 
 
 1 -r" 
 1 +r'" 
 
 c - i^ + Vz ,, 
 
 2V-ce(l -^V) 
 
 = ^VZ 
 
 Tl = 
 
 IC 
 
 1/2 
 
 
 <^i 
 
 0-2 
 
 
THE REAL REPRESENTATION OF THE COMPLEX CONIC. 39 
 
 we bring the functions iv and z to the final form 
 
 w = Wo + /o-i, 
 
 S = So + 1^2- 
 
 We examine these in the reverse order in which they are tabu- 
 lated. Whatever be the Riemann surfaces belonging to ai and a> 
 equations (11) express merely integral rational transformations 
 of the same. The first consists of an expansion 1 1 \ and a rota- 
 tion Z (/) followed by a translation over the vector wq. In 
 the second we have first of all a reflection over the axis of reals 
 of 0-2, followed by transformations identical in nature with those 
 of the first equation. 
 
 In §§ 4 and 5 we discussed functions very similar to ci and a-y- 
 There the constant entering, § 4, (7) and § 5, (3) was a real 
 number, c^. Here we have the complex numbers a = a — c 
 — 2ij3 and y = a — c -\- 2i/3 appearing in the place of c^. Let us 
 examine the Riemann surface belonging to ai. For the moment 
 let us write o-i = u + iv, n = x + iy and y = a — c -\- 'lijS 
 = p -\- iq, and resolve the expression into its real and pure 
 imaginary parts 
 
 1/ qx - py \ 
 
 We seek the orthogonal families of curves in the cri-surface 
 corresponding to concentric circles x~ -\- y~ = r^, and the pencil 
 of rays y = ex in the ri-plane. For the first we have: 
 
 (^4 _ 2pr^ + R2)u2 _ 4qrhii, + (r' -f. 2pr'- + R'y - ^ ^^ = 0, 
 
 where R = Vp- -{- q^ = \y\. This is the equation of a conic 
 for which the discriminant, (r^ — R-)" is a quantity which is 
 never less than zero; hence corresponding to the family of con- 
 centric circles about the origin in the ri-plane is a family of 
 ellipses. 
 
 The curves in the o-i-surface corresponding to the pencil of 
 rays y = ex in the ri-plane are given by the equation, 
 
 2c(ep - g)w- + 2g(l + c')uv - 2(p - eq)v- - [2cp 
 
 - q(l - c2)]V4(l + c') = 0. 
 
40 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 This equation represents a one-parameter family of hyperbolas 
 for the discriminant, — {2cp — q + qc^Y, is a quantity which 
 is never greater than zero. 
 
 We find further that the real foci of both families of curves 
 is given by Vy and furthermore the slope of the principal axis 
 is given by {R — y)jq and this is the tangent of Z ( V7). The 
 (Ti-surface then consists of confocal ellipses and hyperbolas 
 corresponding to concentric circles about and radial lines through 
 the origin of the ri-plane. 
 
 Thus we see that the Riemann surface belonging to the func- 
 tion ci = ^(n + (t/ti)) is got from the surface belonging to 
 ^(n + (l/ri)) by a magnification in the ratio 1 : | V7I and a 
 rotation through the angle of V7, Similar results hold for the 
 function <J2 using a instead of 7. 
 
 The transformations 
 
 c - r/3 + VZ „ , c + 2/3 - VZ „ 
 Ti = r-j7^ r and ti = r-r^ r 
 
 transform the ry and T2-planes into themselves, merely inter- 
 changing the concentric circles among themselves and the pencil 
 of rays among themselves. 
 
 Thus as t" describes a circle about the origin in its plane ri 
 and T2 describe circles about the origins in their respective planes 
 and in turn ai and <X2 describe ellipses in their respective planes 
 with centers at the origin; while finally iv and z describe ellipses 
 about M'o and 20 in their respective planes. Then as r" describes 
 a ray of the pencil orthogonal to the concentric circles w and 2 
 describe hyperbolas in their respective planes confocal with the 
 ellipses just described. 
 
 The Riemann surface belonging to the function 
 
 ""K^+O 
 
 is two-sheeted, the cross-cut of the surface extending from — 1 
 to + 1- The unit circle of the r-plane maps into this line con- 
 sidered as double. Circles outside the unit circle and concentric 
 with it map into ellipses in the upper sheet of the o--surface. 
 Circles inside the unit circle and concentric with it map into 
 ellipses lying in the lower sheet. So for our functions cri and a-^. 
 
THE REAL REPRESENTATION OF THE COMPLEX CONIC. 41 
 
 The Riemann surfaces belonging to them are two-sheeted, the 
 sheets being joined along the lines joining — V7 and + V7 or 
 — Va and + Va as the case may be. To the circle of radius 
 I V7 1 of the n-plane corresponds this line joining the branch 
 points ± V7 counted twice. Ellipses in the ci-surface, corre- 
 sponding to circles lying outside the circle of radius \ ^y\ in 
 the n-plane lie in the upper or first sheet of the surface, while 
 those corresponding to circles within the circle of radius | V7I 
 lie in the lower or second sheet. As we have already noted, 
 
 iiri 
 
 >C 
 
 / 
 
 Fig. 3. 
 
42 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 the IV- and z-surfaces are integral rational transformations of 
 the ci- and <72-surfaces respectively. 
 
 With proper precautions we may consider all these surfaces 
 superposed and with origins common with that of the (^, 77)- 
 plane. The surfaces discussed are to be considered in no wise 
 organically connected with each other. A schematic diagram of 
 quantities considered is set forth in drawings. 
 
 The Case Z = 0. 
 
 17. So long as we require that the complex coefficient jS in our 
 so-called canonical equation of the conic be a general complex 
 quantity, that is /3 = 61 + ib2 with 61 and bo different from zero, 
 this case, Z = cannot occur; for we must have 
 
 Z = ac - i82 = ac - bi" + 62' - 216162 = 0, 
 
 which requires that either 61 = or 62 = 0. 
 
 This case requires, then, that either 61 or 62 vanish. If 62 = 0, 
 the conic becomes real and we have already considered this 
 case. Chapter I. Again Z may vanish for 61 = 0, that is, for 
 jS = ib2, pure imaginary. For this case we have 
 
 Z = ac - /3- = ac + 62- = 0. 
 
 Hence if |acl = 62" and a and c have opposite signs, Z will be 
 zero. Since bo = ±i '^ac, the terms of second degree for this 
 case, just as for the real case, form a perfect square: 
 
 a^2 _t- 2ib>i^r} - erf = ( Va^ ± ^ <cr}f. 
 
 Choosing the plus sign we have for the equation of the conic 
 with Z = 0: 
 
 a^2 -f 2i Vac^?7 - erf- + 2d^ + 2er) + T = 0. 
 
 The equation in z and w giving the real representation is 
 aV + 2b'wz + c'z" + 28io + 2€z + i' = 0, 
 
 where 
 
 a' = a + c -^2^ae = i-^-}- ^ef, 
 c' = a + c — 2 ^ = ( Va — Vc)2, 
 h' = a — c, 
 
THE REAL REPRESENTATION OF THE COMPLEX CONIC. 43 
 
 8 = 2{d- ie), 
 
 e = 2{d + ie) = 5, 
 
 We resort to the use of the parameter in this case as in the 
 preceding, with the same method of introducing it. 
 
 The quadratic giving the points where the tangent is vertical 
 reduces for this case to — 2A^ + A = 0, which with 
 
 i3^ - C77 + e = 
 gives for the coordinates of the point 
 
 A 
 
 ^1 = 
 
 2A* 
 
 irr vei 
 
 Introducing the parameter as in the preceding section we have 
 
 77 - Tji = r(^ - ^i), T = s -\- it, 
 
 which considered with (1) gives for ^, 77, w and z expressed as 
 functions of r 
 
 ^ = ^1 + 
 ^ = m + 
 
 2A 
 
 {cr - ^f 
 2Ar 
 
 = ^1 + 
 
 = T/l -}- 
 
 2a/^ 
 
 2 -Veer 
 
 Making the transformations 
 
 T = 
 
 cr - /3 = r', 
 
 w = wi -\- 
 
 2 = Sl + 
 
 2(c - il3) 
 Ti — i 
 
 2(0 - i^) 
 
 T2 — i 
 
 2A(1 + Jt) 
 (cr - ^f 
 
 2A(1 + ir) 
 
 of the parameter and applying them respectively to the func- 
 
44 COMPLEX CONICS AND THEIR REAL REPRESENTATION. 
 
 tions 20 and z we have after reduction 
 
 A Ve 
 
 A ^f 
 
 where 6 = ( Vc(^ + i Vae)^ is the determinant of the conic, 
 z' -> w' is the picture of the finite focus. The functions <ti and era 
 are well known and have been discussed in Chapter I. The 
 functions w and z are linear integral transformations of the 
 surfaces belonging to a-i and 0-2. 
 
 18. The Case, h = 0. — In § 13 we noted that in the case 
 b = a -\- c = our variables iv and z were rationally separable. 
 We have 
 
 mv^ + 28iv = - yz- - 28z - 4^, 
 
 Let us put 
 Then 
 
 ,2 2 -/2 
 
 W = p-^ — 2 . 
 
 This case is similar to the one discussed by Holzmiiller, 
 
 ?(J = Vl — s^. 
 
 The complex quantity p however excludes the involutorial 
 property of the case just cited. The simplification of this case 
 is noted by observing equations (9) of section (16). 
 Jackson, Mississippi. 
 
VITA. 
 
 Benjamin Ernest Mitchell, son of B. W. and Rachel Mitchell, 
 born Oct. 16, 1879, Morrisville, Mo. Graduated, Scarritt-Mor- 
 risville College, A.B., 1900. Student, University of Missouri, 
 1902-03. Professor of Mathematics, Scarritt-Morrisville Col- 
 lege, 1903-06. Scholastic Fellow, 1906-07; Teaching Fellow, 
 1907-08; A.M., 1908; Instructor in Mathematics, 1908-12; Van- 
 derbilt University. Student, Columbia University, 1912-14. 
 Tutor in Mathematics, College of the City of New York, 1913. 
 Instructor in Extension Teaching, Columbia University, 1913-14. 
 Professor of Mathematics, Millsaps College, 1914-. Member 
 of American Mathematical Society. My sincere thanks to the 
 members of the Department of Mathematics of Columbia Uni- 
 versity and in a special degree to Professors Kasner and Keyser. 
 
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