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 The theory of the imaginary in geometry. 
 
 3 1924 001 523 665 
 
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THE THEORY 
 
 OF THE 
 
 IMAGINARY IN GEOMETRY 
 
CAMBRIDGE UNIVERSITY PRESS 
 
 C. F. CLAY, Manager 
 
 LONDON : FETTER LANE, E.C. 4 
 
 NEW YORK : G. P. PUTNAM'S SONS 
 
 BOMBAY •) 
 
 CALCUTTA I MACMILLAN AND CO., LTD. 
 
 MADRAS J 
 
 TORONTO : J. M. DENT AND SONS, LTD. 
 
 TOKYO : MARUZEN-KABUSHIKI-KAISHA 
 
 ALL RIGHTS RESERVED 
 
THE THEORY 
 
 OF THE 
 
 IMAGINARY IN GEOMETRY 
 
 . TOGETHER WITH 
 
 THE TRIGONOMETRY OF THE 
 IMAGINARY 
 
 BY 
 
 J. L. S. HATTON, M.A. 
 
 PRINCIPAL AND PROFESSOR OF MATHEMATICS, EAST LONDON COLLEGE 
 (UNIVERSITY OF LONDON) 
 
 CAMBRIDGE 
 
 AT THE UNIVERSITY PRESS 
 
 1920 
 ft 
 
 ) 
 
PREFACE 
 
 rriHE position of any real point in space may be determined by 
 - 1 - means of three real coordinates, and any three real quantities 
 may be regarded as determining the position of such a point. In 
 Geometry as in other branches of Pure Mathematics the question 
 naturally arises, whether the quantities concerned need necessarily be 
 real. What, it may be asked, is the nature of the Geometry in which 
 the coordinates of any point may be complex quantities of the form 
 as + ix, y + iy , z + iz' ? Such a Geometry contains as a particular case 
 the Geometry of real points. From it the Geometry of real points may 
 be deduced (a) by regarding x', y\ z as zero, (b) by regarding x, y, z t as 
 zero, or (c) by considering only those points, the coordinates of which 
 are real multiples of the same complex quantity a + ib. The relationship 
 of the more generalised conception of Geometry and of space to the 
 particular case of real Geometry is of importance, as points, whose 
 determining elements are complex quantities, arise both in coordinate 
 and in projective Geometry. 
 
 In this book an attempt has been made to work out and determine 
 this relationship. Either of two methods might have been adopted. 
 It would have been possible to lay down certain axioms and premises 
 and to have developed a general theory therefrom. This has been 
 done by other authors. The alternative method, which has been 
 employed here, is to add to the axioms of real Geometry certain 
 additional assumptions. From these, by means of the methods and 
 principles of real Geometry, an extension of the existing ideas and 
 conception of Geometry can be obtained. In this way the reader is 
 able to approach the simpler and more concrete theorems in the first 
 instance, and step by step the well-known theorems are extended and 
 generalised. A conception of the imaginary is thus gradually built up 
 and the relationship between the imaginary and the real is exemplified 
 and developed. The theory as here set forth may be regarded from 
 
vi Preface 
 
 the analytical point of view as an exposition of the oft quoted but 
 seldom explained " Principle of Continuity." 
 
 The fundamental definition of Imaginary points < is that given by 
 Dr Karl v. Staudt in his Beitrage zur Geometrie der Lage; Nuremberg, 
 1856 and I860.' The idea of («, /3) figures, independently evolved by 
 the author, is due to J. V. Poncelet, who published it in his Traiti des 
 Proprietes Projectives des Figures in 1822. The matter contained in 
 four or five pages of Chapter II is taken from the lectures delivered by 
 the late Professor Esson, F.R.S., Savilian Professor of Geometry in the 
 University of Oxford, and may be partly .traced to the writings of 
 v. Staudt. For the remainder of the book the author must take the < 
 responsibility. Inaccuracies and inconsistencies may have crept in, but 
 long experience has taught him that these will be found to be due 
 to his own deficiencies and not to fundamental defects in the theory. 
 Those who approach the subject with an open mind will, it is believed, 
 find in these pages a consistent and natural theory of the imaginary. 
 Many problems however still require to be worked out and the subject 
 offers a wide field for further investigations. i 
 
 To Professor Whitehead, F.R.S., the author has to tender his 
 sincerest thanks for much valuable help and assistance. To Professor 
 G. S. Le Beau and Mr S. G. Soal of the, East London College he is 
 indebted for very valuable advice and criticism. To Mr J. B. Peace of 
 the Cambridge University Press he desires to tender his warmest 
 thanks for the great assistance which he has rendered in connection 
 with this book. 
 
 J. L. S. H. 
 
 Chideock, 
 
 September, 1919. 
 
CONTENTS 
 
 CHAPTER I 
 
 PAGE 
 
 Imaginary points and lengths on real straight' lines . 1 
 
 Imaginary straight lines 13 
 
 Properties of semi-real figures . . 20 
 
 CHAPTER II 
 The circle with a real branch .... . .41 
 
 The conic with a real branch .53 
 
 CHAPTER III 
 
 Angles between imaginary straight lines 70 
 
 Measurement of imaginary angles and of lengths on imaginary 
 
 straight lines 76 
 
 Theorems connected with projection. - . . ■ .116 
 
 CHAPTER IV 
 The general conic 125 
 
 CHAPTER V 
 The imaginary conic . . 141 
 
 CHAPTER VI 
 Tracing of conies and straight lines 164 
 
 CHAPTER VII 
 The imaginary in space 196 
 
 Index of theorems 213 
 
 Index of terms and definitions 216 
 
NOTE 
 
 The reference numbers of articles in square brackets, such as 
 Art. [22], refer to articles in the author's Principles of Projective 
 Geometry. In this book, as in the Principles of Projective Geometry, 
 the term " line " is used as an abbreviation for straight line. 
 
CHAPTER I 
 
 IMAGINARY POJNTS AND LENGTHS ON REAL STRAIGHT LINES. 
 IMAGINARY STRAIGHT LINES. PROPERTIES OF SEMI-REAL FIGURES 
 
 1. Imaginary points on a real straight line. 
 
 Axiom I. Every overlapping involution range determines a pair of 
 points as its double points, each of which has a definite position on the base. 
 
 This axiom does not imply that the position of these points can be 
 graphically determined with reference to real points on the base. 
 
 Def. I. Each of these double points is termed an imaginary point 
 and, Considered as a pair of points, they are termed a pair of conjugate 
 imaginary points. 
 
 On reference to Art. [51] it will be seen that, if is the centre of an 
 overlapping involution of which A and A' are a pair of conjugate points 
 —situated on opposite sides of — then the double points of the involu- 
 tion are given as the positions of a point P which satisfies the relation 
 
 OP*=OA.OA' = -K* 
 
 Hence, if P r and P. 2 be these points, 
 
 OP^ + J^K and OP, = ;- V^l K. % 
 If 0' be any other real point on the base such that 0'0 = L, then the 
 positions of P 1 and P 2 relative to 0' are given by the relations 
 
 aP 1 =L + 4~^\K and O'P, = L r- V^TZ. . 
 
 The axiom therefore states that a definite position may be assigned 
 to the points P, and P 2 on the base, and consequently a definite magni- 
 tude to the lengths V^l K and - V - 1 K measured along the base. 
 Such lengths cannot be equal to any real lengths and therefore the 
 points Pi and P 2 cannot coincide with any real points situated on the base. 
 
 Similarly every other overlapping' involution on the same base, which 
 has its centre a.t 0, has a pair of imaginary double points situated on 
 the base at distances V -IK' and -V-lA" from 0, where A" is 
 different for each involution and may have any real value from to ao . 
 
 h. i. e. 1 
 
2 The Imaginary in Geometry 
 
 Hence it follows that, given any real straight line I and any real 
 point on it, there are on the straight line an infinite .number of 
 pairs of imaginary points. Such a system of points may be termed an 
 imaginary system, base I, centre or mean point 0. 
 
 A quantity of the form V - 1 K is termed a purely imaginary 
 quantity and a quantity of the form, L + V — \K, an imaginary or a 
 complex quantity, where L and K are real. 
 
 Imaginary lengths. The position of an imaginary point P may be 
 determined by its distance O'O + OP from a real point 0', where O'O is 
 a real length and OP an imaginary length. A point P' is termed the 
 graph of P if it is at a distance O'O + OP' from 0', where 
 
 V^T OP' = OP. 
 
 Hence, if P' be the graph of P, the distance of P from 0' is 
 
 O'O + V^l . OP'. 
 
 A real length V + \K and an imaginary length V — 1 K are incom- 
 mensurable, and do not in themselves involve any relative magnitudes. 
 Like two real incommensurable quantities V2 and 3, whose squares are 
 commensurable, the squares of a real and an imaginary quantity may 
 be commensurable. 
 
 The position of a real point on a given base can only be determined 
 graphically when the unit is known in which its distance from a given 
 point on the base is expressed. Similarly, the position of an imaginary 
 point is only known when the unit in which its imaginary distance is 
 expressed is also known. The units in the two cases may be regarded 
 as V + 1 and V — 1. As there is no inherent relation as to magnitude 
 between these units, the relative position of real and imaginary points 
 on the base is indeterminate. 
 
 Lengths of the form V — 1 K and V — 1 E' which determine the 
 positions of imaginary points may be combined like real lengths. (See 
 Art. 3.) So long as the lengths considered are all real or all purely 
 imaginary each system may be graphed in the same way, the quantities 
 V + 1 and V — 1 being regarded as units in which the lengths are 
 expressed, the only difference between these units lying in the fact that 
 in one case the square on a line is regarded as positive and in the other 
 negative. Thus in accordance with the conventions of Algebra the 
 imaginary point at a distance 2K V — 1 from is at double the distance 
 
Imaginary Points 3 
 
 from of the point at a distance JK V — 1, and in accordance with the 
 conventions of coordinate geometry the points at distances + V — IK 
 and — v—lK from are at equal distances from on opposite sides 
 of 0. 
 
 If a second' system of overlapping involutions, which have any other 
 real point 0' on the base for centre, is considered, a second system of 
 imaginary points is obtained, which are determined by imaginary lengths 
 measured from 0'. This is the system, base I, centre 0', and a similar 
 system exists for every real point on the base. No 'two imaginary points 
 can coincide, when they are thus determined from different real base 
 points. Otherwise a real and an imaginary length would be equal. 
 
 If a definite position has been assigned to an imaginary point, 
 distances may be measured from such an imaginary point as origin. 
 If is a real point on a straight line and P a point on the straight 
 line at a distance V — IK from 0, the point P may be taken as the 
 centre of an involution. As in the previous case there are on the 
 base an infinite number of imaginary points, centre P, and an infinite 
 number of points real with reference to the centre P. None of the real 
 points, centre P, can coincide with the real points centre 0, but the 
 imaginary points centre P will be a repetition of the imaginary points 
 centre 0. 
 
 Hence the following conception of a straight line is arrived at. 
 
 On any real straight line some point may be taken which may be 
 termed a base point. An infinite number of real and an infinite number 
 of imaginary points njay be obtained by measuring real or purely 
 imaginary distances from this base point. Any one of these real points 
 may be taken as a new base point and an infinite number of real and an 
 infinite number of imaginary points may be obtained by measuring real 
 and purely imaginary distances from it. The real points so obtained are 
 a repetition of the real points first obtained. The imaginary points form 
 a new system of points. An imaginary point of the first system may be 
 taken as a base point and an infinite number of points real with respect 
 to this base point may be obtained and also an infinite number of points 
 imaginary with respect to this base point. The real points with respect to 
 this centre are imaginary points with respect to the base point first taken 
 and are distinct from those obtained from the original base point. The 
 imaginary system obtained from this second centre is a repetition of the 
 imaginary points obtained from the first base point. 
 
 1—2 
 
4 The Imaginary in Geometry 
 
 2. Conjugate imaginary points. 
 
 It follows from definition (1) that every imaginary point has one and 
 only one conjugate imaginary point. Given an imaginary point, its con- 
 jugate imaginary point is the other double point of the involution of 
 which it is a double point. The conjugate of a given imaginary point 
 may be obtained by changing the sign of V — 1 in the length or lengths 
 ,by which the position of the point is determined. Hence imaginary 
 points occur in pairs, viz. in pairs of conjugate imaginary points, and 
 the connector of any .pair is the base of the involution of which they are 
 the double points. When an imaginary point is given, the involution of 
 which it is a double point is completely determined, for the centre and 
 constant of the involution are known. 
 
 This is true so long as the same points are regarded as real points, that is as 
 long as the origin is not moved through an imaginary distance. Thus OK+i.KP 
 and OK-i.KP give two positions of P which ' correspond to conjugate imaginary 
 points. If however the origin be moved through a distance i. KP, these become 
 OK and OK - 2 . i . KP, which distances give a pair of points which are not conj ugate 
 imaginary points according to the definition. In fact by moving the origin any two 
 points whose distances from the origin are a+ifi and a + i[}' may be made into a 
 pair of conjugate imaginary points. Hence conjugate imaginary points are such 
 with respect to a given origin or with respect to certain given real points. This 
 ought always to be stated, but, when there is no risk of misunderstanding, the- 
 limitation in question will be omitted. 
 
 3. Measurement of distances. 
 
 The positions of two imaginary points P' and Q' which have the 
 same centre and whose, graphs are P and Q are given by the relations 
 
 OP' = V^l OP and OQ' = V -I OQ. 
 The distance Q'P' may be defined as OP' -0Q' = V^l QP. 
 
 Hence,since QP + PQ = 0, Q'P' + P'Q'=0 (1) 
 
 If R' be a third imaginary point with the same centre, it also- 
 follows that 
 
 P'Q'+Q'R' + R'P'=0 (2) 
 
 The relations' (1) and (2) are those on which all the theorems of Art. [7], 
 which refer to points on a straight line, depend. Hence these theorems 
 hold for imaginary points with a common centre. 
 Two imaginary points P', centre lt 
 
 and Q', centre 2 , whose graphs are re- 
 
 spectivelyP and. Q may be determined ° "s °i Q ' p ' 
 
 with reference to any origin by 
 
 OP' = 00, + V^T 0,P and OQ' = 00 2 + V ^1 2 Q. 
 
Imaginary Distances 5 
 
 The distance, Q'P' may be defined as OP' - 00/, hence 
 Q'P' = 00, + V^T o,P - 00, - V^I 2 Q 
 = 2 1 + V^1{0 1 P-0 2 Q} 
 = 0,0, {VT- V~^l} + V~=T QP. 
 
 ■ From this it follows that if P', Q', PJ be any three points on a straight 
 line, real or imaginary, , 
 
 P'Q'+Q'P' = 
 
 and P'Q' + Q'P' + R'F = 0. 
 
 Hence in the most general case all the theorems of Art. [7] which refer 
 to distances of pointy on a straight line are true for imaginary points. 
 
 If 0, P, and Q are three collinear real points, the point M given by 
 the relation OM = \ [OP + OQ} is defined as the middle point of PQ. 
 
 Similarly, if be any real point and P' and Q' a pair of imaginary 
 points, centres 0, and 2 , whose graphs are P and Q, M' the middle 
 point of P'Q' is defined as being the point given by 
 
 QM , : 00 1 + 00 2 m/ — ^P + Q.Q 
 
 This point will be shown, Art. 6> to be the harmonic conjugate of the 
 point at infinity on the base with respect to P' and Q'. 
 
 The product of the distances of a pair of conjugate imaginary points 
 P' and Qf, mean point M, from any real point on the line P'Q' is a 
 positive real quantity. For this product is 
 
 (OM+i . MP) (OM- i . MP) = OM* + MP 2 . ' 
 
 4. Determination of the position of imaginary points by means of ratios. 
 
 I. Let A' be any point on a given real P' 
 
 ine and B a point at a purely imaginary ^~ i q, 
 
 distance A'E from A', and P' a point at a 
 purely imaginary distance A'P' from A'. Then B'P'= A' P' -A' '£'. 
 
 A'P' A'P' 
 
 The ratio of P' with reference to A' and B' is -g^ or 
 
 The ratio in' this case is a real quantity. 
 
 ,B'P' A'P'-A'B'' 
 
 II. Let A' be any point on a given real p , 
 
 line, B and points at real distances A'B — g r— 
 
 and A'O from A', and P' an imaginary point 
 
 at a distance A'O + OP' from A', where OP' is imaginary. 
 
6 The Imaginary in Geometry 
 
 Then A'P'= A'O+OP' 
 
 and BP'=BA' + A'O+OP' 
 
 = BO+OP', 
 A'P' A'O+OP' A'O+OP' 
 
 " BP' BO+OP' A'O + OP'-A'B' 
 
 The ratio of P' with respect to A' and B in this case is a complex quantity. 
 
 In either case when the ratio of a point and the positions of the reference points 
 are given, the position of the point is uniquely determined. The ratio of the con- 
 jugate imaginary point of P' is obtained by changing the sign of the imaginary part 
 of the ratio. 
 
 If lengths are expressed in imaginary units a pair of conjugate imaginary 
 points, whose positions would ordinarily be determined by lengths a + J — la' and 
 a — J -la', are determined by lengths J -la -a' and J — la + a', i.e. the distances 
 which determine the points differ only in the sign of the real part. 
 
 5. Anharmonic ratios of real and imaginary points on a real 
 straight line. 
 
 In Art.[ll] the anharmonic ratio of four real collinear points A,B,C,D 
 was defined as being, with origin 0, 
 
 ,Avnns OC-OA OP - OA . 
 
 (AECD) = q C _qb ■ 0D _ B = ( su PP ose )- 
 
 The range (ABGD) was defined as being harmonic if X had the value — 1. 
 
 Def. 2. If A', B', C, P' be the graphs of four collinear points real or 
 imaginary whose positions are determined by OA, + iA,A' ; 0B 1 + iB,B'; 
 etc., their anharmonic ratio is defined as being 
 
 OC, + iC,C'-OA,-iA,A' OP, + iP,P' - OA, - iA,A' _ ^ \ 
 
 00, + iO,C - OB, - iB,B' : OP, + iP,P' - OB, - iB,F~ ^suppose). 
 
 Under certain circumstances this anharmonic ratio may have the value 
 — 1, in which case the range is said to be harmonic. 
 
 Let 0' and D' be real points, i.e. let 0,0' = B^D' = 0. Let A and B 
 be a pair of conjugate imaginary points and let be their centre. Then 
 OA, = OB, = 0, A,A' = OA', B,B' = OB' and OA' = - OB'. Hence 
 
 OO'-iOA' OP 1 - iOA' _ 
 OC' + iOA' : OP' + iOA' 
 
 OO'-iOA' OP'-iOA' 
 \- l, 0C' + i0A' + OP' + iOA'~ V - 
 
 .-. -00' .0P' = 0A'* = 0B'>- 
 
Anharmonic Ratios 7 
 
 Therefore if C and D' are on different sides of so that - OC . OB' is 
 positive, then OA' and OB' are real and i.OA' and i. OB' give a pair of 
 conjugate imaginary points, which are the double points of an over- 
 lapping involution of which is the centre and of which C and B' are 
 a pair of conjugate points. 
 
 Conversely the imaginary double points of an overlapping involution 
 are harmonic conjugates of every real pair of conjugate points of the 
 involution. Hence it is seen that every pair of conjugate imaginary 
 points may be determined as the common harmonic conjugates of two 
 pairs of real points. 
 
 A given pair of real points are conjugate points of an infinite number 
 of involutions with imaginary double points. The double points of these 
 involutions are harmonic conjugates of the given pair of points. Hence 
 a given pair of real points has an infinite number of pairs of harmonic 
 conjugates which are pairs of conjugate imaginary points. 
 
 From the definition of the anharmonic ratio of a range of real and 
 imaginary points it follows that if three points of a range are given and 
 likewise the value of the anharmonic ratio of these points with a fourth 
 point — which anharmonic ratio may have a real, an imaginary or a com- 
 plex value-r-the position of the fourth is uniquely determined. 
 
 It likewise follows that the theorems, Art. [12], in regard to the 
 change of the value of an. anharmonic ratio, when the order of the points 
 is changed, are true when the range is wholly or in part imaginary. 
 Also if (A'B'C'D')=(ABCD), and (A'B'C'E')=(ABOE) it follows that 
 (A'B'E'B') = (ABEB). 
 
 6. Projective ranges. 
 
 Def. 3. Two ranges of real or imaginary points are said to be 
 projective when the anharmonic ratio of four points of one range is 
 equal to the anharmonic ratio of the four corresponding points of the 
 other range. 
 
 If A, B, C, three points of one range are given, and three corre- 
 sponding points A', B', C of the other, it is always possible, given a 
 fourth point P of the first range, to determine uniquely a fourth point P' 
 of the second, such that (ABGP) = (A'B'C'P'). Hence two sets of three 
 points determine two projective ranges and to each point of one range 
 corresponds one and only one point of the other. 
 
 As a particular case there is in each of two projective ranges, one 
 point termed the vanishing point, which corresponds to the point at 
 
8 The Imaginary in Geometry 
 
 infinity in the other range. The vanishing points may be real or 
 imaginary. 
 
 To find the harmonic conjugate of the point at infinity on a straight line with 
 respect to two imaginary points on the line. 
 
 Let F and Q' be the imaginary points, centres 0, and {? 2 -, and let P and Q be 
 their graphs. Take any real origin and let X' be the required point. Then 
 
 OP -OX' OP-<x> = 
 
 oq-ox' ' oq->x>~ ' 
 
 .-. 0P'+0Q' = 2.0X', 
 . go 1 + 00 2 ,j— i 1 P+0 2 Q 
 
 ..ox= — ^ — +v ~ 1 -%—■ 
 
 This point was defined (Art. 3) as the middle point of PQ'. 
 
 If a real point 0, the point at infinity and a pair of conjugate imaginary points 
 A and A' form a harmonic range, OA = — OA', and therefore must be the mean 
 point of A and A'. 
 
 Any two conjugate imaginary points are harmonic conjugates of their graphs. 
 
 Let P', Pj, centre 0, be the two points and P and Pi their graphs. 
 
 Then 
 
 (PP.'PPO 
 
 OP' -OP OP' 
 ~OP,'-OP • 0P{ 
 
 -OP, 
 -OP, 
 
 
 i.OP-OP i 
 
 .OP+OP 
 
 
 ~ -i.OP-OP " ^ 
 
 i.OP+OP 
 
 
 i-1 , _ i+l 
 
 
 
 -i-\ ' -i + l 
 
 
 
 =■-1. 
 
 
 It follows that if P', P, be a pair of conjugate imaginary points, the^r mean 
 point, P and P 1 their graphs and oo the point at infinity on the line, then 
 
 (PP l O<x>)=.(P'P 1 '0<x,)=(PP 1 P'P 1 ')=-l. 
 
 7. Extended conception of a real involution. 
 
 Every real involution contains two branches, one real and one purely 
 imaginary. In one branch pairs of conjugate points overlap. In the 
 other they do not. In the latter branch, there are a pair of double points, 
 which, according to the nature of the branch in which they occur, are real 
 or imaginary. 
 
 If the first definition of an involution (Art. [51]) be taken.and be 
 the point corresponding to the point at infinity, and P and P' a pair of 
 conjugate points, then the product OP . OP' is constant. There are two 
 cases to be considered according as this constant is positive or negative. 
 
Involution 9 
 
 (1) Let OP.OP' = K*. 
 
 (a) For real lengths, OP and OP, jthere are pairs of conjugate 
 points which do not overlap, and a pair of real double points E and F 
 given by OE = -OF = K. 
 This is the real branch. 
 
 (6) If OP = i.OP 1 and OP' = i.OP;, 
 
 then OP 1 .OP:=~K\ 
 
 The "points then belong to an imaginary branch of the involution, 
 which has no double points and in which corresponding segments overlap. 
 
 (2) Let OP.OP' = -IC-- I 
 (a) For real lengths, OP and OP', there are pairs of conjugate 
 
 .points which overlap, and this branch of the involution has no double 
 points., 
 
 (6) If OP = i.OP 1 and OP = i.OP 1 ', 
 
 then OP 1 .OP l ' = + K\ 
 
 The points then belong to an imaginary branch of the involution, 
 in which there are a pair of double points whose graphs, E 1 and F lt are 
 given by 0E t — — OF! = K. The corresponding segments of this branch 
 do not overlap. Hence in an involution, the constant and centre of 
 which are real, to a real point P corresponds a real point P' and to a 
 purely imaginary point corresponds a purely imaginary point. 
 
 The double points real or imaginary of a real involution are harmonic 
 conjugates of every pair of conjugate points, real or purely imaginary, 
 of the involution. 
 
 (1) If the double points and the pair of conjugates are all real this 
 has been proved in Art. [51]. 
 
 (2) If the double points are imaginary and the pair of conjugates 
 are real the result is that of Art. 5. 
 
 (3) If the double points are imaginary and the pair of conjugates 
 are also purely imaginary the case resolves itself into case (1) where each 
 length is expressed in imaginary units of length. 
 
 If the second definition of an involution, Art. [51], be taken the pre- 
 ceding may be deduced as follows : 
 
 If two superposed projective ranges A, B, G ... and A', B', C ... are 
 such that (ABC A') = (A'B'G'A), then every pair of corresponding points 
 mutually correspond and the ranges form an involution. 
 
10 The Imaginary in Geometry 
 
 The point P of the first corresponding to the point B of the second 
 is given by 
 
 (AA'BP) = (A'AB'B), 
 
 but (A'AB'B) = (AA'BB'), 
 
 .-. (AA'BP) = (AA'BB'), 
 P coincides with B'- 
 Let be the poiiit which corresponds to the point at infinity, then 
 (AB' oo 0) = (A'BO oo ), 
 .• . OA . OA' =OB.OB'=a, constant. 
 This constant may be any quantity real, imaginary, or complex in the 
 most general case. If it is real and- the point is also real, the involution 
 is real and the preceding results are obtained. 
 
 The condition that a system of points may form an involution may 
 also be expressed in ratios. Let b, V, c, c', d, d' be the ratios of B, 
 B', C, C, D, D' referred to A and A' as origin. Then since 
 (AA'BG) = (A'AB'C), 
 bb' = cc = a constant. 
 If the points A, A' ; B, B' ; . . . are real but' the involution is overlapping, 
 the position of the double points is given by 
 X°- = -K* = bb' = cc'=..., 
 .-. X^^J^IK and X 2 = ->J^lK. 
 Conversely if (AA'BC) = (A'AB'C'), then AA', BB', CC form an 
 involution. 
 
 Every point on a line has a conjugate in every involution on the 
 line. If the conjugate of a point a + id with respect to an involution, 
 
 constant K, is sought, this point is found to be — ; „ (a — id). 
 
 An imaginary involution, i.e. an -involution in which the constant is 
 a complex quantity and ivhose centre may be an. imaginary point, has a 
 pair of imaginary doable points, which are equally distant from the 
 centre. ' 
 
 If K + iK' be the constant of the involution, the distances of the 
 double points from the centre are 
 
 ± ^| {J*JK* + K* + K + i JTiW+E'^-K} , 
 where the positive sign has to be given to the square roots. 
 
Principal Coordinates 
 
 11 
 
 8. Analytical expressions. 
 
 Every imaginary point in a real plane whose coordinates are imaginary is u. 
 double point of an involution on a real base. 
 
 Let the coordinates of the point be a + ib and c+id. The equation of the 
 straight line joining this point to the point whose coordinates are a-ib a.nd e- id is 
 
 x y 
 a + ib c + id 
 a — ib c — id 
 
 
 x y 1 
 a c 1 
 b d 
 
 = 
 
 = 0. 
 
 This is the equation of a real line passing through the imaginary point. 
 
 The real point (a, c) is on this straight line and the distance of the point a + ib, 
 c + id from this point is i ,Jb 2 + d 2 . Therefore the point a + ib, c+id is a double 
 point of the overlapping involution on the straight line 
 
 = 0, 
 
 whose centre is the point (a, c) and whose constant is — (b 2 + oP). 
 
 The equation of the real line'through the point a+ib, c+id may also be written as 
 
 x y 
 
 1 
 
 a c 
 
 1 
 
 b d 
 
 
 
 x — a 
 
 ~T~ : 
 
 y — c 
 '' d " 
 
 The preceding shows that the connector of a pair of conjugate imaginary points, 
 denned as points whose coordinates differ only in the sign of the imaginary part, is 
 a real straight line. 
 
 Coordinates of an imaginary point. 
 
 Let the coordinates of an imaginary point P 
 be x x + ix 2 , yi+iyi- Construct the real point Q 
 (xi, y\). Then P may be found by drawing 
 through Q, QN equal to ix 2 and from N, PN 
 equal to iy 2 . Since 0M and MQ are real 
 0Q= Jx x 2 +y x 2 and since QN and NP are purely 
 imaginary QP=i >Jx£+y£. 
 
 Hence P may be constructed by measuring 
 
 along 0Q, which makes an angle v tan _1 — with 
 
 , x \ 
 
 the axis of x, a length Jx x 2 +yi 2 , and afterwards measuring along QP, which makes 
 an angle tan -1 ^ with the axis of x, an imaginary length i Jx^+y? 2 . 
 
 X<l 
 
 J 2 . 
 
 X 2 
 
 Let tana= — , tan B= — and tany = — so that /3 + y= 
 
12 The Imaginary in Geometry 
 
 Then the coordinates of P may be written 
 
 Xi+ix 2) Xy tan a + ix 2 tan /3, 
 or Xj + iy 2 tan y, X\ tan a + iy 2 ■ 
 
 OQ and §P may be termed the principal coordinates of the point P. 
 
 Analytically the representation of all the points on a straight line (real), is ,as 
 follows : , 
 
 If %$! and x^y% are any two real points which determine a straight line, the 
 coordinates of any real point on the line may be obtained by giving all real values 
 
 to X in *\ + f* , ^ + ? 2 . If X has the value a+ib, where a and' b may have any 
 
 1 -{- A 1 + A * < 
 
 real values, the coordinates of all points real or imaginary on the line are of the form 
 l + {a+ib) \ + {a + ib) ' 
 
 If A, A' and B, B' be pairs of conjugate imaginary points and and D a pair of 
 real points, then the involution determined by 
 
 (1) A, A' and B, B', as pairs of conjugate points, is real and has real double 'points. 
 
 (2) A, A' and C, D, as pairs of conjugate points, is real and has real double points. 
 
 (3) A, B and A', B, as pairs of conjugate points, is real. 
 
 Let A, A', B, B', C, D, be determined by a + ia', a — id, b+.ib 1 , b — ib', c, d. 
 Let the centre of the involution be determined by x. 
 
 (1) In this case OA . OA' = OB . OB'. 
 
 Therefore x= =-. jt . 
 
 2 (a — 6) 
 
 Hence the centre of the involution is real and, as the product of the distances of a 
 pair of conjugate imaginary points from any real point on the line, is real and 
 positive, the constant of the involution is positive and the, double points are real. 
 
 (2) In this case OA . 0A' = 0O. OB. 
 
 *v2 _1_ ft a do 
 
 Therefore x= — j . 
 
 ' 2a — a — c 
 
 Hence the centre of the involution is real and as in case (1) the double points are real. 
 
 (3) In this case OA . OB = OA' . OB'. 
 
 Therefore oo= --.- ~rr- . 
 
 a +b' 
 
 Hence the centre of the involution- is real. The constant of the involution is 
 (a — x)(b-x)- a'b'. This is real. Hence the involution is real. 
 
 From (1) it follows that two pairs of conjugate imaginary points have a\ways a 
 pair of real harmonic conjugates. (See also Art. 31 (3).) 
 
 From (2) it follows that a pair of conjugate imaginary points and a pair of real 
 points have always a pair of real harmonic conjugates. (See Art. 31 (3).) 
 
 From (3) it follows that two imaginary points have always one pair of harmonic 
 conjugates, which are either a pair of real points or a pair of conjugate imaginary points. 
 
Imaginary Straight Lines 13 
 
 9. Imaginary straight lines. 
 
 Def. 4. The locus of the double points (imaginary) of the over- 
 lapping involutions in which an overlapping involution pencil, (real) is 
 cut by real transversals is a pair of imaginary straight lines. Each 
 part of this locus, which is continuous, constitutes an imaginary straight 
 line. 
 
 The two straight lines considered as a pair of straight lines are 
 termed a pair of conjugate imaginary straight lines. 
 
 Hence it follows that an imaginary straight line is determined by 
 an imaginary point, which is a double point of an involution, and a 
 real point; the vertex of the involution pencil. The conjugate imaginary 
 line is determined by the conjugate imaginary point, the other double 
 point of the involution, and by the same real point, the -vertex of the 
 involution pencil. Hence two conjugate imaginary straight lines differ 
 as to their determining elements only in the sign of the imaginary part. 
 An imaginary line considered in reference to the involution pencil is 
 spoken of as a double ray of the pencil. 
 
 From the definition it also follows that only one imaginary straight 
 line passes through a given real and a given imaginary point. 
 
 The straight line joining the centre of an involution range to the 
 vertex of the pencil is called the mean line of the two conjugate 
 imaginary lines, which join its double points tcr the vertex of the pencil, 
 This mean line is real. As there are an infinite number of involutions 
 on the same base ■ which have the same centre, there are an infinite 
 number of pairs of imaginary straight lines through a given real point 
 which have the same mean line. 
 
 Hence given any real poinfi and any real straight line o through it, 
 there are an infinite number of pairs of imaginary straight lines which 
 pass through L and which have o as their mean line. Such a system of 
 straight lines may be termed an imaginary systjem centre L, mean 
 
 line o. 
 
 It also follows from the definition that every imaginary straight line 
 passes through a real point, viz. the vertex of the involution pencil of 
 which it is a double ray,, and that a pair of conjugate imaginary straight 
 lines intersect in a real point, viz. the vertex of the pencil of which 
 they are the double rays. 
 
 10. To determine the straight line which joins an imaginary point to 
 a real point. 
 
14 The Imaginary in Geometry 
 
 Construct the involution of which the imaginary point is a double 
 point. (1) If the real point lies on this straight line the required line 
 is the real base of the involution. (2) If the real point does not lie on 
 the base construct an involution pencil by joining pairs of real conjugate 
 points of the involution to the real point. The required line is one of 
 the imaginary double rays of this involution pencil. 
 
 To determine the point of intersection of a given imaginary straight 
 line with a given real straight line. 
 
 Let the given imaginary straight line be determined as the connector 
 of one of the imaginary double points E of an involution, centre V, con- 
 stant — K", situated on a straight line s, with a real point S. Let S, be 
 the given real straight line. Let V and IF/ be the points at infinity 
 on s and «! respectively and let ssj be 0. 
 
 Construct F, and F,' the projections of V and V from $ on s,. 
 Draw S W parallel to s x to meet s in IF'. Take IF the conjugate of IF' 
 in the involution on s. Join IF to 8 to meet s 1 in IF X . Then IF, is the 
 centre of the involution into which the involution on s is projected from 
 S on Si . Also IF! F/. W 1 V 1 = — M*, the constant of this involution on s x . 
 
 Hence, if E be the double point of the involution on s remote from 0, 
 the point E x in which 8E meets s x is at a distance V-fE^ from V( such 
 that VJE 1 = TV IF, - V W l V 1 . W\ V/, the negative sign being given to 
 the square root since F/ IF, is drawn in the opposite direction to VW. 
 
 The following relations hold between the lengths in the figure : 
 
 (a) VW .VWm- K\ IF, F, . Ttyi =- M\ 
 
 (b) Since(F'F'FlF) = (F 1 'IF 1 T 1 IF 1 )and F'and F/ are at infinity, 
 
 vw _ VW _ WW 
 W^rTWF'F' 
 
Imaginary Straight Lines 15 
 
 VW _WW _VW WW -K- s -m*_^. 
 and rw ~ ww -=k*~ ^W ~ vw> ~ WW ~ (su PP ose > 
 
 (c) Since (V'W'OV) = (V 1 'W 1 'OV i ), 
 
 OW . OVS = VW . VW = WW . WW = AW. AW, 
 where A and A l are any pair of real points on s and s a col linear with S. 
 
 If a range composed wholly or partly of imaginary points be pro- 
 jected from a real point by imaginary and real straight lines upon any 
 other real straight line so that a second range is formed on this straight 
 line, the anharmonic ratios of any four corresponding points of tliese 
 ranges are equal. ' 
 
 It will be proved that if, as in the preceding, V be the centre of the 
 involution which determines an imaginary point E, and W x the centre 
 of the involution which determines the corresponding point E lt then 
 WE . ViEt = a constant, where W and F/ are the points on the two 
 bases which in a projection from S correspond to the points at infinity. 
 It will also be shown that this constant is equal to the product of the 
 distances of any pair of real corresponding points from W and F/ re- 
 spectively. Hence since W and F/ are fixed points the theorem is 
 true. 
 
 In the figure 
 
 W'E= WV+*/VW. VW V^E,= V.'W.-^/WW- WW 
 
 ■"('-^ =™('Vi) 
 
 = W V (1 - Vx), = F,' W 1 (1 + A). 
 
 Therefore WE . V^E, = W V . V{ W 1 (1 - X). 
 
 ww vw 
 
 But i_x = i__^ / = _^_. 
 
 Therefore WE. V.'E, = WV. F/F = WO. F/0 = WA . V.'Ai- 
 
 This theorem may be stated as follows : 
 
 If through any real point a system of real and of imaginary straight 
 lines be drawn, these straight lines determine equianharmonic ranges on 
 all real transversals. 
 
 It also follows that projective ranges as defined in Art. 6 are also pro 
 jective according to the usual conception. 
 
16 
 
 The Imaginary in Geometry 
 
 Analytical verification of Art. 10. 
 
 Let S be the vertex of the pencil and V the centre of the involution on OV. 
 Take OF for axis of x and the given real line 0V r for axis of y. 
 
 Let V and'W 7 ! be the points at infinity on the axes. Project these from /J into 
 FY and W. Let S V meet Oy in V x . 
 
 Then Fand V are conjugate points of the involution on ftc/and therefore V x and 
 V{ are conjugate points of the involution on Oy. 
 
 Let W{ be the centre of the involution on Oy. 
 
 Then TPx'and W x are conjugate points of the involution on Oy, and IF and W ' are 
 conjugate points of the involution on Ox. 
 
 Since S is a given point, OW{l) and V{ (m) are given. 
 
 Also the constant of involution on 0x( — ft 2 ) is given and the point V{a). 
 
 Hence OW'*=a + — I ,.:OW 1 '=- l . i 
 
 a-V 1 b i +{a-lf 
 
 + a i — al) , „_ ma 
 and OV, = — ,. 
 a-l 
 
 Therefore V X W{ = - 
 
 -ndV 
 
 and Vi W{ 
 
 -(a-l){b* + (a-m 
 Hence — M 2 the constant of the involution on Oy is 
 
 ml (a - I) 
 
 1 6 2 +(a-J) 2- 
 mHW 
 
 {6 2 + (a-«) 2 } 2 ' 
 
 Hence the distances, of the double points of the two involutions from are re- 
 spectively a + ib, a — ib, and 
 
 mlb 
 
 
 m(b 2 + a 2 — al) 
 
 mlb 
 
 b* + (a-l)» "tf + ia-lf 
 
 But the line joining S (I, m) to (a + ib, 0) meets Oy in a point distant 
 
 m (a 2 4- b 2 — al) — imbl 
 (a-l)*+b* 
 
 from 0. This shows that the geometrical construction of Art. 10 agrees with the 
 analytical conception of an imaginary straight line. 
 
Imaginary Straight Lines 17 
 
 If E and E\ be collinear double points 
 
 Y{ E 1 = ml ^~ l) ~t> and F£=a-! + ib. 
 V* + (a — iy 
 
 .-. ViE l . WE=ml, which is a constant for all involutions. This confirms the 
 result of Art. 10. ' 
 
 11. To determine the straight line connecting any two imaginary 
 points. 
 
 Construct the two involutions of which the two given imaginary 
 points are each respectively a double point. 
 
 (1) If, the two involutions have the same base this real line is the 
 connector of the two imaginary points. 
 
 (2) If the two involutions have different bases, these involutions are 
 in real perspective in two ways (Art. [60]) and in one of these perspectives 
 the two given imaginary points are corresponding points. The required 
 imaginary straight line is the connector of the centre of this perspec- 
 tive, which is real, with either of the two given imaginary points. 
 
 From this it follows that only one straight line can be drawn to join 
 two imaginary points and that it passes through a real point, viz., one of 
 the centres of perspective of the involutions, and therefore no two imaginary 
 straight lines can include a space. 
 
 To determine the point of intersection of any two imaginary straight 
 lines. 
 
 Construct the two involution pencils of which the two given imagin- 
 ary straight lines are each respectively a double ray. 
 
 (1) If the two involution pencils have a common vertex, this vertex, 
 which is real, is the point of intersection of the lines. 
 
 (2) If the two involution pencils have different vertices, these invo- 
 lutions are in real perspective in two ways (correlative of Art. [60]) and 
 in one of these perspectives the two given imaginary lines are corre- 
 sponding rays. The point of intersection of these lines is therefore a 
 double point of the overlapping involution determined by the two invo- 
 lution pencils on one of their real axes of perspective. 
 
 Hence two imaginary straight lines intersect in only one point and 
 this point lies on a real straight line, viz., one of the axes of perspective 
 of the determining involutions. 
 
18 
 
 The Imaginary in Geometry 
 
 12. Summary of properties 
 lines. 
 
 ( 1 ) Every imaginary point con - 
 tains one and only one real straight 
 line. 
 
 This line is the base of the in- 
 volution of which the imaginary 
 point is a double point and it is 
 the connector of the point and its 
 conjugate imaginary point. There 
 can be no other real, straight line 
 through the point for, if there were, 
 the point would be a real point. 
 
 (2) The connector of a pair of 
 conjugate imaginary points is real. 
 
 (3) Every real straight line, 
 that contains an imaginary point, 
 contains its conjugate. 
 
 (4) An imaginary straight line 
 meets all real straight lines in ima- 
 ginary points except those which pass 
 through its one real point. 
 
 (5) The connector of a pair of 
 imaginary points is real or imagin- 
 ary; if real it contains the conjugates 
 of both the points. 
 
 (6) A system of real and ima- 
 ginary points on a real straight line 
 is projected from a real point, not 
 on the line, upon another real 
 straight line, into a system in which 
 real points correspond to real points 
 and imaginary points to imaginary 
 points. 
 
 (7) The connector of a pair of 
 imaginary points and the connector 
 of their conjugate imaginary points 
 
 of imaginary points and straight 
 
 Every imaginary straight line 
 contains one and only one real 
 point. 
 
 This point is the vertex of the 
 involution pencil of which the ima- 
 ginary line is a double ray and it is 
 the point of intersection of the line 
 and its conjugate imaginary line. 
 There can be no other real point on 
 the straight line for, if there were, 
 the line would be a real straight line. 
 
 The point of intersection of a pair 
 of conjugate imaginary straight lines 
 is real. 
 
 Every real point, that contains 
 an imaginary' straight line, contains 
 its conjugate. 
 
 The only real points, the con- 
 nectors of which to an imaginary 
 point are real, are those which lie 
 on the one real line through the point. 
 
 The intersection of a pair of 
 imaginary lines is real or imagin- 
 ary; if real it contains the conjugates 
 of both the lines. 
 
 A system of real and imaginary 
 straight lines through a real point is 
 cut by a real transversal in points 
 which, when connected to another 
 real point, give real straight lines 
 corresponding to real straight lines ' 
 and imaginary straight lines to 
 imaginary straight lines. 
 
 The point of intersection of a pair 
 of imaginary straight lines and the 
 point of intersection of their con- 
 
Projective Pencils 19 
 
 are conjugate imaginary straight jugate imaginary lines are conjugate 
 
 lines. imaginary points. 
 
 (8) The imaginary double points The imaginary double rays of an 
 of an involution are harmonic con- involution pencil are harmonic con- 
 jugates of every pair of conjugate jugates of every pair of conjugate 
 points of the involution. rays of the involution pencil. 
 
 (9) If a pair of conjugate ima- If a pair of conjugate imaginary 
 ginary points coincide, they coincide lines coincide, they coincide in a real 
 in a real point. line. 
 
 13. In two projective pencils with real vertices, which have two pairs 
 of real corresponding rays, all pairs of corresponding rays, real or 
 imaginary, intersect on a straight line {real) if the ray joining the vertices 
 of the pencils is a self-corresponding ray. 
 
 Let »S and S' be the vertices of the pencils. Then by Art. [34], the 
 real pairs of self-corresponding rays intersect on a straight line s. Let 
 an imaginary ray e of the first pencil meet s at E and the corresponding 
 imaginary ray of the second pencil e' meet s at E'. Let the two pairs 
 of real corresponding rays meet s in A, B and let SS' meet s in C. 
 
 Then (ABGE) = (ABCE 1 ). 
 
 Therefore by Art. 5, E and E' coincide and e and e' meet on s. 
 
 Correlatively it may be proved that if the point of intersection of the 
 real bases of two projective ranges, which have two pairs of real correspond- 
 ing elements, is a self-corresponding point, the connectors of all pairs of 
 corresponding points of the ranges (real or imaginary) pass through a 
 real point. 
 
 If two imaginary lines are, corresponding rays of two projective 
 pencils with three pairs of corresponding real rays, the two imaginary 
 straight lines of which they are conjugate imaginary lines are also cor- 
 responding rays of the pencils. 
 
 Let e and e' be the first pair of corresponding imaginary lines of the 
 pencils. By Art. [37], displace the pencils so that they are in per- 
 spective. Then e and e' will intersect on s, the axis of perspective. The 
 point ee' is then an imaginary point since the vertices of the pencils are 
 the real points on e and e . But s is a real straight line. Hence it 
 passes through the conjugate imaginary point of ee'. Hence the con- 
 nectors of this point to the vertices of the pencils, i.e., the conjugate 
 imaginary lines of e and e ', are corresponding rays of the pencils. 
 
 2—2 
 
20 
 
 The Imaginary in Geometry ', 
 
 Correlatively it may be proved that if two imaginary points are 
 corresponding points of two projective ranges with three pairs of corre- 
 sponding real points, their conjugate imaginary points are also corre- 
 sponding points of the projective ranges. 
 
 14. The triangle. 
 
 The different types of triangles which may occur are as follows, viz. : 
 (1) Real Triangle. A triangle Real Triangle. A triangle con- 
 
 consisting of three real vertices de- sisting- of three real lines deter- 
 
 termining three real sides. 
 
 (2) Semi-real Triangle. A tri- 
 angle consisting of one real vertex 
 and two conjugate imaginary ver- 
 tices (on a real line) determining a 
 pair of conjugate imaginary lines 
 (for two sides) and a real line for 
 the third side (on which the pair 
 of conjugate imaginary vertices are 
 situated). 
 
 (3) An imaginary Triangle, 
 type (a). A triangle consisting of 
 one real vertex and two imaginary 
 points situated on a real line de- 
 termining one real .side (on which 
 the imaginary vertices are situ- 
 ated) and a pair of imaginary sides 
 passing through the real vertex. 
 
 (4) An imaginary Triangle, 
 type (6). A triangle consisting of 
 a pair of conjugate imaginary ver- 
 tices and a third imaginary vertex, 
 determining one real side (joining 
 the pair of conjugate imaginary 
 vertices) and a pair of imaginary 
 sides. 
 
 (5) An imaginary Triangle, 
 type (c). A triangle consisting of 
 three imaginary vertices, deter- 
 mining three imaginary sides. 
 
 mining three real vertices. 
 
 Semi-real Triangle. A triangle 
 consisting of one real side and two 
 conjugate imaginary sides (meeting 
 in a real point) determining a pair 
 of conjugate imaginary vertices 
 (lying on the real side) and one 
 real vertex (being the point of in- 
 tersection of the pair of conjugate 
 imaginary lines which form a pair 
 of sides). . 
 
 An imaginary Triangle, type (a). 
 A triangle consisting of one real 
 side and a pair of imaginary sides 
 passing through a real point, de- 
 termining one real vertex (through 
 which the pair of imaginary sides 
 pass) and a pair of imaginary ver- 
 tices on the real side. 
 
 A n im aginary Triangle, type (6). 
 A triangle consisting of a pair of , 
 conjugate imaginary sides and a 
 third imaginary side, determining 
 one real vertex (being the point of 
 intersection of the conjugate ima- 
 ginary sides) and a pair of imagin- 
 ary vertices. ' * 
 
 A n imaginary Triangle, type (c). 
 A triangle consisting of three im- 
 aginary sides, determining three 
 imaginary vertices. 
 
The Triangle 21 
 
 In all the above cases, except case (4), the triangle and that given 
 by the correlative construction are of the same nature. 
 
 15. Extension of Menelaus' theorem. 
 
 I. If an imaginary straight line meets one side of a real triangle in 
 a real point and the other two, sides in imaginary points, the product of 
 the ratios of these points, with respect to the triangle, is unity. 
 
 This may at once be deduced from Art. 10. For if a real straight 
 line be drawn through 8 to meet s and Sj in A and A u the triangle OAA 1 
 will be a real triangle and, the imaginary straight line SEE^ is a straight 
 line which meets two of the sides in imaginary points E and E lt and 
 the third side in a real point S. 
 
 It follows from Art. 10 that 
 
 (OAEW') = (OA 1 E 1 oo). 
 
 Therefore OE . OW = W 
 
 xnereiore AE ' AW A& 
 
 But . w , = -p~ , by similar triangles. 
 
 „, . OE AS A,E, 
 
 Therefore __._.__ = L 
 
 The theorem may also be proved independently as follows : 
 
 Take a real triangle ABC. Let A b 
 and A c be the conjugate points of B 
 and Cin an involution (with imaginary 
 double points) on BC. 
 
 Take any real point Son A B. From 
 .S project A c and Ab into B c and B a . / ~T^~~4§>- 
 
 Then the involution B,A b , C,A c is 
 projected from S into A, B a , C, B c . 
 
 Let the ratios of A c , At be d' and a 
 and of B c , B a be V and b. Then the 
 ratios, x and y, of the double points of the involutions are given by 
 
 x 2 — 2xa' + aa' = 
 
 ao d f-%/b +66' =0. (See Art. [61] Ex. (7).) 
 
 If a be the ratio of S then 
 
 cab=\ and oa'b'=l (1) 
 
 The double points are given by 
 
 (% — a') 2 =a< (a' — a) 
 (3-6)2 = 6(6-5'), 
 .-. x = a' ±*]a' (a! -a), y = b±Jb(b-b'). 
 
22 
 
 The Imaginary in Geometry 
 
 But from (1) 
 
 Let 
 
 Similarly if 
 then 
 
 a V 1 . 
 
 - = T = -Tj- = X suppose, 
 
 a b cab 
 
 x=a'±Ja' 2 (l-\), y = b±Jb*(l-\)- 
 
 ^ 1 = a'(l + . v /l :: ^). yi=&(l-Vl-X); 
 
 .-. ^ 1 y 1 = a'6(l-l+X)=a'6X = ^, 
 
 * 2 =o'(l- Vl-X), y 2 =6(l + «yi- r X) ) 
 a; 2 y 2 c=l. 
 
 II. //" aw imaginary straight line meets the sides of a real triangle 
 in three imaginary points, then the product of the ratios of these points 
 with respect to the triangle is unity. 
 
 Take any real triangle ABC. Draw through B any real transversal 
 
 to meet AG in A and the imaginary line ^.B^! in G l , where S is real. 
 
 Let the ratios of A lt B ls G r referred to the triangle A BG be x, y, z. 
 
 AS 
 
 Let 
 
 BS 
 
 = c. 
 
 Then from the triangle ABC 
 
 GB, . CBj A B, A.B, ... 
 
 c - x -A^r l0rc - x -A-^ l --AB J =l0TC - x - y -'ABr 1 '-' {1) 
 
 and from ABA e 
 
 BZ A& _ . nr 1 A t B, _ 
 C, 'A t O l ' AB, " °- z' AB, ~ • 
 
 .(ii) 
 
The Triangle 
 
 Dividing the expressions (i) and (ii) 
 
 23 
 
 xy 
 .'. xyz=l. 
 Hence the imaginary line A^G-i meets the sides of the triangle of 
 reference A BG in three points (imaginary) for which Menelaus' 
 theorem holds. 
 
 16. Extension of Ceva's theorem. 
 
 I. If the connector of any imaginary point to one of the vertices of 
 a real triangle is real, the product of the ratios of the three points in 
 which its connectors to the vertices meet the opposite sides of the triangle 
 is — 1. 
 
 B A' 
 
 Let P be the imaginary point and let AP, BP, CP, meet the opposite 
 sides of the triangle in A', B', G'. Let BPB' be real. Draw A Q parallel 
 to BC to meet BB' in Q. Join CQ to meet AB in Q'. 
 
 Then (A.BB'PQ) = (BOA' oo ). 
 
 But « (G.BB'PQ) = (BAG'Q'). 
 
 Therefore (BCA' oo ) = (BA G'Q'). 
 
 BA^ = BCT B$ 
 
 ■'• GA'~ AC' : AQ" 
 
24 The Imaginary in Geometry 
 
 Therefore 
 
 BG' GA' BQ' BG GB' 
 AC' BA'~ AQ'~ AQ B'A 
 
 Therefore 
 
 BG' GA' AB' 
 AC BA' GB 7 ~ 
 
 II. The connectors of an imaginary point to the vertices of a real 
 triangle meet the opposite sides in three points the product of whose ratios 
 with reference to the triangle is — 1. 
 
 Let P be the imaginary point- and let the connectors of this point 
 with the vertices of the triangle ABC meet the sides in A', B', C', 
 
 respectively. Let the real line through the point P meet the sides in 
 A , B , C . Join GG and AA . Let a, b, c be the ratios of A', B', C' 
 respectively. Let GP and BP meet AA in Q and R respectively. 
 Then from the triangle AA G by I. 
 
 AQ A A' G& = , 
 A Q- GA' • AB l ' 
 
 and from the triangle AA B by Menelaus' theorem 
 
 A Q BG AG' 
 ' AQ A C B&~ + 
 
 Therefore c.^ .g.|| = -l (I) 
 
 Similarly from the triangle A A B by I. 
 
 AJR AC, BA' 
 AR'BC A A' 
 
The Quadrangle 
 
 and from the triangle AA G by Menelaus' theorem 
 
 25 
 
 AR A B GB' 
 
 Therefore 
 
 b. 
 
 A R- GB AB 
 BA' AC A B 
 
 -, = + 1. 
 
 = -1. 
 
 'A A'-BC : GB 
 Hence multiplying together (I) and (II) 
 
 , BA' GB AC A B 
 
 . C • 
 
 .(II) 
 
 = + 1. 
 
 ' CA' AB, BG A G 
 Hence a . b . c = — 1. 
 
 Therefore Ceva's theorem holds for a real triangle. 
 
 It may be shown as follows from the first case of Menelaus' theorem, which has 
 been proved independently of this result (Art.' 15), that the anharmonic ratios of 
 two partly imaginary ranges obtained by pro- ,g 
 
 jection from a real point are equal. 
 
 Let two imaginary lines be drawn through 
 a real point C on the side AB of a real triangle 
 ABO to meet the two other sides in B', A' and 
 B", A". 
 
 Let the ratios of C", B', B" be 
 c, b' + ibi, b" + ibi", 
 respectively. 
 
 Then the ratios of A' and A" (by the exten- 
 sion of Menelaus' theorem) are 
 
 1 
 
 Therefore 
 
 c(&'+iV) 
 
 and 
 
 c{b" + ib{Y 
 BA" 
 CA" 
 1 1 
 
 c(b'+ibi') ' c(6"+iV) 
 = (6" + iV) : (b'+iW) 
 
 CB" CB^ 
 = AB" ' AB' 
 , ={ACB'B'). 
 
 ' It does not follow, and it is not true, that the real portions of the determining- 
 elements of points on one range are projected into the real portions of the determining 
 elements of points on the other range. This is only true when the cutting lines are 
 parallel. 
 
 17. The quadrangle and quadrilateral. 
 
 There are various kinds* of quadrangles and quadrilaterals which 
 differ with the nature of the four points and the four straight lines 
 which determine them. 
 
26 
 
 The Imaginary in Geometry 
 
 The most important of these are the real quadrangle and quadri- 
 lateral and the semi-real quadrangle and quadrilateral. The latter are 
 constructed as follows : 
 
 Semi-real Quadrangle. 
 1st Kind. 
 
 Two pairs of conjugate imagin- 
 ary points (A, A' and B, B') deter- 
 mine 
 
 (1) a pair of real lines and two 
 pairs of conjugate imaginary lines; 
 
 (2) three real points, being the 
 points of intersection of the three 
 pairs of lines. 
 
 In the figure the lines 
 
 A A', BB' are real, 
 
 AB', BA' are conjugate 
 imaginary lines, 
 
 AB, A'B' are conjugate 
 imaginary lines. - 
 
 E, F, G are real, since the inter- 
 sections of conjugate imagin- 
 ary lines are real, and are the 
 diagonal points of the quad- 
 rangle. 
 
 Semi-real Quadrilateral. 
 
 Two pairs of conjugate imagin- 
 ary lines (a, a' and b, V) determine 
 
 ( 1 ) a pair of real points and two 
 pairs of conjugate imaginary points; 
 
 (2) three real lines, being the 
 connectors of the three pairs of 
 points. 
 
 In the figure the points 
 ad, bb' are real, 
 ab', ba' are conjugate 
 
 imaginary points, 
 ab, a'b' are conjugate 
 
 imaginary points. 
 e,f,g are real, since the connectors 
 
 of conjugate imaginary points 
 
 are real, and are the diagonals 
 
 of the quadrilateral. 
 
The Quadrangle 
 
 27 
 
 2nd Kind. 
 
 A pair of imaginary conjugate 
 points (A, A') and a pair of real 
 points (B, G) determine 
 
 (1) a pair of real lines and two 
 pairs of imaginary lines ; 
 
 (2) three points, one of which 
 is real, the other two conjugate 
 imaginary points. 
 
 (A) 
 
 A pair of conjugate imaginary 
 lines (a, a') and a pair of real lines 
 (b, c) determine 
 
 (1) a pair of real points and 
 two pairs of imaginary points ;, 
 
 (2) three lines, one of which is 
 real, and the other two a pair of 
 conjugate imaginary lines. 
 
 In the figure 
 
 A A' and GB are real, 
 AB and A'G are imaginary, 
 A'B and AC are imaginary. 
 The point E is real. 
 
 The lines BA, BA' and CA, GA' 
 are pairs of conjugate imaginary 
 lines and therefore their points of 
 intersection G and F are conjugate 
 imaginary points. Hence the line 
 GF is real and EG, EF are con- 
 jugate imaginary lines. 
 
 If in the previous case the pair 
 of conjugate imaginary points B 
 and B' and the pair of real points 
 G and F are looked jipon as the 
 determining points, the diagonal 
 points triangle is AA'O. 
 
 In the figure 
 
 aa! and be are real, 
 
 ab and a'c are imaginary, 
 
 a'b and ac are imaginary. 
 
 The line e is real. 
 
 The points ba, ba' and ca, ca' 
 are pairs of conjugate imaginary 
 points and therefore their con- 
 nectors g and /are conjugate ima- 
 ginary lines. Hence the point gf 
 is real and eg, ef are conjugate 
 imaginary points. 
 
 If in the previous case the pair 
 of conjugate imaginary lines b and 
 b' and the pair of real lines g and 
 / are looked upon as the determin- 
 ing lines, the diagonal triangle con- 
 sists of a, a and the connector of 
 gf to bb'. 
 
28 The Imaginary in Geometry 
 
 For the construction, of the diagonal points triangle of a semi-real 
 quadrangle and of the diagonal triangle of a semi-real quadrilateral, see 
 Art. [60]. 
 
 18. Harmonic property of a semi-real quadrangle. 
 
 In a semi-real quadrangle the ranges determined on a real side or on 
 a real side of the diagonal points triangle are harmonic. 
 
 Case I. Let A, A' and B, B' be two pairs of conjugate imaginary 
 
 points and let 
 
 AA'.BB'beG, 
 
 AB' .BA'beE, 
 AB.A'B'beF. 
 ' Let EF meet AA' and BB' in L and M. 
 The ranges G, L\ A, A' and G, M, B', B are in perspective with the 
 real point E as centre of perspective. 
 
 Therefore (GLAA') = (GMB'B). 
 
 Similarly the ranges G, M, B', B and G, L, A', A are in perspective 
 with the real point F as centre of perspective. 
 
 Therefore (GMB'B) = (GLA'A). 
 
 Hence (GLAA') = (GLA'A) and therefore the range GLAA' is 
 harmonic. 
 
 Since E and F are the centres of perspective of the involutions on 
 GL and GM the range LMEF is harmonic. (Art, [60].) 
 
 Case II. The real points E, F and the pair of conjugate imaginary 
 points B, B' determine a semi-real quadrangle of the second kind in 
 which the ranges on the real lines are as above harmonic. 
 
 The harmonic property of a semi-real quadrilateral may be proved 
 in a similar manner. 
 
The Quadrangle 29 
 
 19. Involution property of a semi-real quadrangle. 
 
 The three pairs of opposite sides of a semi-real quadrangle are cut by 
 any real transversal in three pairs of conjugate points of an involution. 
 
 Case I. Let the quadrangle be determined by a pair of real points 
 A and B and a pair of conjugate imaginary points A' and B'. Then in 
 the figure the points G and F are a pair of conjugate imaginary points. 
 
 Let the transversal meet the pairs of opposite sides of the quadrangle 
 as in the figure in XX', YY', ZZ'. The point E is a real point. Pro- 
 jecting the range A'B'EY' from the real points A and B it follows that 
 
 (XZ' YY') = (ZX'YT) = (X'ZY' Y). 
 Hence (Art. 7), XX', YY', ZZ' form an involution. 
 By Art. 8 the involution is real. 
 
 Case II. The quadrangle determined by A'B'GF is a quadrangle 
 determined by two pairs of conjugate imaginary points. Let GF meet 
 the transversal in W. 
 
 By Case I the quadrangle A, B, G, F, determines an involution XZ, 
 X'Z', YW on the transversal, therefore (XZX' Y) = (ZXZ' W). 
 
 By Case I the quadrangle A, B, A', B' determines an involution XX', 
 YY', ZZ' on the transversal, therefore (XZX'Y) = (X'Z'XY'). 
 
30 The Imaginary in Geometry 
 
 Hence (X'Z'X Y') = (ZXZ' W). 
 
 Therefore ZX', XZ', Y'W form an involution. But these are the 
 three pairs of points in which the opposite sides of the quadrangle A'B'GF 
 are met by the transversal. Hence the theorem is proved. 
 
 By Art. 8 this involution is real and has real double points. 
 
 The involution property of a semi-real quadrilateral may be proved 
 in a similar manner. 
 
 Conjugate points with respect to a semi-real quadrangle and conjugate 
 lines with respect to a semi-real quadrilateral. 
 
 From the preceding the construction for" the conjugate points of 
 real points with respect to a semi-real quadrangle may be, deduced as 
 in Art. [57]. The conjugates of real lines with respect to a semi-real, 
 quadrilateral may be constructed by the correlative method. 
 
 20. Any two real involutions of different kinds in the same plane 
 are in imaginary plane perspective, the real branch of each corresponding 
 to the imaginary branch of the other. 
 
 Let B t and B 2 be the real double points of one involution and C 
 and C" the conjugate imaginary points which are the double points of 
 the other involution. 
 
Projective Ranges 
 
 31 
 
 Then, from the properties of the semi-real quadrangle B^Bfi'C", the 
 points S and S' are a pair of conjugate imaginary points and the line 
 SS' is real.- 
 
 Take the real triangle ABC as triangle of reference. 
 
 Let P', P" be two real points on AC, whose ratios p' and p" are such 
 that p'p" =K' i , K- being the constant of the involution of AC. Let — K" 1 
 be the constant of the involution on AB. Then the ratios of S and S' are 
 
 + 
 
 and — 
 
 1 
 
 >J-K*K'*' 
 
 (Art. [60].) 
 
 Let the imaginary lines SP' and SP" meet AB at Q' and Q". 
 Then the ratios of Q' and Q" are 
 
 ;— - and 77 ( by Menelaus theorem). 
 
 p p x J ' 
 
 Therefore the product of the ratios of Q' and Q" is 
 
 -K*K 
 
 P'P" 
 
 = - K'\ 
 
 Therefore Q', Q" are a pair of imaginary conjugate points of the involu- 
 tion on AB. 
 
 21. Projective ranges and pencils. 
 
 (a) If a range of points, real If a pencil of rays, real and 
 and imaginary, situated on a real imaginary, passing through a real 
 straight line be projected from an vertex, be cut by an imaginary trans- 
 imaginary point upon a real straight versal and the range so found be pro- 
 line the two ranges so obtained are jected from a real point, the two 
 equianharmonic. pencils so formed are equianhar- 
 
 monic. 
 
32 
 
 The Imaginary in Geometry 
 
 Let the range of points situated 
 on a real line s be G, A', A", A'" — 
 Let S be the imaginary point which 
 is the centre of projection. Draw 
 BA the real line through S to meet 
 s in B. Let the projections of A', 
 A",A'",B...on s',beB',B", B'", 
 A,... and let the ratios of the points 
 A', A", A'". . . with respect to BG be 
 a!, a", a". . ., those of B', B", B'". . . , 
 with respect to CA, V , b", b'"..., 
 and that of S with respect to 
 AB, c. 
 Then by Art. 15 
 
 a'-6'c = l = a"6"c, 
 
 •'• a"~b" 
 .-. {BGA'A") = (GAB"B') 
 = (AGB'B"), 
 so (BGA'A'") = (ACB'B'"), 
 . •. (GA'A"A'") = {OB'B"B'"), 
 .-. (A' A" A'" A"") = (B'B"B"'B""). 
 
 Let the pencil of rays through 
 the real point $ be o, a', a", a" ..., 
 Let s be the imaginary line which 
 cutsthemin 0, C, G", C" . ..,0 being 
 the real point on this line. Let the 
 connectors of 0, C, C", C". . . to S' 
 be t, b', b", V". . . . Take OS'S as tri- 
 angle of reference. Let the ratios of 
 the intersections of a, a", a'". . . with 
 OS' be Oj', a/', a-i" . . . and the ratios 
 of the intersections of 6', b", V" '. . . 
 with OS be W, 6,", W" ... and the 
 ratio of the point of intersection of 
 OC with SS' be Sl . 
 
 Then a.'Ws, = - 1 = a/'i/'s, by 
 Art. 16, 
 
 " ' < ~W 
 therefore as on the left-hand side 
 therangesdeterminedbythepencils , 
 on the real transversals OS' and OS 
 are equianharmonic and therefore 
 the pencils are equianharmonic. 
 
 If a system of real and imaginary points on a real straight line be 
 joined to a real point (Art. 10) or to an imaginary point (Art. 21) the 
 pencil so formed is cut by real transversals in projective ranges. Two 
 pencils so cut by real transversals in projective ranges are said to be 
 projective. 
 
 As in Art. [34] it can be shown that if two projective ranges on real 
 bases have the point of intersection of their bases for a self-corresponding 
 point, the ranges are in plane perspective. Correlatively, if two projective 
 pencils with real vertices have the connector of their vertices for a self- 
 corresponding ray the. pencils are in plane perspective, i.e. the pairs of 
 corresponding rays intersect on a straight line. These are extensions of 
 Art. 13. 
 
 (b) Two superposed projective ranges on the same real base, and also 
 two superposed projective pencils with the same real vertex, have two self- 
 corresponding elements. 
 
The Involution Pencil 33 
 
 Consider two superposed ranges determined by points A, B, G and 
 A', B', C, real or imaginary. Let Vbe the point of the first, which 
 corresponds to the point at infinity of the second, and W the point of 
 the second which corresponds to the point at infinity of the first. 
 
 Then (ABV ao ) = (A'B' oo W), 
 
 AV_ B'W 
 ' " BV~ A'W" 
 
 .-. AV. A' W' = B V.B'W' = constant = K + iK' (suppose). 
 
 If P be a self-corresponding point of the ranges, then 
 
 PV.PW' = K + iK'. 
 
 Let the distances of V, W and P from a given point of the base be 
 a + ia', b + ib' and x respectively. Then 
 
 (x - a - ia') (x-b- ib') = K + iK'. 
 
 As this is a quadratic equation in x, there are two values of x and 
 therefore two self-corresponding points. Since two superposed projective 
 pencils are cut by a real transversal in two superposed projective ranges, 
 two superposed pencils have two self-corresponding rays. 
 
 22. Real involution pencil. 
 
 If the pairs of conjugate points, real and imaginary, of a real involu- 
 tion range on a base s be joined to a real or an imaginary point S, the 
 pencil so formed is cut by real transversals in ranges, which are pro- 
 jective with the ranges formed by the corresponding points of the given 
 involution (Art. 21). Hence the pencil is cut by all real transversals in 
 involutions and is therefore said to form an involution pencil. 
 
 If a pair of conjugate points A, A' of the involution on s coincide, 
 their projections on any Teal transversal coincide, and therefore the 
 double rays of the involution pencil are obtained by joining to the vertex 
 of the pencil the double points of the involution range and are therefore 
 real or imaginary. 
 
 If the vertex S of the pencil is real, the involution pencil always has 
 a real branch and is said to form a real involution pencil. 
 
 If two pairs of the real conjugate rays of an involution pencil with 
 a' real vertex S are at right angles every pair of real conjugate rays of 
 the pencil are at right angles (Art. [58]). If through another real point S' 
 two other pairs of rays, parallel to the rays which meet at S, be drawn 
 they will be at right angles. The rays of the pencil vertex S' being 
 
 H. i. g. 3 
 
34 The Imaginary in Geometry 
 
 parallel to the rays of the pencil vertex $ meet the line at infinity in 
 the same two pairs of points and will therefore determine the same 
 involution on the line at infinity. The double rays of both pencils are 
 obtained by joining the double points of this involution^termed the 
 circular points at infinity — to S and S'. The lines joining /S and S' to 
 the circular points at infinity are termed the critical lines of S and S'. 
 Real lines, which are at right angles, are therefore harmonic conjugates 
 of the critical lines through their point of intersection. There are also 
 through S and S' pairs of imaginary conjugate rays of the same involu- 
 tion pencils. These pairs of rays are also harmonic conjugates of the 
 critical lines and are defined as imaginary lines which are at right angles. 
 Hence, if an imaginary line passes through S, there is through S one 
 imaginary line perpendicular to it, viz. its harmonic conjugate with 
 regard to the critical lines through S. 
 
 Similarly, if S be an imaginary point, there are a pair of critical lines 
 through it, which are the connectors of $ to the circular points at infinity. 
 Pairs of lines through S which are at right angles are defined as being 
 harmonic conjugates of these critical lines. 
 
 If y = (m + im') x be an imaginary line through the origin and y = Mx be the line 
 perpendicular to it, then since these lines are harmonic conjugates of the critical 
 lines, whose equation is # 2 +y 2 =0 (see Art. 78), 
 
 l+M(m+im') = 0, 
 
 1 
 
 .-. M= 
 
 m+im' 
 
 23. Anharmonic ratio of pencils subtended by points in the same plane 
 at a real or purely imaginary point. 
 
 (1) If four real points A, B, C, D are joined to a real point S the anharmonic 
 ratio of the pencil so formed is real. This is obvious. 
 
 (2) If two pairs of conjugate imaginary points A, A', B, B' are joined to a real 
 point S, the anharmonic ratio of the pencil so formed is real. 
 
 Let the lines A A', BR meet at 0. Then OA and OA' are of the form a + ia' 
 and a — ia'. 
 
 From 8, B and B project into a pair of conjugate imaginary points B x and B{ 
 on OAA'. Hence the form of 0B X and 0B{ is b+ib', b-ib'. 
 Hence the anharmonic ratio of the pencil (S. AA'BB') is 
 
 a+ia' -b-ib' a+ia'-b + ib' 
 a-ia' — b-ib' ' a — ia' — b+ib' 
 
 _ {a-bf + {a' -b'f 
 (a-Vf+{a' + b'f 
 
Anharmonic Ratios 35 
 
 (3) If a 'pair of conjugate imaginary points A, A' and a pair of real points B, C 
 be joined to a real point S, the anharmonie ratio K + iK' of the pencil (S . AA'BG) is 
 such that K 2 + K' 2 = 1. 
 
 If in the preceding B and B' are a pair of real points they projeGt from S into a 
 pair of real points B t and Cj on OA. i 
 
 Let {AA'B l C 1 )=K+iK. 
 
 Then, since A and A' are conjugate imaginary points, 
 
 (A'AB^^K-iK'. 
 
 Therefore (^'B,C 1 )= ( ^ m = ^,^K+iK' ; 
 
 .-. l = (K-iK')(K+iK')=K*+K' 2 
 
 (4) If four purely imaginary points A, B, C, D are joined to a purely imaginary 
 point S, the anharmonic ratio of the pencil so formed is real. 
 
 This is obvious since the ratio of two purely imaginary quantities is real. 
 
 (5) If two pairs of conjugate imaginary points A, A' and B, B' are joined to a 
 purely imaginary point S, the anharmonic ratio of the pencil so formed is real. ' 
 
 This result follows from (2) by substituting ia, ia', ib, iV etc. for a, a', b, V etc. 
 
 When looked at from the purely imaginary point of view the determining quan- 
 tities for a pair of conjugate imaginary points are ia — b and ia + b, i.e. the imaginary 
 parts are the same and they differ in the sign of the real part. 
 
 (6) If a pair of conjugate imaginary points A, A' and a pair of purely imaginary 
 points B, C be joined to a purely imaginary point S, the anharmonic ratio K+iK' of 
 the pencil (S. AA'BG) is such that K 2 +K' 2 =l. 
 
 The values of (AA'B^) and (A'ABjCi) will in this case be the same as in case ' 
 (3) except that all the quantities determining the position of the points are multiplied 
 by i. This quantity will divide out in the expressions for the anharmonic ratios and 
 therefore K a,nd K' are as in (3), i.e. K 2 +K' 2 =\. 
 
 A quantity K+iK', where K 2 +K' 2 = \, can be expressed as the quotient of two 
 conjugate imaginary quantities. 
 
 A+iB (A 2 -B 2 )+2iAB 
 
 Let K+UC-- 
 
 A-iB l A 2 +B* 
 
 A 2 — S 2 2/1 R 
 
 Then if K =A>+-m mdK '=A*+&' &+K' 2 =l. 
 
 But 
 
 K(A 2 +B 2 )=(A 2 -&), ■■■§, = — ■ 
 
 Since 2T<1, because K 2 + A" 2 =l, this expression is positive whether A' is positive 
 or negative. Therefore -5 = ,<V _ „ is real. 
 
 *Jl + K+i«Jl-K 
 
 Hence K+iK' = 
 
 *Jl+K-i\/l-K' 
 
 3—2 
 
36 The Imaginary in Geometry 
 
 24. Condition that the anharmonic ratio of four collinear points' may be real. 
 Let the four points A, B, C, D be determined by distances a + ia', b + ib', c + ic', 
 d+id' measured from an origin on the straight line. 
 
 T , a-c D b-c a-d _ b-d 
 
 Let A=-, — „B= Tl — ,, Cm- — -j„ D=ti — 37. 
 
 a-c b-c a-d' b—d 
 
 Then ( ABOB>-{£< : £$ . "g±J = gf= M < (suppose). 
 
 If K'=0 and J/' represents the expression in the first bracket 
 
 It' {A +i){D+ i) =K{B+ i) (C+i). 
 Hence M'{AD-\) = K(BC-1) and M' {A+D) = K(B+C). 
 
 Therefore 
 
 A.B.C.B.\l + l-^-^B + C-A-D. (1) 
 
 Hence it may be shown that the required condition is 
 
 1111 
 
 abed 
 
 a' V c' d' 
 
 a? + a' 2 &2 + 6' 2 c*+c'* d* + d'* 
 
 A = 
 
 =0. 
 
 25. Relations connecting, the anharmonic ratios of collinear imaginary points. 
 In the general case with the notation of Art. 24 
 
 M'(A+i)(B+i)=(K+iK')(B + i)(C'+i). 
 Therefore IT (AD - 1) =K{BC- 1)-K'(B + C) 
 
 and M'{A + D)=E(B+C)+K'(BC-1). 
 
 Hence it may be proved that 
 
 A A 
 
 K ' = {{b-cf+{b' -c , Y}{(a-dy+^ -d'f} = f{b,c).f{a,d) ( su PP 0Se )-(2) 
 
 and ^ + ^ = /(M)-/(a,c) ' 
 
 f(b,c).f(a,d) , —*/ 
 
 Hence K- +^_L_ f) <Jf(b, c ) f{a, d) /(ft, d) f(a, c) - A* (4) 
 
 If K' = Q the anharmonic ratio of the four points and their conjugate imaginary 
 points are equal and 
 
 f(b,c).f(a,d)- 
 
 This is the result obtained by making A zero in (4). 
 
 Lei f(a,b).f(c,d) = S, f{a,c).f(b,d) = S u f{a,d).f(b,c) = S 2 . 
 
 ,_, _ St +'S a -S „,„ 2S 1 S i +2SSi + 2SS a -S 1 *-Sf-S» 
 Then K= 2S 2 ' K = : 48? ' 
 
 and since K' = -§■ , 4A 3 = 2^ S 2 + 2SS t + 2SS 2 - S? - *S 2 2 - £ 2 . 
 
 "2 
 
Real Projection 37 
 
 Hence it is seen that 
 
 and so on, where 4A 2 = 25-S' I + 2OT 2 +2^ 1 /S 2 -5 2 -<S' 1 2 -»S2 2 - 
 
 If a' = b' = c' = d' = Q the corresponding formulae for real points are obtained. 
 
 26. Projection from a real point on a real plane. 
 
 (a) Any pair of real points and any pair of conjugate imaginary 
 points can by a real projection be projected into any pair of real points 
 and any pair of conjugate imaginary points, if the two systems are each 
 coplanar. 
 
 Let A, B, C 1 and C 2 be a pair of real points and a pair of conjugate 
 imaginary points in a plane a- and A', B', C/ and C 2 ' any pair of real 
 points and any pair of conjugate imaginary points in a plane cr. 
 
 Take any real plane cr" through A' and any centre of projection 
 (real) S on A A'. Project the figure in the plane cr from S on cr". Thus 
 a figure in plane a" is obtained, viz. A', B", G", C 2 " in which C" and G 2 " 
 are a pair of conjugate imaginary points. 
 
 Let A'B' and G t 'C 3 ' meet at E' and let A'B" and 0"G 2 " meet at E". 
 Then E' and E" are real points. 
 
 Since A'B'E' and A'F'E" are coplanar B'B" and E'E" meet at a real 
 point S'. Project the figure in a-" from S' upon a plane a'" through the 
 line s A'B'E'. Then a figure A', B', E', G"' , 2 "' is obtained in the 
 plane a". 
 
 Since 0/, C 2 ' and (7/", C 2 '" are two pairs of conjugate points in a plane 
 (CiCi and Q("Gi" are concurrent at E') therefore C/C/" and G % 'G 2 '" meet 
 at a real point S". 
 
 Projecting the figure in cr'" from S" ,on a the figure A'B'E'G{G* is 
 obtained. Hence the first system of points has been projected into the 
 second. 
 
 It should be noticed that C^ may be projected into G-[ or G 2 ', and the 
 point (? 2 into the other point. 
 
38 The Imaginary in Geometry 
 
 The preceding is equivalent to proving that any two points and an in- 
 volution in one plane may be projected into any two given points and a 
 given involution in another plane. 
 
 (b) To project two pairs of conjugate imaginary points in one plane - 
 into any two pairs of conjugate imaginary points in another plane. 
 
 Let A lt A 2 and B u B» be any two pairs of conjugate imaginary points 
 in the plane a and AJ , A 2 and 2?/, B 2 anytwo pairs of conjugateimaginary 
 points in the plane cr'. 
 
 Let A^A 2 and B X B 2 meet at E and A-[A 2 ' and BJB 2 meet at E'. 
 Then E and E' are real. Take a centre of projection S on EW and 
 project the figure in the plane <r upon a plane cr" which passes through E'. 
 
 Then A 1} A 2 become a pair of conjugate imaginary points A", A 2 " in 
 a" and B 1 , B 2 become a pair of conjugate imaginary points B", B 2 " in cr" 
 and E becomes E'. 
 
 Since B", B 2 " and BJ, B 2 ' meet at E' they are in a plane and the lines 
 Bj'B" and B 2 'B 2 " being conjugate imaginary lines in this plane meet at 
 a real point S'. Take a plane v'" through the line B^B^E' and project 
 the figure in the plane cr" from 8' upon this plane. 
 
 Then A", A 2 ", collinear with E', become a pair of conjugate imaginary 
 points A"' , A 2 " in a" collinear with E'\ B", B 2 " become the pair of con- 
 jugate imaginary points B{, B 2 ' and E' remains the point E'. 
 
 Since A"', A 2 " and Af, A 2 are collinear with E' they lie in a plane 
 and therefore the conjugate imaginary lines A"'A( and A 2 "A 2 meet in 
 a real point S". 
 
 If the figure in the plane cr'" be projected from 8" upon the plane , 
 cr', the pair of conjugate imaginary points A-l", A 2 " are projected into 
 the points Ai, A 2 . 
 
 Hence the required real projection has been performed. 
 
 27. Semi-real square. 
 
 A semi-real square of the first kind is determined by two pairs of conjugate 
 imaginary points situated on two real straight lines, which 
 are at right angles, the distances, imaginary, of the four points JP* 
 
 from the point of intersection of these lines being equal. 
 
 In the figure A and A' and B and B are the pairs of con- / 
 
 jugate imaginary points, and EA = A'E=EB=B'E. The (/ft\~ 
 lengths of the sides are all equal and are purely imaginary, 
 since EA 2 +EB 2 =AR 2 . It follows that the triangle BEA 
 is equal in all respects to the triangle AEB' and that the 
 angles BAB', ABA' are right angles. Opposite sides of the figure are conjugate 
 imaginary lines. 
 
 ^A) 
 
 B') 
 
..<■'' 
 
 B 
 
 \ 
 
 <*k 
 
 E 
 
 yfo) 
 
 v \ 
 
 C 
 
 , 
 
 -Rea? Projection 39 
 
 ,4 semi-real square of the second kind is determined by apairof conjugate imaginary 
 points and a pair of real points situated on two real straight 
 lines, which are at right angles, the distances of the pair 
 of imaginary points from the point of intersection of these 
 straight lines being a/ - 1 the distances of the real points from 
 the same point. 
 
 In the figure A and A' are the pair of conjugate imaginary 
 points and B and C the pair of real points, 
 
 BE=EC=*J^\ . A'E=\J~^\.EA. 
 
 The lines GA and GA' and also the lines BA and 2L1' are pairs of conjugate imaginary 
 lines. The properties of this figure are further discussed in Art. 86. 
 
 It follows from the last Art. that a semi-real quadrangle of the first kind can be 
 projected into a semi-real square of the first kind and that a semi-real quadrangle of 
 the second kind can be projected into a semi-real square of the second kind, the pro- 
 jections in both cases being real. 
 
 EXAMPLES 
 
 (1) If P and Q be two points on a real straight line and the points R and It' 
 divide the distance between P and Q in the ratios iX and ip, prove that 
 
 (PQRR')=*. 
 
 (2) Prove that the condition that the four points determined by distances 
 n + ia', b + ib', a-ia', b — ib' should form a harmonic range is 
 
 (a - bf + (a' + 6') 2 + 4a'6' = 0. 
 
 (3) Prove that no imaginary line can meet a pair of real lines in conjugate 
 imaginary points. 
 
 This follows from the fact that the connector of a pair of conjugate imaginary 
 points is a real line. 
 
 (4) If the angles between the real rays a, b, c of one pencil are equal to the 
 angles between the corresponding rays a', b', c' of a pencil projective with it, then 
 the angle between any pair of rays p and q of the first is equal to the angle between 
 the corresponding rays p' and q' of the second. 
 
 (5) If V and W are the vanishing points of two real superposed projective 
 ranges of which ^-1 and A' are corresponding points so that VA . W'A'=K, pr6ve 
 that the condition that the ranges should have a pair of real self-corresponding 
 points is that V0 2 + K should be positive, where is the middle point of VW. 
 
 (6) If, in the involution {ax* + 2hx+b)+\ (a'« 2 + 2A'a-+6') = 0, A, A' and B, B 
 are the pairs of conjugate points given by a^ + ^hx + b — O and a'x 2 +2h'x + b'=0 
 and X x and X 2 are the values of X corresponding to the double points of the involu- 
 
40 The Imaginary in Geometry 
 
 tion, then A^X^O is the condition that (A A' BE) should be harmonic; and 
 generally if p = {AA'BB') 
 
 /1+ M y fri + *& 
 \l-p) 4X X X 2 ■ 
 
 From the condition for equal values of x 
 
 . . ab' + ba! - %hh' , . ab-h? 
 
 ■ • -Mr = (o6-A»)(«'y-A'» r^ (w) by Art - [14]> Examp (3) 
 
CHAPTER II 
 
 ■ THE CONIC WITH A REAL BRANCH 
 
 28. In the Principles of Projective Geometry (Art. [92]) a conic 
 was defined as the locus of the points of intersection of a pair of corre- 
 sponding rays (real) of two projective pencils (with real vertices). In 
 Art. 95 (k) it was proved that such a conic determines on every straight 
 line (real) an involution by means of pairs of collinear conjugate points, 
 and that, when the double points of this involution are real, they are the 
 points of intersection of the line and conic. 
 
 The consequences of Axiom I, which are set forth in the preceding- 
 chapter, render it possible to enlarge and extend what was proved in 
 the Principles of Projectine Geometry. The assumption of Axiom I is 
 that when there are no real double points of a real involution- there 
 are a pair of imaginary double points, which are called a pair of conju- 
 gate imaginary points, and, as a consequence, a pair of conjugate 
 imaginary lines are defined as the double rays of a real overlapping 
 involution pencil. It follows (Art. 6) that two real projective ranges 
 and two real projective .pencils have, in addition to pairs of real 
 corresponding elements, pairs of corresponding imaginary elements, and 
 it is proved that the anharmonic properties of real and of imaginary 
 elements are similar. A pair of self-corresponding elements always 
 exist, when the pencils or ranges are superposed, and (Art. 21 (&)) they 
 are either a pair of real, coincident, or conjugate imaginary elements. 
 The same is true of a real involution, which is only a particular case of 
 two superposed projective ranges or pencils. 
 
 Hence taking into account the imaginary corresponding elements of 
 two real projective ranges or pencils it is seen that 
 
 (1) There are on a conic an infinite number of imaginary points, 
 viz. the points of intersection of pairs of imaginary corresponding elements 
 of the real generating pencils. 
 
 (2) On, every real straight line in its plane the conic determines 
 a real involution of which the double points are a pair of real, co- 
 incident, or conjugate imaginary points, and these are the points of inter- 
 section of the real line and the conic. Consequently every real line 
 in the plane of a real conic meets the conic in a pair of points 
 
42 The Imaginary in Geometry 
 
 real, coincident or conjugate imaginary, and, if a conic passes through 
 an imaginary point it also passes through the conjugate imaginary 
 point. 
 
 (3) Every line through a given real point meets its polar in a 
 point, such that this point, the given point and the points of intersection 
 of the line with the conic form a harmonic range. 
 
 Similarly the theorems correlative to the above may be shown to be 
 true. 
 
 In the extension obtained in this manner it must be borne in mind 
 that the vertices of the pencils and likewise the base's on which the 
 ranges are situated must be real as well as the anharmonic ratios of both 
 ranges and pencils. In Chapter IV the fundamental theorems for the. 
 conic are proved for the conic in general, including of course the case 
 of a conic with a real branch, which in this chapter is termed a real 
 conic. The restriction that the vertices of the pencils and the bases of 
 the ranges considered must be real does not apply to the proofs in 
 Chapter IV. 
 
 A circle is only a particular case of a conic, so that the preceding 
 applies to a circle with a real branch. 
 
 29. Circle with a real branch. 
 
 Construction of the involution determined by a real circle on a straight 
 line. 
 
 (a) If a line p meets a circle in real points E and F, these points 
 are the double points of the involution determined 
 by the circle on the line. P and P', any pair of 
 conjugate points of the involution, are harmonic 
 conjugates of E and F. If be the foot of the 
 perpendicular from C, the centre of the circle, on 
 p, is the conjugate of the point at infinity on p 
 and consequently 
 
 OP.OP' = OE*=OF*. 
 
 A circle described on PP' as diameter cuts the given circle ortho- 
 gonally (Art. [82]). 
 
 If CO meets the circle in R and K, then 
 
 OR.OK=OE.OF=-OP. OP'. 
 
The Circle 
 
 43 
 
 (b) If a line p meets a circle in imaginary points, and the perpen- 
 dicular from 0, the centre of the circle, 
 meets p in 0, then is the conjugate 
 in , the involution of the point at in- 
 finity on p, and therefore 
 
 OP. OP' = constant, 
 where P and P' are a pair of conjugate 
 points of the involution on p. 
 
 Let the polar of P meet OC in M 
 and OP in N; then the triangles 
 OP'M and OOP are similar, 
 
 OM OP 
 ' ' 0P'~ OC 
 
 and OP. OP' = - OM. 00 = -OR. OK, 
 
 since (OMRK) is harmonic. 
 
 But OR.OK=OV\ where OF is the tangent from to the circle. 
 Hence if a circle be described with centre and radius F to meet 00 
 in $, this circle will cut the given circle at right angles and PP', QQ, ... 
 pairs of conjugate points of the involution will subtend right angles 
 at S (Art. [53]). 
 
 This circle also determines on p the graphs of the double points of 
 the involution, which are the points of intersection of the line and circle. 
 
 S is either of the pair of common harmonic conjugates of MO 
 and RK. 
 
 30. Conjugate loci or Poncelet figures for a circle. 
 
 By taking the points of intersection of a series of parallel chords 
 with a circle it is possible to obtain a graph of the imaginary portion 
 of a circle. Such a figure may be termed a conjugate locus or a 
 Poncelet figure. Figures of this nature were first given by Poncelet in 
 his Traite des Propriety Projectives des Figures. (See Art. 39.) 
 
 Let be the centre of the circle, and consider the points in which 
 lines parallel to any diameter US meet the circle. 
 
 As long as the distance of these lines from is less than the radius, 
 they meet the curve in the part of the locus which is drawn in a con- 
 tinuous line. 
 
 If however a line of the system such as A' MB' meets NON' in M on 
 the sides of N and N' remote from 0, the involution determined on 
 
44 
 
 The Imaginary in Geometry 
 
 this line is overlapping and the double points, in which the circle 
 intersects the line, are a pair of conjugate imaginary points A' and B . 
 M is the real mean point of A' and B! and the distances MA', MB' 
 
 expressed in imaginary units are equal to the tangent from M to the 
 real branch of the circle. Hence, lengths measured parallel to OM 
 being regarded as real and lengths parallel to OS as imaginary, the 
 locus of A' and B' is a rectangular hyperbola, which touches the real 
 branch at N, has for centre and ON for semi-transverse axis. 
 
 For the system of lines parallel to any other diameter there is an 
 exactly equal rectangular hyperbola touching the real part of the curve 
 at the ends of the diameter. 
 
 31. Theorems concerning pairs of conjugate imaginary points determined 
 as the points of intersection of a circle and straight line. 
 
 (1) Any pair of conjugate imaginary points may be determined as the points of 
 intersection of a real circle and a real straight line. 
 
 Let M be the centre and P, P' any pair of conjugate points of the involution of 
 which the given pair of conjugate imaginary points are the double points. On 
 
The Circle 
 
 45 
 
 PP' as diameter describe a circle PKP'. Draw MO perpendicular to PP' and take 
 any point G on this line external to the 
 circle. Join PC to meet the circle described 
 on PP' as diameter in K. Join KP'. With 
 centre G and radius, the square of which is 
 equal to CK. OP, describe a circle. This circle 
 determines the given pair of conjugate imag- 
 inary points Q and Q' on PP'. This follows 
 from the fact that it determines on PP' the 
 involution of which M is the centre and P, P' 
 are a pair of conjugate points. The two circles 
 obviously cut orthogonally since CU 2 = CK. CP. 
 
 If Q, <2' a%e the graphs of a given pair of conjugate imaginary points, and a circle 
 be described on QQ' as diameter, any circle, with centre on MO the perpendicular 
 through the mean point of Q and Q', which cuts this circle orthogonally, determines 
 the given pair of conjugate imaginary points on PP'. This follows also from 
 Art. 29. 
 
 The circle, centre G, since its centre is on the radical axis of circles described on 
 the lines joining pairs of conjugate points of the involution as diameters and cuts 
 one of these circles orthogonally, cuts all circles of this coaxal system orthogonally. 
 Hence all circles with centres on MC, which cut one circle of the first system ortho- 
 gonally, form a coaxal system. If the distance of the limiting points of the first 
 system from M is V— 1A', the distance of the limiting points of the second system 
 from M is K. 
 
 This result may be put into a slightly different form in which it is an extension 
 of that given in Art. 82. 
 
 In this form it is as follows : 
 
 (2) (a) If two circles cut each other orthogonally, each determines inverse points 
 upon every diameter of the other. 
 
 (b) If one circle passes through inverse points with respect to another, they cut 
 orthogonally. 
 
 For if the circle, centre C, Which is orthogonal to the circle PKP', does not 
 meet PP' in real points,, it meets this line in a pair of conjugate imaginary points 
 which are harmonic conjugates of P and P' and, since PP' is a diameter of the 
 circle PKP', these points are a pair of imaginary inverse points with respect to 
 this circle. 
 
 The converse follows from the fact, that when the circle, centre O, passes through 
 the imaginary double points of the involution, it is orthogonal to all circles described 
 on the lines, joining pairs of conjugate points of the involution, as diameters. 
 
 The theorem may also be easily proved from a graphical figure. 
 
 (3) To construct the common harmonic conjugates of two pairs of collinear points, 
 either conjugate imaginary or real, determined as the intersections of a straight line 
 and a pair of circles. 
 
 Let the radical axis of the pair of circles meet the base in 0. The circle with 
 centre cutting the given circles orthogonally meets the base in the required points. 
 
46 
 
 The Imaginary in Geometry 
 
 This construction fails if is within the circles. In this case the two pairs of 
 points, say A, A' and B, B, are real and the segments overlap. In this case describe 
 on AA' and BE as diameters circles intersecting in P. On the perpendicular 
 from P on the given line take any point E outside the circles. The circle with 
 centre E cutting the two circles orthogonally will determine the pair of common 
 harmonic conjugates, which are in this case a pair of conjugate imaginary points. 
 
 From the nature of this construction it follows that there can be only one pair 
 of common harmonic conjugates of the two pairs of points. 
 
 Also the only case when the common harmonic conjugates of two pairs of collinear 
 points, real or conjugate imaginary, are imaginary, is when both pairs are real and 
 the segments overlap. 
 
 (4) Given a pair of points (P, P'), real or conjugate imaginary, as the points of 
 intersection of a straight line and a circle, to determine those harmonic&onjugates of 
 the pair, which have a given mean point M. 
 
 Describe a circle with centre M to cut the given circle orthogonally. This circle 
 determines the required points. 
 
 This construction fails if M is inside the given circle, in which case P and P' 
 are real and M is between them. Describe a circle on PP' as diameter (Figure of 
 Art. 31 (1)). Draw MC perpendicular to PP' and with centre C, any point on MC 
 external to the circle, describe an orthogonal circle. This meets PP' in the required 
 points Q and Q' which are imaginary. 
 
 32. (1) If through a real point P a real line be drawn to meet a circle in a pair 
 of imaginary points Q and Q' then PQ. PQ 1 is equal to the square of the tangent from 
 P to the circle. 
 
 If be the centre of the circle and M the foot 
 of the perpendicular from on the real line through 
 P, then the points Q and Q' are at imaginary 
 distances from M equal to V — \MT, where MT is 
 the tangent from M to the circle. 
 Hence 
 
 PQ .pq=(PM-i. MT) (PM+ i.MT) 
 =PM 1 +MT 2 
 
 = PM* + OM 2 ~(OM*-MT*) 
 = OP 2 -OT 2 =PZ 2 
 where PL is the tangent from P to the circle. 
 
 Certain important results follow from this 
 theorem. 
 
 (as) If a straight line meet a circle of a coaxal 
 system in a pair of imaginary points, these points are a pair of conjugate points of 
 the involution determined on the line by the system of coaxal circles. 
 
 This follows from the fact that, if the straight line meet the radical axis at P, 
 the squares of the tangents from P to all the circles are equal, and therefore the 
 products of the distances from P of the pairs of points in which the line meets the 
 circles of the system are equal. 
 
The Circle 47 
 
 (b) The ratio of the tangents from a variable point on a circle to two other circles 
 with which it is coaxal is constant. 
 
 Take P and P' any two points on the circle from which the tangents are drawn. 
 Join PP' to meet the other circles in Q, Q' and 
 R, R. Then P, P', Q, $', R, R' are pairs of con- 
 jugate points of an involution. Therefore 
 (PP'QR) = (P'PQR'). 
 
 Therefore PQ-PQ' _ P'Q- P'Q ' 
 inerelore pR pRI - piR plR ,- 
 
 Hence the squares of the tangents from P to 
 the two circles are in the same ratio as the 
 squares of the tangents from P'. 
 
 By taking P and P' on the line of centres it 
 follows at once that the ratio of the squares of 
 
 CC 
 these tangents is ~^ where C, C\ and C 2 are the 
 OO2 
 
 centres of the circles. 
 
 (c) Carnofs theorem holds for the circle even when one or more of the sides of the 
 triangle meets the circle in pairs of imaginary points. 
 
 This can be proved at once as in Art. [89]. 
 
 (2) If a chord QPQ' be drawn through a real internal point P to meet a circle 
 in Q and- Q', the square of the tangent (imaginary) from P to the circle is equal to 
 PQ.PQ. 
 
 Join the centre of the circle to P. Through P erect a perpendicular to OP to 
 meet the circle in L and L'. Then PQ.Pq=PL.PL'= -PL\ Draw RME the 
 
48 
 
 The Imaginary in Geometry 
 
 polar of P, which will meet the circle in a pair of conjugate imaginary points R and 
 R'. Then the square of the tangent PR is PR 2 and 
 PR?=PM 2 -RM 2 
 
 = PM 2 - MI?, since LPL is the polar of M 
 = -PL 2 =PQ.PQ'. 
 Hence generally if through a real point P any real chord be drawn to meet a 
 circle in Q and Q', PQ . PQ' equals the square of the tangent from P to the circle. 
 
 To draw a real circle through a real point A and an imaginary point P. 
 
 Let P' be the conjugate imaginary point of P. Then the line PP' is real. Let M 
 be the mean point of P and P' Draw MON perpendicular to PP'. Draw AN, the 
 perpendicular from A on MON. 
 
 Construct Pj and iY the graphs of P and P' On P 1 Pi as diameter describe 
 a real circle. Let CO, the radical axis of this circle and A, meet MN in 0. Then 
 is the centre of the required circle and, if OS 7 be the tangent from to the circle 
 on P-i.P-1, then OA = OT is the radius. 
 
 33. The points of intersection of two circles. 
 
 Two circles determine the same involution (a) on their radical axis, 
 and (b) on the line at infinity. 
 
 (a) Let P be any point on the radical axis of the circles. Then 
 the tangents from P to the circles are equal and, therefore, a circle 
 
 P 
 
 with centre P and radius equal to the length of these tangents cuts the 
 two circles orthogonally. 
 
 The chords TT' and NN' are the polars of P with respect to the 
 two circles. These lines are also radical axes of the circles taken in 
 pairs. Hence, since the three radical axes of three circles taken in 
 pairs are concurrent, TT and NN' meet in a point P' on the radical 
 axis of the first two circles. 
 
The Circle 49 
 
 Hence the conjugate of P on the radical axis with respect to both 
 the circles is P'. Hence the two circles determine the same involution 
 on their radical axis. 
 
 (6) Let G and C be the centres of the circles. Consider a point A 
 at oo . Its polar with respect to the 
 first circle passes through G, the pole of 
 the line at infinity, and is perpendicular 
 to GA. 
 
 Similarly, the polar of J. with respect 
 to the second circle is the line through 
 G' perpendicular to C'A. Therefore 
 these polars of A are parallel lines 
 through G and C". They therefore meet, 
 the line at infinity in the same point A'. 
 
 Hence the two circles determine the same involution on the line at 
 infinity. 
 
 i 
 
 34. (1) Every circle meets the line at infinity in the same pair of 
 conjugate imaginary points. 
 
 Connected with every pair of circles there are two chords on which 
 the circles determine the same involution. These are their radical axis 
 and the line at infinity. The radical axis is a different line for different 
 pairs of circles, but the line at infinity is the same line. On the line at 
 infinity every circle determines the same involution, namely, the invo- 
 lution obtained by drawing pairs of conjugate (i.e. orthogonal) diameters 
 through the centre. Hence every circle passes through the double 
 points of this involution, which since the involution is an overlapping 
 involution are a pair of conjugate imaginary points. These points are 
 termed the circular points at infinity, or the critical points. 
 
 The lines joining -the circular points at infinity to the centre of a 
 circle, which is the pole of the line at infinity with regard to the circle, 
 are imaginary tangents to the circle. These tangents are the double 
 rays of the involution pencil made up of pairs of conjugate (or orthogonal) 
 diameters of the circle. They are the critical lines through the centre. 
 (Art. 22.) . 
 
 (2) All conies through the circular points at infinity are circles. 
 
 Draw a circle through the circular points at infinity and a conic 
 through them which is supposed not to be a circle. Since these curves 
 h. i. a. 4 
 
50 
 
 The Imaginary in Geometry 
 
 intersect the line at infinity in the same pair of points they determine 
 the same involution on it. 
 
 Draw parallel diameters of the circle and the conic and their conju- 
 gate diameters. These pass through the same pairs of conjugate points 
 of the involution on the line at infinity. In the case of the circle, these 
 pairs of conjugate diameters are at right angles. Therefore the conjugate 
 diameters of the conic are at right angles. Therefore all the pairs of 
 conjugate diameters of the conic are at right angles and consequently 
 it is a circle. 
 
 35. If the self-corresponding rays of two superposed projective pencils are the 
 lines joining tlie vertex to the circular points at infinity the angle between pairs of 
 corresponding rays is constant. 
 
 Describe a circle through the vertex S to meet the rays of one pencil in A, B 
 C, ... and the corresponding rays of the x 
 
 other in A', E, C, . . . . Join A B and A'B 
 to meet at K and AC and A'C to meet 
 at L. 
 
 Then the self-corresponding rays of 
 the pencils are the lines joining the 
 points of intersection of KL with the 
 circle to S. (Art. £109].) 
 
 If these points are the circular points 
 at infinity, KL must be the line at 
 infinity, and therefore AB and A'B are 
 parallel, as are also AC and A'C. 
 
 Since AB and A'B are parallel the 
 arcs AA' and BB are equal. Hence the 
 angles A SA' and BSB are equal. Simi- 
 larly the angles between other pairs of 
 corresponding rays are equal to the angle 
 ASA'. 
 
 Conversely if the angles ASA', BSB, CSC are equal, the lines AB, A'B ; 
 AC, A'C; ... will be parallel in pairs and KL will be the line at infinity. This line 
 will meet the circle in the circular points at infinity, and therefore the self-corre- 
 sponding rays of the pencils are the connectors of the circular points at infinity to S.- 
 
 36. Every pair of circles intersect in the circular points at infinity 
 and in a pair of points on their radical axis which may be either a pair 
 of real points or a pair of conjugate imaginary points. 
 
 Hence their four points of intersection are either : 
 
 (1) Two pairs of conjugate imaginary points, 
 or (2) A pair of conjugate imaginary points and a pair of real points. 
 
 These form a semi-real quadrangle of the 1st or 2nd kind (Art. 17). 
 
The Circle 
 
 51 
 
 If in the 1st figure of Art. 17 
 A, A' are taken as the circular 
 points at infinity, il and O', and 
 B and B' as the other pair of 
 imaginary points of intersection of 
 the circles, then the line BB' is 
 real, viz. the radical axis. 
 *E 
 
 If in the 2nd figure of Art. 17 
 A, A' are taken as the circular 
 points at infinity, O and O', and 
 B and G as the real pair of points 
 of intersection of the circles, then 
 the line BC is real, viz. the radical 
 axis. 
 
 F and G are the real common 
 harmonic conjugates of the two 
 pairs of points in which the line 
 of centres meets the circles. (Art. 
 [84].) They are also harmonic con- 
 jugates of the points where -the 
 radical axis and the line at infinity 
 meet the line of centres. 
 
 E is the real point of inter- 
 section (at infinity) of the radical 
 axis and tne line at infinity. 
 
 EFG is a real triangle. 
 
 BB'FQ is a semi-real square of 
 the 2nd kind. 
 
 F and G are a pair of conjugate 
 imaginary points, namely, the com- 
 mon harmonic conjugates of the 
 points where the line of centres 
 meets the circles. (Art. [84].) 
 
 E is the real point of inter- 
 section (at infinity) of the radical 
 axis and the line at infinity. 
 
 The triangle EFG has a real 
 vertex • E and a real side FG. 
 F and G are a pair of conjugate 
 imaginary points and the lines 
 EF, EG are a pair of conjugate 
 imaginary lines. 
 
 BCFG is a semi-real square of 
 the 2nd kind. ; ; 
 
 4—2 
 
52 
 
 The Imaginary in Geometry 
 
 Hence every pair of circles have a common self-conjugate triangle 
 which may be real or semi-real. The vertices are the common inverse 
 points of the two circles and the point at infinity perpendicular to their 
 common diameter. The circles have also three pairs of common chords, 
 viz. (i) the radical axis and the line at infinity, (ii) the critical lines 
 through one common inverse point, (iii) the critical lines through the 
 other common inverse point. 
 
 37. Poneelet figure of two circles. 
 
 The fact that two circles always intersect in four points may be 
 illustrated by a Poneelet figure. 
 
 rH 
 
 fli 
 
 A 
 
 n 
 
 \ X 
 
 v>^ 
 
 / 
 
 // 
 
 \ \ 
 \ \ 
 \ \ ^-- — 
 
 A/ 
 
 // 
 
 
 
 
 \ \ yS^ 
 
 \ / / 
 
 // 
 
 \ y 
 
 x/ / 
 
 \\ 
 
 // 
 
 \ /\ 
 
 /\ ' 
 
 \\ 
 
 // 
 
 / \ ' 
 
 \\ _— 
 
 . // 
 
 \ 1 \ 
 V \ 
 
 / V 
 
 1 \ 
 
 / V 
 
 \* / 
 
 XI U 
 
 m In, / 
 
 D, )l, 
 
 A x 
 
 1 \ s 
 
 \ h 
 
 / X. 
 
 j4 \ 
 
 1 \ / 
 1 /\ 
 
 N. / \ 
 
 / / ^- - 
 
 ' \ \ 
 
 \y \ 
 
 / / 
 
 \\ 
 
 ?\ , \ 
 
 
 \^ 1 
 
 / / \_ 
 
 ^^^ N, \ 
 
 
 NX 
 
 / / — ' 
 
 \ V 
 
 dV 
 
 
 'a 
 
 at 
 
 *n 
 
 The above Poneelet figure of the curves is constructed by taking as 
 real axis the line joining the centres of the circles. The hyperbolae 
 (rectangular) intersect in two points A' and B! and the lengths MA' 
 and MB' give the imaginary coordinates of the points of intersection, 
 while MO and MO^ give the real coordinates. A'B' is of course the 
 radical axis of the circles, which is the real connector of the conjugate 
 imaginary points A' and B'. 
 
 It is obvious from the figure that the two circles determine the 
 same involution on their radical axis, viz. the involution of which M is 
 the centre and of which the constant is minus the square of the tangent 
 from M to either of the circles. 
 
 The principles on which the figure, is constructed are explained in 
 Chapter VI.' 
 
The Conic 53 
 
 EXAMPLES 
 
 (1) "If two circles cut each other orthogonally each determines inverse points 
 \ipon every diameter of the other.'' Prove this in the case where the orthogonal 
 circle meets the diameter in real points. 
 
 Give any justification of the extension of this theorem to the case in which the 
 diameter meets the orthogonal circle in imaginary points. (L. U. 1904.) 
 
 (2) " We have in the plane a special line, the line at infinity ; and on this line 
 twp special (imaginary) points, the circular points at infinity. A geometrical 
 theorem has either no relation to the special line and points and it is then descrip- 
 tive ; or it has a relation to them and it is then metrical." 
 
 Explain and comment on this statement. (L. U. 1904.) 
 
 (3) If the points P and Q are a pair of conjugate imaginary points, which are 
 also a pair of conjugate points with respect to a circle, prove that the line PQ meets 
 the circle in real points. 
 
 (4) Prove that any two straight lines at right angles are harmonic conjugates of 
 the lines joining their point of intersection to the circular points at infinity. 
 
 (5) Show that the three poles of a straight line with respect to the three pairs 
 of points of intersection of four given straight lines lie upon another straight line 
 conjugate to the first straight line with respect to each of the three pairs of points. 
 
 When the four given lines are the connectors of two given real points with the 
 circular points at infinity, construct the conjugate of a given straight line. . 
 
 (6) Show that every circle in a given plane may be regarded as passing through 
 the same two imaginary points at infinity. 
 
 (7) Prove that no two pairs of conjugate imaginary points can be pairs of 
 harmonic conjugates. 
 
 (8) Show that a circle which passes through a real point and a pair of conjugate 
 imaginary points is real. 
 
 - 38. Conic with a real branch. 
 
 Construction of the pair of conjugate imaginary points in which a 
 real line meets a conic. 
 
 The imaginary points in which a line meets a conic are the double 
 points of the involution which the conic determines on the line. 
 
 Let the conic determine on the line I an involution of which KK', 
 LI! are pairs of conjugate points. Let CD be the diameter parallel to 
 I and let its conjugate diameter meet the curve in A and B and I in 0'. 
 
 Then the pole of I is on ABs 
 
54 
 
 The Imaginary in Geometry 
 
 Hence (Art. [136]) 
 
 0'K.O'K' = 0'L.O'L' 
 
 = + m .O'A.O'B. 
 
 Therefore if M' and N' are the graphs of the double points of this 
 
 involution 
 
 O'C* _ O'M' 2 
 
 + 
 
 = 1, 
 
 GA^ CD" 
 
 where N' may be written for M' and the — or + sign must be taken 
 according as the conic is an ellipse or hyperbola. Hence the locus of 
 ■the graphs of the imaginary double points on systems of parallel chords 
 is a conic touching the given conic at A and B. 
 
 The figure so obtained may be termed a Poncelet figure. 
 
 39. Conjugate loci or Poncelet figures for a conic. 
 
 This result may be illustrated as follows : 
 
 (a) The ellipse. If I meet the conic in imaginary points M( and N{, 
 
The Conic 
 
 55 
 
 and AB, the diameter conjugate to the diameter parallel to I, in 0', 
 then 
 
 ilf/O' 2 _ CO'* 
 GJ> '" ~ CB* 
 
 = a positive quantity, since CO' > CB. 
 
 •(i) 
 
 Therefore J//0' must be a purely imaginary quantity. Let M' and 
 N' in the figure* be the graphs of the points M{ and i^'. Then 
 
 _ wo'* cjy* _ 
 
 CD 2 + ~CB>~ 
 
 Hence for a system of chords parallel to I the locus of the graphs is 
 a hyperbola touching the original ellipse at A and B. 
 
 (b) The hyperbola. If in the figure of (a) the hyperbola is given it 
 will be found in exactly the same way that the ellipse is the graph of 
 the points of intersection of chords parallel to CD with the hyperbola. 
 
 The parabola. If PB is any chord of the parabola, S the focus, and 
 AQ the diameter corresponding to PR, then 
 
 PQ a = 4 . A3 . AQ. 
 Draw a line I as in the figure parallel to PR. 
 
 * This figure is a reproduction of Figure (6) in Poncelet's Traits des Proprietes Projec- 
 t/ties des Figures. Paris, 1822. 
 
56 
 
 The Imaginary in Geometry 
 
 Then if i¥/ be on the curve, 
 
 M 1 '0' 1 =4,.AO'.AS, 
 where AO' is negative. 
 
 .-. MiO'^-i.O'A.AS. 
 Hence M^O' is imaginary. Let M' be its graph. Then 
 M'0' i = ^.0'A.AS. 
 
 Hence, if a point 8' be taken on SA such that S'A = AS, the locus 
 of M' for a system of chords parallel to I is a parabola equal to the 
 given one having its focus at 8' and touching the given parabola at A. 
 This parabola is the graph of the imaginary points of the parabola for 
 chords parallel to PR. 
 
 From the preceding it is seen that a Poncelet figure gives the 
 graphs of the intersections of a conic and a system of real parallel lines, 
 distances measured parallel to one direction being real and those parallel 
 to another direction being purely imaginary. In fig., page 55, the tangent 
 at M' to the graph represents a tangent to the imaginary branch. 
 
The Conic 57 
 
 It meets AB in a real point 0. This is the one real point on the 
 tangent at M'. It is the pole of the line M^N^. 
 
 The figure given in this Art. may be obtained from that of Art. 30 
 by projection from a real point on a parallel plane, since in such a 
 projection real lengths are projected into real lengths and imaginary 
 lengths into imaginary lengths. A more complete figure is given in 
 Art. 127. 
 
 40. The, following, which is a particular case of the anharmonic property of a 
 conic, may be deduced from Art. [150]. 
 
 If two fixed real points on a conic and a pair of conjugate imaginary points on 
 the same conic are joined to a variable real point on the conic, the pencil so formed is 
 cut by any real transversal in two real and two imaginary points whose anharmonic 
 ratio is constant. 
 
 Let A and B be any two fixed real points on the conic and X^X^, Y X Y 2 any two 
 pairs of conjugate points of the involution determined by the conic on any real line 
 in its plane. If the points A and B are projected from any point S on the conic 
 into the points A u B x on the real line, then, by Art. [150], for all positions of S 
 
 (MfhM™)}'« onstat . 
 
 Take X u A" 2 as the double point J£, and 1\, F 2 as the double point F, of the 
 involution X x , X%, Y it Y 2 . These may be the imaginary points in which the line 
 meets the conic. 
 
 Then {{AiEFB^-^FEB^Y is constant. 
 
 Let (A 1 EFB 1 ) = \. Then {2X - 1} 2 is constant. 
 
 Therefore X the anharmonic ratio of the pencil formed by joining the double 
 jioints of the involution — which may be any pair of conjugate imaginary points on 
 the curve — and the pair of real' points — A and B — to any real point on the curve is 
 constant. 
 
 In the case of a real conic it is seen that : 
 
 (1) No imaginary line can touch the conic at a real point and no real line can 
 touch the conic at an imaginary point on the conic. 
 
 A. tangent at a real point passes through two real points on the curve, and since 
 it passes thtough two real points it must be a real line. 
 
 No real line can touch a conic at an imaginary point, for as it meets the conic in 
 an imaginary point it must meet the conic in the conjugate imaginary point, and an 
 imaginary point and its conjugate can only coincide in a real point.. 
 
 (2) There is one real point on an imaginary tangent at an imaginary point. If 
 the imaginary point of contact is joined to the conjugate imaginary point on the 
 curve, a real line is obtained. The tangents at this pair of conjugate imaginary 
 points are conjugate imaginary lines and pass through the pole of their connector 
 which being a real point is the one real point on the two imaginary tangents; 
 
58 The Imaginary in Geometry 
 
 (3) If A, A' and B, B' be pairs of conjugate imaginary points in which real 
 chords from G meet a conic, it is seen from the property of the semi-real quadrangle 
 (Art. 17) that AB' . A'B and AB . A'B are real points on the line joining L and M 
 the harmonic conjugates of G with respect to AA' and BB', i.e. on the polar of G. If 
 the points A and B coincide and likewise A' and B', then the lines AB and A'B' 
 become the imaginary tangents at a pair of conjugate imaginary points and F the 
 pole of OAA' is on the polar of G. This is otherwise obvious. 
 
 Consider the inscribed quadrangle A, A', B, B' of which A, A' and B, B' are pairs 
 of conjugate imaginary points and EGF the real diagonal points triangle. The 
 tangents at A and A' intersect in a real point on EF as also do the tangents at B 
 and B'. These real points are the real points on these tangents. The tangents at 
 A and B intersect in an imaginary point, which is the pole of the imaginary 
 line FBA. 
 
 Join EG meeting FBA and FA'B' in R and S. Then since 
 (FRAB)=(FSA'B')=-1, 
 RS is the polar of F (Art. 18) and the imaginary tangents at A and B intersect in 
 an imaginary point on the real line EG. 
 
 Hence if through a real point an imaginary line is drawn to meet a conic in a 
 pair of imaginary points, the imaginary tangents at th%se points intersect- in an 
 imaginary point on the real polar of the real point. 
 
 41. Diameters of a conic. 
 
 Every real line through the centre of an ellipse or a parabola — the centre of the 
 latter curve being at infinity — meets the curve in real points. This is not however 
 the case with the hyperbola, A diameter may meet the curve in a pair of real 
 points A and A'. It may however meet the curve in a pair of conjugate imaginary 
 points B and B'. These are of course the double points of an overlapping involution 
 the centre of which is the centre of the curve. The lengths CB and OB' are imaginary 
 and (7S 2 =C J B' 2 = the product of the distances from C of the pair of equidistant 
 conjugate points of the involution, or the product of the distances of any pair of 
 conjugate points of the involution. In Art. [136], where the case of the hyperbola 
 was considered, B and B' were taken as the pair of equidistant conjugate points' of 
 the involution on BOB'. If however the imaginary semi-diameter CB be used to 
 determine B, it follows that in that Art. both for the ellipse and the hyperbola 
 
 XY. XY' = - XA .XA'.~ 
 
 CA 2 
 
 , XP 2 , CX 2 , 
 
 and _ + _ = 1 . 
 
 42. (i) IfPhe a fixed point through which a variable real line PO is drawn and CB 
 
 be the parallel semi-diameter of an ellipse or hyperbola, and its conjugate semi-diameter 
 
 PO 2 OC 2 
 C'A meets the line OP in 0, then -=5^ + —^ is constant. 
 , BC 2 AC 2 
 
 Describe through P a similar and similarly situated conic and let CA and CB 
 
 CA CB 
 meet this conic in A t and B v Then -sj-j- =-= r5 - = \ (a constant). 
 » C4j CxJx 
 
The Conic 
 
 59 
 
 Hence 
 
 PO 2 PC 2 
 
 BC 2 + CA 2 ~~ 
 
 1 (PO 2 0G 2 \ 1 , . , r/111 
 
 (ii) If P be a fixed point through which a variable real line PO is drawn to meet 
 an ellipse or hyperbola in imaginary points Q, q, and if CB be the semi-diameter 
 parallel to the line and its conjugate diameter 
 meet the line in and the polar of P meet the 
 line in P', then 
 
 PQ.pq po.pp' 
 
 pjx — = — TV™ — = a constant. 
 
 The points P, P' are a pair of conjugate 
 points of the involution of which Q and Q' are 
 the double points (imaginary) and is the 
 centre. Hence PQ . Pq equals PO . PP'. 
 
 Also 
 PO.PP' P0 2 -OP.OP' 
 
 CB 2 
 
 CB 2 
 P0 2 +OA.OA 
 
 ,0b 2 
 CA* 
 
 PO 2 
 
 CB 2 
 C0 2 -OA 2 
 
 (Art. 38) 
 
 PO 2 CO 2 
 
 CB 2 + CA 2 
 
 -1 
 
 CB 2 ' CA 2 
 = a constant (by (i)). 
 From this result it follows that : 
 
 (a) Carnots theorem holds, when the conic meets one or more sides of the triangle 
 in imaginary points. 
 
 (b) Newton's theorem (Art. [104 a]) holds when the points of intersection of the 
 chords with the conic are imaginary, also the deduction (i) holds in this case. Hence 
 the imaginary tangents from any real point to a conic are in the ratio of the parallel 
 semi-diameters. 
 
 These results follow from the extension of Carnot's theorem contained in (a) in 
 the same way that the corresponding results follow from Carnot's theorem. 
 
 (e) If a system of conies be described through four points, a conic of the system 
 which meets a straight line in imaginary points determines on the line a pair of con- 
 jugate points of the involution determined on the line by the conies of the system which 
 meet it in real points. 
 
 This follows from (6) by the method of Art. [101 (6)]. 
 
 (d) If three conies intersect in the same four points, the ratio of two tangents from 
 a variable point on one conic to the two other conies is ,in a constant ratio to the ratio 
 of the diameters of the two latter conies, which are parallel to the tangents. 
 
 Consider any two points P and P' on the conic from which the tangents are 
 drawn. Let PP' meet the other conies in Q, q and It, R'. Then PP', Qq, RR' are 
 pairs of conjugate points of an involution. 
 
 , PQ.pq _P'Q.P'q 
 
 Therefore 
 
 PR. PR' P'R.P'W 
 
60 
 
 The Imaginary in Geometry 
 
 Let d and I be the semi-diameters of the conies through Q, Q' and if, B' which 
 are parallel to PP' 
 
 PQ.PQ P'Q.P'Q' 
 
 d 2 d 2 
 
 Then _ _ _, = p ,„ plR , =& constant for different positions of P. 
 
 P P 
 
 Let t x and t 2 be tangents from P to the two conies and d\ and d 2 the semi- 
 diameters of these conies parallel to these tangents. 
 
 PQ.pq tj_ 
 
 Then 
 
 PR PR' = Ti =a ' cons * an ^ f° r different positions of P. 
 d? 
 
 I 2 
 
 h t? 2 
 
 — . ~f= a constant for different positions of P on the conic. 
 
 43. The corresponding theorems for the parabola are as follows : 
 
 (i) If P be a fixed point through which a variable real line PO is drawn, and the 
 
 diameter of a parabola at the point A, 
 
 at which the tangent is parallel to OP, 
 
 meet PO at 0, and S be the focus of the 
 
 , , ., P0 2 +4SA.0A . 
 
 parabola, then =-, is con- 
 
 &A 
 
 Draw an equal parabola through 
 the point P, having its axis in the 
 same straight line as the given para- 
 bola. Let S' be the focus of this 
 parabola and A' the point where OA 
 meets this parabola. 
 Then 
 P0*+4SA . OA 
 SA 
 
 ^ P0 2 +4S'A'.(A'A-A'Q) 
 
 SA 
 _ P0 2 -4S'A'. A'0+4:S'A'. A' A 
 SA 
 
 But by Art. [137] (1), PO^-iS'A . A'O is zero. 
 
 _, „ P0 2 +4SA.OA .,,,,.,. 
 Therefore =-j = 4A'A, which is constant. 
 
 (ii) If P be a fixed point through which a variable line PO is drawn to meet a 
 parabola in imaginary points Q, Of, and if A be the point of contact of the parallel 
 tangent, S the focus, and the diameter through A meets the line PO in and the polar 
 of P meets it in P', then 
 
 PQ.Pq PO.PP' 
 
 SA 
 
 SA 
 
 = a constant. 
 
The Conic 
 
 61 
 
 The points P and P' are a pair of conjugate points of the involution of which 
 Q and q are the double points (imaginary) and is the centre. Hence PQ.PQ! 
 equals PO . PP'. 
 
 But 
 
 PO.PP' PQZ-QP.QP' PQz + lSA.AO 
 
 SA 
 
 iSA 
 
 SA 
 = a constant by (i). 
 
 (Art. [137]) 
 
 Hence it follows : 
 
 _ (o) That Carnot's theorem holds for a parabola when one or more sides of the 
 triangle meet the parabola in imaginary points. 
 
 _ (6) Newton's theorem (Art. [104(a)]) holds for the parabola when the points of 
 intersection of the chords with the parabola are imaginary. Hence also the tangents, 
 real or imaginary, from any real point to a parabola are in the ratio of the square 
 roots of the distances of their points of contact from the focus. 
 
 44. If by means of common tangents the self-conjugate triangle of two conies 
 can be constructed, Art. 42 (ii) renders 
 the construction of chords of intersection 
 of the conies possible. 
 
 Let A be a vertex of the common 
 self-conjugate triangle and ABB' a com- 
 mon chord of the conies which passes 
 through A. Let C and C be the centres 
 of the conies and let AC, AC and CC 
 meet the conies in D, If, K and K'. Let 
 d and d' be the semi-diameters of the 
 conies parallel to ABB'. 
 
 Then 
 
 AB.AB' AC 2 - CD* AC 2 , 
 
 and 
 
 AB.AB' 
 d' 2 
 
 CD" CD 2 
 
 AC' 2 -CD' 2 AC' 2 
 
 cm 
 
 AC 2 
 d' 2 CD 2 ' 
 
 ■ cm 
 
 -1, 
 
 Similarly 
 
 d 2 AC' 2 
 
 cm 
 
 -1 
 
 OB. OB' OC 2 -CK 2 
 
 d 2 
 
 CK 2 
 
 PC 2 
 CK 2 ' 
 
 and 
 
 OB. OB' OC' 2 -CK' 2 
 
 d' 2 
 AC 2 
 d' 2 CD 2 
 
 CK' 2 
 
 -1 
 
 OC 2 
 
 -1 
 
 d 2 AC 2 
 
 cm 
 
 -1 
 
 CK' 2 
 
 This relation, combined with the fact that C0 + 0C'=CC, enables the values of 
 CO and CO to be found. 
 
62 
 
 The Imaginary in Geometry 
 
 45. Given two straight lines 
 a and b as the double rays real 
 or imaginary of a real involution, 
 to determine the two other pairs of 
 lines «!, bi and a 2 , & 2 connecting their 
 points of intersection with a conic. 
 
 Given two points A and B as the 
 double points real or imaginary 
 of a real involution, to determine 
 the two other pairs of points A l ,B 1 
 and J. 2 , B 2 in which the fangents 
 from these points to a conic intersect. 
 
 Let a and b meet the conic in 
 K, L, M, N. Let EFG be the 
 diagonal points triangle of this 
 quadrangle and let e, f g be its 
 sides. 
 
 Draw HH' a tangent at any 
 point H on the conic ; every point 
 on HH' is a conjugate of H with 
 respect to the conic. 
 
 Let the tangents be k, I, m, n. 
 Let efg be the diagonal triangle 
 of this quadrilateral and let E, F, G 
 be its vertices. 
 
 Draw h ■ any tangent to the 
 conic. Every line through its point 
 of contact is a conjugate of h with 
 respect to the conic. 
 
The , Conic 
 
 63 
 
 Let the conjugate of EH with 
 respect to a and b meet HH' in 
 H'. Then H' is the conjugate of 
 H with respect to another conic 
 through K, L, M, M, viz. the lines 
 a, b. Therefore H and H ' are con- 
 jugates with respect to every conic 
 through K, L, M, N, and therefore 
 with respect to the lines a lt b x and 
 a 2 , b 2 . 
 
 Hence the following is the 
 construction : 
 
 The line GF (e) is determined 
 as the polar of E. EG and EF are 
 determined as the common conju- 
 gates of the given involution and 
 of that determined by the conic 
 at.E. 
 
 The lines a 2 and b 2 are deter- 
 mined as lines through G harmonic 
 conjugates of f and e and also of 
 the lines joining H and H' to G. 
 
 Similarly c^ and b± are deter- 
 mined. 
 
 Let the conjugate of he with 
 respect to A and B be joined to the 
 point of contact of h by ti. Then 
 h' is the conjugate of h with respect 
 to another conic touching k, I, m, n, 
 viz. the points A, B. Therefore h 
 and h' are conjugates with respect 
 to any conic touching k, I, m, n, 
 and therefore with respect to i^ 
 jBj and A 2 , J5 2 . 
 
 Hence the following is the 
 construction : 
 
 The point gf(E) is determined 
 as the pole of e. eg and fe are 
 determined as the common conju- 
 gates of the given involution and 
 of that determined by the conic 
 on e. 
 
 The points A^ and B 2 are deter- 
 mined as points on g harmonic 
 conjugates of F and E and of the 
 points where h and h' meet g. 
 
 Similarly A 1 and B x are deter- 
 mined. 
 
 Particular case : 
 
 If A and B (on the right-hand side) are the circular points at 
 infinity, the construction for A lt B x and A 2 , B^ is as follows : 
 
 E, the pole of the line e (AB) now the line at infinity, becomes the 
 centre of the conic. 
 
 f and g, by the harmonic property of the quadrilateral, are harmonic 
 conjugates of the lines joining A and B to E and are, therefore, at 
 right angles. / and g (since efg is self-conjugate) are conjugate 
 lines with respect to the conic. They are therefore a pair of con- 
 jugate diameters of the conic which are at right angles, i.e. the axes 
 of the conic. 
 
64 The Imaginary in Geometry 
 
 A 1 and B x are harmonic conjugates of E and the point where / 
 meets the line at infinity. They are therefore at equal distances on an 
 axis of the conic from the centre. 
 
 h and -h! are conjugate lines of an involution whose double elements 
 are obtained by joining their point of intersection to A apd B, the 
 circular points at infinity. Therefore h and h' are at right angles and 
 h' is the normal at the point where h touches the conic. Hence A 1 and 
 Bi are harmonic conjugates of the points where any tangent and normal 
 meet the axis of the conic. They are, therefore, according to the usual 
 definition, a pair of foci of the conic. 
 
 Hence the following definition of the foci of a conic is arrived at. 
 
 The foci of a conic are the four points real or imaginary in which 
 the tangents from the circular points at infinity intersect. 
 
 A directrix is the polar of the corresponding focus. Hence a directrix 
 is the chord of contact of a pair of tangents from the circular points at 
 infinity to the conic. 
 
 Conjugate lines through a focus are at right angles. The lines joining 
 a focus to the circular points at infinity are tangents to the curve and 
 they are therefore the double elements of the involution on the line at 
 infinity, determined by pairs of conjugate lines through the focus. But 
 this must be an involution made up of pairs of lines at right angles, 
 because its double elements pass through the circular points at infinity. 
 Therefore the conjugate lines through a focus must be at right angles. 
 
 46. The foci may also be constructed by means of a Poncelet 
 figure. 
 
 (a) Let the curve be an ellipse. Form the Poncelet figure for the 
 major and minor axes looking upon lines parallel to the minor axis as 
 measured in imaginary units. The imaginary branch of the curve is a 
 hyperbola touching the ellipse at A and B, the ends of the major axis. 
 The tangents to these branches from the circular points at infinity are 
 lines inclined at angles of 45° to the axis AB, such lines replacing in 
 the graph those which are inclined at an angle tan -1 1 to the axis of x. 
 From symmetry, they must form a square with two vertices, F and F', 
 on the axis AB and two, F x and F^, on the axis BE. Looking 
 upon the hyperbola as real, F, F', F-^ and F^ lie on the director circle 
 so that the distances GF, GF', CF X and CFJ are each equal to the 
 square root of the difference of the squares of the semi-axes of the 
 hyperbola. As GF and GF' are real the points F and F' are real. As 
 
The Conic 
 
 65 
 
 CFy and CF-[ are expressed in imaginary units, F^ and F-! are imaginary 
 points. FFiF'Fi is a semi-real square of the second kind (Art. 27). 
 
 \\ \ 
 \\\ 
 s ^sX 
 
 S XN. 
 
 \\\ 
 \\\ 
 
 a \ 
 
 X 1 
 
 
 
 X / ^\/ 
 
 V X / ^\ 
 
 ./ X sx 
 
 
 
 V x ^>C 
 
 
 
 
 ^n 
 
 X 1 
 
 ^ V. 
 \^ X> 
 
 ^s. X *• 
 
 vx% 
 
 (6) Let the curve be a hyperbola. In this case the imaginary 
 branch is the ellipse of (a). Tangents making angles of 45° with the 
 major axis determine the foci, two on AB and two on ED. Their 
 distances from G will be equal to the square roots of the sum of the 
 squares of the semi-axes of the ellipse. Those on AB will be real and 
 those on DE imaginary. 
 
 47. The existence of the foci of a conic may be explained as follows. 
 Through a real point A (see figure, Art. 45) a pair of conjugate imaginary lines 
 oil and a 2 may under certain circumstances be drawn so as to touch a real conic. 
 
 Similarly from a second real point B a second pair of conjugate imaginary lines 
 &! and 6 2 ma 7 b e drawn to touch the same conic. 
 
 For the real points A and B a pair of conjugate imaginary points may be substi- 
 tuted. In this case a x and a 2 and also 6j and b 2 are no longer pairs of conjugate 
 imaginary lines, but a Y and b 2 are a pair of conjugate imaginary lines, as are also 
 a 2 and 6 1 . Hence the points aib 2 and a^ (A x and B t in the figure) are real. Also 
 the point B 2 given by aA is the conjugate imaginary point of the point A 2 given by 
 the lines a 2 and b 2 , which are the conjugate imaginary lines of «x and b t . Hence A 2 
 and B 2 are a pair of conjugate imaginary points. Hence the lines AB, AiB t and 
 A 2 B 2 are real. Also {A l B l EG)={A 2 B 2 EF)=(ABGF)= - 1, and therefore E is the 
 pole of AB. 
 
 h. i. a. 5 
 
66 The Imaginary in Geometry 
 
 Let GFbe the line at infinity. Then 2? is the centre of the conic and A l E=EB 1 
 and B 2 E=EA 2 . EG and EF being conjugate lines through E are a pair of conjugate 
 diameters. 
 
 Let A and B be the circular points at infinity. Then EG and EF, which are 
 harmonic conjugates of EA and EB, are at right angles. Hence, since a 1 and a 2 are 
 parallel, as are \ and b 2 , A x AJi x Bz is a semi-real square of the second kind (Art. 27), 
 and it is seen that the foci are two real points A Y and B x and a pair of conjugate 
 imaginary points A 2 and B 2 , such that EA t = iEA 2 . 
 
 48. Intersections of two conies. 
 
 (1) Two conies cannot intersect in more than four points. 
 
 If possible let them intersect in A, B, C, D, P. Then the pencil (P . ABCD) 
 must have the same anharmonic ratio for both conies. Therefore every point on 
 both conies subtends a pencil of the same anharmonic ratio at A, B, C, D. There- 
 fore the conies coincide. 
 
 (2) Every two conies with real branches have two real chords of intersection which 
 may be coincident. * 
 
 In Art. [125] it was shown that in the case of every pair of conies there are two 
 real lines (which may be coincident) on which the conies determine the same invo- 
 lution. The double points of these involutions are the four points of intersection of 
 the conies. Therefore, excluding the special case when the chords are coincident, 
 there are three 9ases for consideration : 
 
 (i) When the conies intersect in four real points, 
 
 (ii) When the conies intersect in two real and a pair of conjugate imaginary 
 points, 
 
 (iii) When the conies intersect in two pairs of conjugate imaginary points. 
 
 (i) In this case, the common inscribed quadrangle is real and its diagonal points 
 triangle is a real common self-conjugate triangle of the two conies. 
 
 (ii) In this case, two of the points of intersection are real and two a"re conjugate 
 imaginary points. They form a semi-real quadrangle of the second kind. One 
 vertex and the opposite side of the common self-conjugate triangle are real. The 
 other vertices are a pair of conjugate imaginary points and the other sides a pair of 
 conjugate imaginary lines. 
 
 (iii) In this case, the vertices of the quadrangle are two pairs of conjugate 
 imaginary points. They form a semi-real quadrangle of the first kind. The diagonal 
 points triangle of this quadrangle is real and is a common self-conjugate triangle of 
 the two conies. 
 
 49. (1) To construct graphically the imaginary points of intersection of two 
 conies with real branches. 
 
 (a) Let the conies intersect in a pair of real points. Then by Art. [125] the 
 real chord joining their pair of imaginary points of intersection can be constructed. 
 Let this chord be a. Let a' and a" be the diameters of the conies parallel to a. 
 Construct the diameters b and c of the conies which are conjugate respectively to 
 a' and a" (see figure, Art. 99). 
 
 For the conjugate diameters a' and b construct the graph of the first conic in 
 which imaginary distances are measured parallel to a'. 
 
The Conic 67 
 
 Similarly for the conjugate diameters a" and c construct the graph of the second 
 conic in which imaginary distances are measured parallel to a". 
 
 These two graphs intersect in the required points. 
 
 Let be the mean point of the two conjugate imaginary points of intersection 
 L and M. Then the imaginary coordinates of these points are measured from 
 along a. Since this is the case the diameters b and c must intersect on a at 
 the point 0. 
 
 (b) Let the conies intersect in two pairs of conjugate imaginary points. Then 
 by Art. [125] a pair of common real chords of the conies can be constructed. The 
 diameters of the two conies conjugate to the diameters parallel to these chords will 
 intersect in pairs on the chords in question and the two pairs of graphs correspond- 
 ing to these two pairs of conjugate diameters will intersect in the required points. 
 In the two graphs imaginary lengths must be measured parallel to the two common 
 chords. 
 
 (2) To construct the points of intersection of a conic, having a real branch, with 
 an imaginary straight line. 
 
 Denote the conic by S and the imaginary straight line by I. Let V be its 
 conjugate imaginary line. Then I and V are the double rays of a real overlapping 
 involution pencil. They will intersect the conic in two pairs of conjugate imaginary 
 points which form a semi-real quadrangle. The connectors of the pairs of conjugate 
 imaginary points are two real straight lines. These may be constructed by means 
 of Art. 45. 
 
 Construct the diameter of the conic parallel to either of these lines and its 
 conjugate diameter. The Poncelet figure for the conic may be constructed for these 
 diameters. 
 
 Construct the line parallel to this same line through the point of intersection of 
 the conjugate imaginary lines, and its conjugate in the involution which determines 
 the pair of conjugate imaginary lines. The Poncelet figure of the pair of imaginary 
 lines, for these lines, may be constructed (Arts. 76 and 133). The intersections of 
 the two figures in question give two of the four points of intersection of the conic 
 and the pair of imaginary straight lines. The other two points may be similarly 
 constructed. 
 
 50. Prom the preceding pages it will be seen that the effect of giving an inter- 
 pretation to the imaginary is to do away with restrictions, which are imposed in 
 ordinary geometry. One of these is that any two sides of a triangle must be greater 
 than the third. A triangle may, taking the imaginary into account, have one side 
 greater than the sum of the other two. Such a triangle may be constructed as 
 follows. 
 
 Let a and b be the two sides the sum of which is less than the third side c. 
 
 Let A and B be the ends of c. With centre A and radius 6 describe a circle, and 
 with centre B and radius a describe another circle. Let I be the radical axis of these 
 circles. The circles intersect in two points on I which are given as the points of 
 intersection of their hyperbolic branches described with axes parallel to AB and I. 
 Either of these points is a vertex of the required triangle. The distances of these 
 points from the point of intersection of AB and I are purely imaginary quantities. 
 
 5—2 
 
68 The Imaginary in Geometry 
 
 EXAMPLES 
 
 (1) If two conies have four imaginary points of intersection, show that they 
 have a real common self-conjugate triangle and one real pair of common chords. 
 
 (2) If A, A' be two paired elements of an elliptic involution, there is one and 
 only one other pair which divide 'AA' harmonically. Apply this to determine the 
 imaginary line joining two given coplanar imaginary points. 
 
 (3) Two hyperbolae have the same asymptotes : prove that they cannot intersect 
 
 (4) Show that the four directrices of a conic are chords of intersection of the 
 conic and its director circle. 
 
 (5) One side of a triangle is a real line, the other two meet in a real point, and 
 each passes through one of the circular points at infinity ; required the orthocentre. 
 
 (6) Three given conies touch the same pair of straight lines : construct the 
 conic which touches these lines and is such that the points of contact of any 
 common tangent to this conic and one of the given conies are conjugate points 
 with respect to the straight lines. 
 
 Examine the case when the straight lines pass through the critical points. 
 
 (7) Prove that the construction of Art. 37 holds for the imaginary points o 
 intersection of two similar and similarly situated ellipses. 
 
 Project on a parallel plane from a real point. 
 
 (8) If a pair of conjugate imaginary lines are tangents to a real conic, their 
 points of contact are conjugate imaginary points. 
 
 (9) Prove the following construction for the graphs of the imaginary points of 
 intersection of a straight line I (real) with a conic : 
 
 Let M be the pole of,/. Take M' the inverse point of M with respect to the 
 director circle, and let I meet MM' at 0. Then the circle through M, M', whose 
 centre is on the perpendicular to I through 0, determines the required points. (See 
 Gastrin's theorem, Art. [138].) Every circle through M, M' determines a pair of 
 conjugate points on I. 
 
 (10) In example (9) show that the circle whose centre is on the perpendicular 
 to I at and which cuts orthogonally the circle described on PP' as diameter, deter- 
 mines the points of intersection of the conic with I. 
 
 In a real plane Perspective. 
 
 [In the following the pair of conjugate imaginary points which correspond to 
 the circular points at infinity are termed the vanishing circules.'] 
 
 (11) Prove that if the centre of perspective S be the centre of a circle, S is the 
 focus of the corresponding conic and the vanishing line its directrix. 
 
 (12) Prove that a system of rectangular hyperbolae have for their plane perspec- 
 tives a system of conies which determine on the vanishing line an involution of 
 which the vanishing circules are the double points. 
 
The Conic 69 
 
 (13) Prove that the plane perspectives of a system of concentric circles form a 
 system of conies which have double contact at the vanishing circules. 
 
 (14) Deduce by (13) for a system, of conies having double contact at conjugate 
 imaginary points the properties which are proved in Arts. [130pand [131] for 
 conies having dbuble contact at real points. 
 
 (15) Prove that a system of coaxal circles and a system of confocal conies may 
 be looked upon as the correlatives of each other. 
 
 (16) From (15) deduce the properties of confocal conies set forth in Art. [140] 
 from those of coaxal circles. 
 
 (17) Prove that the plane perspectives of a system of similar conies is a system 
 of eonics, which cut the vanishing line in constant anharmonic conjugates of the 
 vanishing circules. 
 
CHAPTEK III 
 
 ANGLES BETWEEN IMAGINARY STRAIGHT LINES. MEASUREMENT 
 
 OF IMAGINARY ANGLES AND OF LENGTHS ON IMAGINARY 
 
 STRAIGHT LINES 
 
 51. Imaginary lines and imaginary angles. 
 
 On reference to Art. 1 it will be seen that a real line contains : 
 
 (1) An infinite number of real points determined by their real 
 distances from some given base point. 
 
 (2) An infinite number of purely imaginary points determined by 
 their purely imaginary distances from the base point. 
 
 (3) An infinite number of infinite systems of imaginary points, 
 whose determining distances with reference to the base point are 
 complex quantities. Each infinite system may be obtained by measuring 
 purely imaginary lengths from some real point of (1) or by measuring 
 real lengths from some purely imaginary point of (2). 
 
 The base point may be either real or imaginary. Any point, real or 
 imaginary, on the line may be taken as this point. Points, real, purely 
 imaginary or complex, are such with reference to the base point. If a 
 different point, real or imaginary, be taken as base point the nature of 
 certain of the points considered will be different. In itself however the 
 base point is neither real nor imaginary. 
 
 The point at infinity on the line is of the same nature as the base 
 point. It can be regarded as either real or imaginary and it belongs to 
 the real and to the imaginary system of points. The determining 
 distance of the base point is and that of the point at infinity on the 
 
 line jr. These quantities are of the same nature. If it is conceivable 
 
 to divide an infinite length into finite portions, then the length from 
 the base point to infinity may be regarded as divided into an infinite 
 number of real units of length and also into an infinite number of purely 
 imaginary units of length. 
 
 Consider any pair of real lines s and s', which intersect in a real 
 point 8. Rotate the line s' round S till s and s' coincide, i.e., through 
 
 an angle s's. In this way the two straight lines are made to coincide, 
 as do also the systems of points, real and imaginary, on them. 
 
Imaginary Angles 7 J 
 
 Consider two imaginary points which do not lie on the same real 
 straight line. By Axiom I. they have definite positions. Hence the line 
 joining them has a definite position (see Art. 11) and the points, since 
 they have definite positions, are at some definite distance apart. This 
 distance generally must be a function of the real and imaginary quan- 
 tities, by which the positions of the points are determined, but at present 
 no attempt is made to define or measure this distance. It follows how- 
 ever that there is a measure of this distance. Hence, as along a real 
 line, distances real and imaginary can be measured along an imaginary 
 line. A base point can be taken, which may be the real point on the 
 line, and from it real and purely imaginary lengths can be measured, and 
 from the points so determined purely imaginary and real lengths may in 
 their turn be measured. Therefore the nature of the systems of points on 
 an imaginary line is the same as on a real line. The difference between a 
 real and an imaginary line does not lie in the nature of the points on- the 
 lines in regard to themselves nor in the lines themselves, but in respect to 
 the relation of the lines to other lines and to points which are not situated 
 on the lines. In fact all lines real or imaginary have the same characteristics. 
 
 Consider two imaginary straight lines s and s' in the same plane. 
 They intersect in a point A, which is generally imaginary but may be 
 the real point on both lines. Consider A as the base point of systems 
 of points on the lines s and s. These lines s and s'have by Art. 11 
 definite positions. 
 
 It is now assumed that by a rotation of s round the point A some 
 point of s' (real with respect to A) can be brought into coincidence with 
 some point on s (real with respect to A). If this be done, the straight 
 lines s and s in the new position of s' must coincide, for (Art. 11) no 
 two different straight lines real or imaginary can join the same pair of 
 points. The measure of the amount of rotation necessary to bring the 
 lines s and s' into coincidence is termed the angle between s and s' in their 
 original position. The measure of this, whatever system of measurement is 
 used, must as a general rule depend on imaginary lengths and being a 
 function of such lengths is termed an imaginary angle. The assumption 
 made in the preceding may be embodied in a second axiom as follows :. 
 
 Axiom II. Hither of two given straight lines, real or imaginary, 
 may be superposed on the other by a motion of rotation through a definite 
 angle about their point of intersection*. 
 
 * Hereafter it will be seen that there is an apparent exception to the principle laid 
 down in this axiom, see Art. 78. 
 
72 The Imaginary in Geometry 
 
 The rotation may be in a positive or a negative direction. After 
 the line s' has been brought into coincidence with the line s, it may be 
 further rotated round A so as to come into coincidence with a third 
 
 line s", which passes through A. Hence if ss' denote the angle between 
 the lines s and s', it follows that 
 
 7s" = s's + ss". 
 Hence angles, real or imaginary, at a 1 point may be measured from a 
 base line through the point, in the same way that distances, real or 
 imaginary, can be measured along a straight line from a base point on 
 the line. 
 
 Note. (1) It does not follow that, if A is a real point and * and s' are imaginary 
 lines, the same rotation round A which brings s' into coincidence with s will bring the 
 conjugate imaginary line of s* into coincidence with the conjugate imaginary line of s. 
 The angle s's will usually be an imaginary angle and its measure will involve "%.'' 
 To bring the conjugate imaginary line of *' into coincidence with the conjugate 
 imaginary line of s the angle of rotation must be measured by a quality, which is the 
 imaginary conjugate of the measure of s's, i.e., the rotation must be through the 
 conjugate imaginary angle of s's, if such a term may be used. After an imaginary 
 displacement along a straight line points which had been conjugate imaginary points 
 cease to be so. This is also the case after a rotation through an imaginary angle. 
 
 (2) The coefficients in an analytical equation in x and y perform a double duty : 
 (i) They determine the dimensions of the curve and incidentally its nature, 
 (ii) They determine its position. 
 
 All curves, which are of the same nature and have the same dimensions, may be 
 looked upon as the same curve. Thus all ellipses with semi-major and semi-minor 
 axes a and b are the same curve, only displaced by a motion of translation, of say 
 the centre, and a motion of rotation, of say the major axis. Invariants (geometrical) 
 are of course functions of the quantities which give the dimensions of the curve. 
 
 In the equation of a real straight line the coefficients are entirely employed to 
 determine position. All straight lines are therefore the same straight line displaced 
 by a motion of translation of some point on the line and a motion of rotation round 
 some point. 
 
 That this is the case with imaginary as well as with real straight lines is the 
 assumption of Axiom II. 
 
 , 52. Without at present attempting to define the measure of an 
 imaginary angle there are certain consequences of the preceding which 
 may be noticed. 
 
 (1) The angle between two imaginary lines depends on lengths 
 some of which as a general rule are real and some imaginary. There- 
 fore in whatever way the measure of this angle is expressed it must be 
 
Imaginary Angles 73 
 
 ¥ 
 
 of the form ufcti, where a is the part of the angle which can be con- 
 structed as a real angle and a* the part which depends on the imaginary. 
 It is clear that no imaginary angle can equal a real angle. The angle a,- 
 must not however be confused with the angle i . a. 
 
 (2) If the angle which an imaginary line makes with a real line 
 be a + o; then the angle which its conjugate imaginary line makes with 
 the same real straight line is a — a { . 
 
 (3) The internal and external bisectors of the angle between a pair 
 of conjugate imaginary lines are real. (See also Art. 66.) 
 
 For if the lines make angles a + cti and a — a, with any real line 
 through their point of intersection, their bisectors make angles 
 
 a + « f + a — «j , a + a, + a — Oi + ir 
 ___ and i 
 
 with the same straight line. 
 
 But these angles are a and a + „ which are real. 
 
 Hence it follows that if the angle between a real line and an imaginary 
 line, which meets it in a real point, be expressed in the form a + a;, then 
 a is a measure of the real angle between the real line and one of the 
 bisectors of the angle between the imaginary line and its conjugate ima- 
 ginary line. (See Art. 66.) 
 
 (4) The sum of the angles of an imaginary triangle is ir. 
 
 (5) The following among other elementary theorems given in Hall 
 and Stevens' Geometry hold when the lines mentioned in the enuncia- 
 tions are imaginary, viz., 1, 2, 3, 4, 6, 17 and 18. 
 
 Among these are the following : 
 
 (a) The vertical and opposite angles between lines, real or ima- 
 ginary, are equal. , 
 
 (6) The sum of the angles at any point on the same side of a line, 
 real or imaginary, is two right angles. 
 
 (c) The external angle of any triangle equals the sum of the internal 
 and opposite angles. 
 
 EXAMPLES 
 
 (1) Prove that the liney (5+ ib')-sc (a+ib') =0 must be turned about the origin 
 through an angle 6, where cot #= "?,_, , - i ", , , to change it into the real line 
 yb-aos=&. 
 
74 The Imaginary in Geometry 
 
 (2) Prove that the bisectors of the angle between the pair of conjugate imaginary 
 lines y 2 + mV = are the pair of real lines xy=0. 
 
 This follows from the fact that the bisectors of the angles between the lines 
 ax 2 +2Lvy + by 2 =0 are given by the equation h (# 2 - y 2 ) = (a -b) xy. 
 
 53. Parallel straight lines. 
 
 Consider a system of points, real and imaginary, on a real line v' in a 
 plane a (cf. Art. [20], etc.). This line may be projected from a real 
 centre S into the line at infinity in a plane a. Through each of the 
 real points on v' pass an infinite number of real and an infinite number 
 of imaginary straight lines. To the real straight lines correspond in o- 
 a system of real parallel straight lines passing through the same point 
 at infinity. The imaginary straight lines will also pass through this 
 point at infinity and, in view of the fact that they do not intersect the 
 system of real lines in any points at a finite distance, they may be 
 regarded as forming a system of lines parallel to themselves and to the 
 real system. 
 
 Through each imaginary point on v' an infinite number of imaginary 
 straight lines pass. These correspond to a system of imaginary straight 
 lines in cr, which all pass through the same imaginary point at infinity. 
 Since these straight lines do not intersect in points at a finite distance 
 they too may be termed a system of parallel imaginary straight lines. 
 Such a system of parallel imaginary lines intersect at an imaginary point 
 at infinity and the one real line of the system is the line at infinity. 
 
 The angle between a pair of straight lines, real or imaginary, which 
 meet at infinity must be infinitely small. Hence, since the sum of the 
 angles of all triangles is equal to 77-, parallel straight lines, whether real or 
 imaginary, make equal angles with every straight line, real or imaginary, 
 in their plane. 
 
 The following among other elementary theorems in Hall and Stevens' 
 Geometry hold, when the lines mentioned in the enunciations are 
 imaginary, viz., 13, 14, 15, 20 and 21. . 
 
 Consider a system of parallel lines, real and imaginary, which pass 
 through a real point at infinity. (Figure, Art. 55.) 
 
 Let a real line RR' of the system meet two real lines OL and OM 
 in R and R', OL being perpendicular to the system, and let an imaginary 
 line of the system meet the same lines in imaginary points Q and Q'. 
 
 00' OR' 
 
 Then, by Art. 10, regarding 8 as being at infinity -^L = -~td = cos @> 
 
Trigonometry of the Imaginary 
 
 75 
 
 where 8 is the real angle between OL and OM. Hence the ratio 
 
 is real. 
 
 oq 
 
 54. Perpendicular lines. 
 
 (1) Through every imaginary point a straight line can be drawn 
 perpendicular to a real line. 
 
 Take any imaginary line through the imaginary point and take the 
 real point on this line. Through it draw a perpendicular to the real 
 line. This line is real. Join the point at infinity on it to the given 
 imaginary point. This is the required line. 
 
 (2) Through every real point a straight line can be drawn perpen- 
 dicular to a given imaginary line. 
 
 Join the given point to the point at infinity on the imaginary line. 
 Let this line be I. Find the line a the harmonic conjugate of I with 
 respect to the lines joining the given point to the circular points at 
 infinity. Then a is the required line. (See Art. 22.) 
 
 (3) Through every imaginary point a straight line can be drawn 
 perpendicular to a given imaginary line. 
 
 Draw any imaginary line through the given point. Take the real 
 point on this line. Draw by (2) a perpendicular through it to the 
 imaginary line. Take the point at infinity on this line — which will 
 usually be imaginary — and join this point to the given imaginary point. 
 This is the required line. 
 
 55. Projection of an imaginary length measured along a real 
 line upon another real line. 
 
 Let P and Q be any two 
 imaginary points upon a real 
 straight line OPQ and let any 
 other real line OL make an angle 
 6 with OPQ. 
 
 Through P and Q draw any 
 two imaginary lines perpendi- 
 cular to OL to meet it in P' 
 and Q'. Then P'Q' is termed 
 the projection of PQ on OL. 
 
 Take R any real point on OPQ and draw RR' perpendicular to OL 
 to meet OL in the real point R'. 
 
76 
 
 The Imaginary in Geometry 
 
 Then, by Art. 10, %: ^ = ^f- 
 
 OR 
 
 = cos 0. 
 
 OP OQ PQ OR 
 Therefore FQ' = PQ. co.s 0. 
 
 If A, B be any two real points on a real straight line and P any 
 imaginary point on the same straight line, the sum of the projections of 
 AP, PB, BA, on any real line is zero. 
 
 If a real line I make an angle with the given line, the sum of the 
 projections in question is 
 
 AP cos + PB cos + BA cos 0. 
 But since, Art. 3, BA = BP + PA, this expression is zero. 
 
 Similarly if A, B, G be any three points on a real line, the sum of 
 the projections of AB, BO, CA on any other real line is zero. 
 
 56. Definition of 
 
 (1) the measure upon a real straight line of an imaginary length 
 along an imaginary line; 
 
 (2) the measure of a length along an imaginary straight line ; 
 
 (3) the sine, cosine and tangent of the angle between a real and an 
 imaginary straight line. 
 
 y o/ 
 
 W 
 
 Let P and Q be any pair of imaginary pointsj the real lines through 
 which, viz., OP and OQ, contain an angle m. 
 
 Let any real straight line SB A meet PQ, OP, and OQ in S, A, and 
 B respectively. Draw straight lines through P, Q, and perpendicular 
 to 8BA to meet it in P', Q', 0'. Of these P' and Q' are imaginary. 
 Let 0! and <£ 2 be the angles that PO and QO make with SBA and let 
 be the angle PSP'. The angle is imaginary. 
 
 Then OQ cos <f> s - OP cos &= O'Q' -O'P' = P'Q' and is defined as 
 the measure of PQ on SBA. 
 
Trigonometry of the Imaginary 77 
 
 Let P"Q" be the measure of PQ on a line, perpendicular to SBA, 
 then 
 
 OQ sin ft - OP sin & = P"Q". 
 Therefore 
 
 P'Q' 2 + P"Q"* = (OQ cos ft - OP cos ft) 2 + (OQ sin ft - OP sin ft) 2 
 = OP 2 + OQ 1 - 2 . OP . OQ (cos ft cos ft + sin ft. sin ft) 
 = 0P*+0Q*-2. OP.OQ cos w. 
 This expression, which is independent of the position of SBA, is 
 denned as the square of the measure of PQ : 
 
 cos 8 is defined as the ratio of the measure of PQ on SBA to the 
 measure of PQ. 
 
 sin 6 is defined as the ratio of the measure of PQ on a line perpen- 
 dicular to SBA to the measure of PQ. 
 
 tan 8 is defined as the ratio of sin 6 to cos 8 provided always that the 
 measure of PQ is not zero. 
 
 Hence* cosfl= , OQ cos ft - OP cos ft 
 
 VOP 2 + OQ 2 - 2 .OP.OQ cos <o 
 OQ sin ft — OP sin ft 
 
 sin# = 
 tan# = 
 
 V0P 2 + OQ 2 - 2 .OP.OQ cos to ' 
 OQ sin $ 2 — OP sin ft 
 
 OQ cos ft — OP cos ft ' 
 It follows at once from the definition that 
 sin 2 6 + cos 2 = 1. 
 
 57. To prove that the tangent of an angle between a real line and 
 an imaginary line has the same value whatever imaginary points Pyind 
 Q (Art. 56) are chosen to determine its value. 
 
 Let the imaginary line be SPQ and the real line SL, and let the 
 angle between them be 0. Let T be the real point on SPQ. Draw 
 through T a parallel TBA to SI^. Then the angle PTA equals the 
 angle 8 and it is obvious from the definition that tan 8 as obtained 
 from the angle QTA is the same as that obtained from the angle QSL. 
 
 Let the real lines through P and Q meet TA and a line perpendicular 
 to TA through T in A and A' and in B and B'. Let these lines inter- 
 sect at 0. Draw the imaginary lines PP', PP", QQ', QQ" through P and 
 Q perpendicular to TA and TA'. Let the angles PAT and QBT be ft 
 
 * There is an ambiguity in the sign of the denominator of these expressions, which is 
 considered hereafter. v 
 
78 
 
 The Imaginary in Geometry 
 
 PP' 
 
 and tf) 2 . If ft when determined by P and T, be ft then tan ft is „„p . 
 
 If ft when determined by Q and T, be ft then tan ft is 
 
 Therefore tan ft = if,* 1 * and tan ft = ^^ . 
 PA cos <^ Qif cos $ 2 
 
 Q"G" 
 
 P' Q' 
 
 Therefore tan ft = sin & cos & PA QF 
 
 tan ft sin # 2 cos ^ ' PA' ' QB 
 OB OA' PA OB 
 OA'OB'-PA'- QB 
 _ {BO BQ\ (AV A'P \ 
 ~\B'0 : B'q)\AO : APJ 
 = (BB'OQ) (A'AOP) 
 
 PP' 00' 00' — PP 
 Therefore tan ft = tan ft = -p-p = -^ = Q l ,Q_ p ,> p = the tangent 
 
 of the angle determined by P and Q. Since, if P is fixed, Q may be 
 any point on the line, the result follows. 
 
 Hence also sin 6 and cos 9 are independent of the positions of P and 
 Q on the given line. 
 
 It follows from the definition (Art. 56) that 
 
 (1) If either of the lines SPQ or TBA is moved parallel to itself 
 the values of sin 9 and cos 9 are not altered. (See Art. 53.) 
 
Trigonometry of the Imaginary 
 
 79 
 
 (2) If the usual convention as to the sigh of a real angle is applied 
 to imaginary angles 
 
 sin (tt — 6) = sin 8, cos (ir — 8) = — cos 0. 
 
 (3) If 8 and 0' are the angles which an imaginary line OG makes 
 with a real line A OB then sin 8 = sin 8'. (See Art. 53.) 
 
 (4) If0 = O, sin0 = Oandcos0 = l. 
 If = ^,sin0 = l andcos0 = O. 
 
 Also sin (8 + 2tt) = sin 8 and cos (8 + 2tt) = cos 8. 
 (5) sin f 8 + ^ j = cos 8 and cos f 8 + ^ J = — sin 8. 
 
 58. TAe sum of the measures of the sides of any plane figure on a 
 real line is zero. 
 
 Consider any triangle ABG. Let the real lines through A, B, G 
 form a real triangle A'B'C. Let A", B", C" be the projections of A, B, 
 G on any real line s. It is necessary to prove that 
 
 A"G"+G"B"+B"A" = 0. 
 Let fa, fa, fa be the angles, real, which the sides of A'B'G' make with s. 
 
 <A) 
 
 
 / 
 
 \~ ~~--~-V) 
 
 
 
 (C5 
 
 / _____ 
 
 i^ML 
 
 
 03^ 
 
 (C 
 
 (/ 
 
 ' (B) 
 
 
 s 
 
 Then 
 
 B"C" = A'Gcos fa - A'B cos fa, 
 A"G" = B'G cos fa - B'A cos fa, 
 B"A" = G'A cos fa - G'B cod fa. 
 
80 The Imaginary in Geometry 
 
 Therefore A"C" + G"B" + B'A" 
 
 = cos & (G'A + AB') + cos tj> 2 (A'B + BC) + cos cf> 3 (FC + 04') 
 
 = cos £, (C.B') + cos <f>, (A'C) + cos <£ 3 (B'A') 
 
 = the sum of the projections of the sides of a real closed figure 
 
 = zero. 
 
 It should be noticed that 
 
 G'B' sin & + A'C sin </> 2 + B'A' sin 0, 
 
 is the sum of the projections of the sides of A'B'C on a perpendicular 
 line and is therefore zero. 
 
 59. Triangle with real lines for two of its sides. 
 
 (1) Let A and B be two imaginary points on real straight lines 
 CA and CB, which intersect at G at 
 
 an angle co. Let the angles GAB and 
 
 GBA be a. and /3. Suppose also that A! 
 
 the measure of AB is not zero and y 
 
 that this measure is -denoted by AB. 
 
 Taking the measures of the sides 
 of the triangle on GA, 
 
 AB cos a = CA — CB cos w. 
 
 Taking the measures of the sides /&> /?/ 
 
 of the triangle on a line perpen- '® 
 
 dicular to C4, 
 
 .42? sin a = CB sin to. 
 
 Therefore AB* = CA* + CB' - 2 . C4 . CB cos eo, (1) 
 
 also CB 2 = CA>+AB*- 2. C4 MB cob o : (2) • 
 
 Similarly CA*= BA* + BC*-2.BA .BCcos (3) 
 
 (2) If a, b, c are the measures of the sides opposite respectively to 
 the angles at A, B, C, then, taking measures on lines perpendicular 
 to the sides, 
 
 a sin co = c sin a and 6 sin a> = c sin y8. 
 
 Therefore - — = = -; — - . 
 
 sm a sin co sm p 
 
 Hence in this case the measures of the sides of a triangle are 
 proportional to the sines of the opposite angles provided none of the 
 measures are zero. 
 
Trigonometry of the Imaginary 81 
 
 60. Formulas for expressing the sine and cosine of the difference — 
 when real — of two imaginary angles in terms of their sines and cosines. 
 
 Let an imaginary line ABT meet two real lines OA and OB, which 
 intersect at at an angle co, in A » 
 
 and B, and let the angles OA T and 
 
 OBT be fa and fa respectively. Then Z^' 
 
 fa — fa = » (Art. 52). 
 
 Take the measures of AB and 
 BO on AO. 
 
 A0 = AB cos fa + BO cos a>. 
 
 Bi , D OA . 
 But AB = — — r sin co, 
 
 sin fa / 
 
 j nz> OA . , 
 
 and OB = —. — - sin fa . ,' 
 
 sin fa <f 
 
 Therefore sin </> 2 = sin &> cos <£ x + sin fa cos « (1) 
 
 Take an imaginary line A'B' at right angles to AB meeting OA 
 
 IT 
 
 and OB in A' and £'. Then for <£j and <£ 2 may be substituted X + ^ 
 
 7T ... 
 
 and $2 + -o- Hence in a similar manner 
 
 sin ( c/> 2 + g J = sin co cos f c/h + ^J + sin ( fa + -r j cos co. 
 
 Therefore cos c/> 2 = — sin co sin fa + cos c/> x cos co (2) 
 
 Multiply (1) by sin fa and (2) by cos fa and add. 
 
 Then cos co = cos (fa — fa) = cos fa cos fa + sin fa sin fa. 
 
 Similarly 
 
 sin co = sin (fa — fa) — sin fa cos c^j — sin fa cos c/> 2 . 
 For these results to be true it is necessary that the measures of AB 
 and A'B' should not be zero. 
 
 61. ' Definition of 
 
 (1) The measure on an imaginary line of an imaginary length along 
 an imaginary line and 
 
 (2) Of the sine, cosine and tangent of the angle between two imagi-* 
 nary straight lines. 
 
 Construct a similar figure to that in Art. 56 but let the line SB A 
 be an imaginary line. 
 
82 The Imaginary hi Geometry 
 
 In this case the points B, A, and 0' are imaginary as are the angles 
 ■0! and 2 , while is the imaginary angle between two imaginary' 
 straight lines. 
 
 Then as before OQ cos 2 - OP cos t = O'Q' - O'P = P'Q' is de- 
 fined as the measure of PQ on SBA. 
 
 Let P"Q" be the measure of PQ on a line perpendicular to SB A. Then 
 
 OQ sin 2 - OP sin X = P"Q". 
 Therefore 
 P'Q' 2 + P"Q" 2 = (OQ cos 2 - OP cos 0J 2 + (OQ sin 2 - OP sin 0O 2 
 = OP 2 + OQ 2 - 2 . OP . OQ (cos 2 cos fa + sin 2 sin X ) 
 = OP 2 + OQ 2 - 2 . OP . OQ cos <b (Art. 60). 
 This expression which is independent of the position of SB A is the 
 same as that found in Art. 56 which has been defined as the square of 
 the measure of PQ. 
 
 cos is defined as the ratio of the measure of PQ on SBA to the 
 measure of PQ. 
 
 sin is defined as the ratio of the measure of PQ on a line perpen- 
 dicular to SBA to the measure of PQ. 
 
 tan is defined as the ratio of sin to cos provided always that 
 the measure of PQ is not zero. 
 
 n OQ COS 2 — OP COS 0j 
 
 Hence cos = Z - • 
 
 slOP* + OQ' - 20P . OQ cos io 
 
 „ _ OQ sin 2 — OP sin X 
 
 ~~ \/OP* + OQ 2 - 20 P . OQ = coT« ■' 
 
 a_ 0Q s i n <fra - OP sin 0i 
 — OQ cos 02 — OP cos 0! " 
 
 It follows at once from the definitions that 
 
 sin 2 + cos 2 0=1. 
 
 To prove that tan0, where .0 is the angle between two imaginary 
 straight lines, is the same whatever pair of points on one of the imaginary 
 lines is taken to determine tan 0. 
 
 This may be proved in the same way as the corresponding theorem 
 is proved in Art. 57, the angles 0j and 2 being in this case angles 
 between a real and an imaginary line. 
 
 Similarly sin and cos are independent of the positions of P and Q. 
 
Trigonometry of the Imaginary 83 
 
 It follows from the definition that 
 
 (1) If either of the lines SPQ or TBA is moved parallel to itsel 
 • the values of sin and cos are not altered. (See Art. 53.) 
 
 (2) If the usual conventions as to sign are applied to imaginary 
 angles 
 
 sin {it — 8) = sin 0, cos (ir — 0) = — cos 0. 
 
 (3) If and 8' are the angles which an imaginary line OG makes 
 with an imaginary line AB, then sin 8 = sin &. (See Art. 53.) 
 
 (4) If = 0, sin0 = O, cos = 1. 
 If = ^, sin 0=1, cos = 0, 
 also sin (0 + 2tt) = sin 0, cos (0 + 2ir) — cos 0. 
 
 (5) sin (0 + |" J = cos 0, cos (0 + ^)=- sin 0. 
 
 The sum of the squares of the measures of PQ on two imaginary 
 lines at right angles gives the square of the measure of PQ. 
 
 Particular case. If'the imaginary points P, Q are on the same real 
 line, or are real, it is seen that the point lies on this line and that 
 the measure on an imaginary line of a real or imaginary length PQ, 
 which lies along a real line, is the length PQ multiplied by the cosine of 
 the angle between PQ and the imaginary line. 
 
 This is the same as saying that 
 
 SQcos8-SPcos8 
 PQ = cos * 
 
 But SP and SQ are already defined and so is the angle. Hence, as 
 is necessary, an identity is arrived at. 
 
 62. The sum of the measures of the sides of any closed plane figure, 
 real or imaginary, on any line is zero. 
 
 Consider the figure of Art. 58 but regard the line s as imaginary. In 
 this case, <p lt <£ 2 , $3 are imaginary angles between real and imaginary 
 lines. 
 
 As in Art. 58 it may be shown that the sum of the projections of 
 the sides of ABC on s is 
 
 cos 0,. CF + cos fa.A'C + cos(j> s ,B'A' (1) 
 
 6—2 
 
84 
 
 The Imaginary in Geometry 
 
 Let S be the real point on s. Through S draw any real straight 
 line a making an angle (j> with s. Let i/r 1( i/r 2 > yjr s be the angles, real, 
 which B'C, G'A', A'B' make with a. 
 
 Then <j>i = ^}fi — <f>, <£ 2 = ^ - </>> <^3 = V r s — < t>- 
 
 Therefore (1) becomes 
 
 CB' cos (^ - <£) + A'C cos (ifr 2 - </>) + 5'^ cos (i/r s - <£). 
 
 But since <£ is the angle between a real and an imaginary line this 
 equals by Art. 60 
 
 cos (j) \C'B' cos i/rj + A'C cos ^ + B'A' cos i/r s } 
 
 + sin {CB' sin ^ + ^.'(7 sin i/r 2 + 5'^.' sin yjr s }. 
 
 But by Art. 58 both the expressions in the brackets are zero and 
 therefore the whole expression is zero. 
 
 63. Relations connecting the measures of the sides of an imaginary 
 triangle and its angles. ^ 
 
 ' ** 
 
 Let the measures of the sides of an / X 
 
 imaginary triangle whose angles are A,B,C (<y/ X (b) 
 
 be a, b, c, none of the latter being zero. / X_ 
 
 Taking measures of the sides of the triangle / 
 on lines perpendicular to AB and AC, L (a > 
 
 -c 
 
 and 
 Therefore 
 
 a sin 5 = 6 sin .4, 
 a sin G= c sin A. 
 a b c 
 
 ■(1) 
 
 sin A sin B sin C ' 
 
 Taking the measures of the sides on a, b and c, 
 
 a = ccosB + bcos C 
 b = a cos C + c cos A 
 c = a cos B + b cos A 
 
 From (1) and (2) it follows that 
 
 c 2 = a* + ¥-2abcosC) 
 a? = b*+c*-2bccosA 
 b* = a? + c*-2ac cos B) 
 
 ■(2) 
 
 •(3) 
 
Trigonometry of the Imaginary 85 
 
 64. To find the formulae connecting the sines and cosines of the 
 difference of two imaginary angles with the 
 sines and cosines of these angles. A y" 
 
 Let two imaginary straight lines OA and y'f 
 
 OB make angles fa and fa, as in the figure, ,"' ! 
 
 with the imaginary line AB. Then the y' / 
 
 angle AOB is fa - fa = &>. ^,''' / 
 
 Take the measures of BO and OA on O'--— Wb- - 
 
 AA. Then «/ B 
 
 But 
 
 -45 = OA cos <£j - 05 cos fa 
 AB OA OB 
 
 sin a> sin $ 2 sin fa ' 
 Therefore sin (0 2 — fa) = sin a> = sin fa cos $! — sin fa cos <£ 2 . 
 
 IT 
 
 Increase fa by 5- . Then 
 
 cos (</> 2 — ^j) = cos fa cos 0! + sin fa sin <^ . 
 If fa is taken as the internal angle at B, then 
 
 co = Tr — (fa + fa) an d -45= 0-4 cos fa + OB cos fa. 
 Therefore 
 
 sin <a = sin' (fa + fa) = sin fa cos <£j + sin fa cos <£ 2 
 and also cos (fa + fa) = cos fa cos fa — sin <£ 2 sin fa. 
 
 For these formulae to hold it is necessary that the measures of the 
 lengths from which the angles are derived should not be zero. 
 
 As these addition and subtraction formulae hold for imaginary 
 angles, all the formulae deduced therefrom for real angles also hold 
 for imaginary angles. 
 
 65. To prove the general cases of Menelaus' theorem and of Geva's 
 theorem. , 
 
 Let the sides of an imaginary triangle ABC meet an imaginary 
 straight line in P, Q, B as in the figure. Let the angles at P, Q, B be 
 denoted as in the figure. 
 
 Then (Art. 63) 
 
 sin /3 _ sin a sin 7 sin ft sin a _ sin 7 
 ~CP~~CQ , ~AQ~lLB' B~B~'BP~' 
 
86 The Imaginary in Geometry 
 
 Therefore 
 
 ... GQ AR BP . . . 
 
 sin a sin B sin 7 = -r^k . ^„ . 77^5 sin a sin a sin 7, 
 H ' AQ BR GP M ' 
 
 Hence, if none of the sines of the angles a, 8, 7 are zero, 
 
 GQ AR 
 AQBR 
 
 BP 
 GP~ 
 
 l 
 
 \(R) 
 
 \y\ 
 
 \ \ 
 
 \ 
 
 \ 
 
 
 \ 
 \ 
 
 \<Q) 
 
 
 Z A . a .:(P) 
 
 (C) (B) 
 
 From the general case of Menelaus' theorem the general case of 
 Ceva's theorem may be deduced as follows : 
 
 Join the vertices of any triangle ABC to A 
 
 any point P and let the lines AP, BP, GP 
 meet the opposite sides in A', B' , C respec- 
 tively. 
 
 Then by Menelaus' theorem for the tri- Q'X / ,\B' 
 
 angle ABB', / % ^'"' \ 
 
 AG' BP B'C 
 
 BC'B'P'AG L 
 
 ■». \ 
 
 Similarly from the triangle GBB', B A' C 
 
 BA' GA B'P 
 CA"B'A' BP~ 
 
 ™ , AG' BA' GA B'C . 
 
 Therefore _._.__. __ = L 
 
 rp. - A.G' BA 1 GB' 1 
 
 Therefore _._ _ = _i. 
 
Trigonometry of the Imaginary 
 
 87 
 
 Analytical verification of the fact that the sum and difference formulae hold for a 
 real and an imaginary angle. 
 
 Let c, a + ib, be the coordinates of any 
 point P. Let OQ = c, NQ = a and NP = ib. 
 Drop PM perpendicular to ON, and MK 
 perpendicular to OX. Let QOJY=a and 
 MOP=p ( , so that NPM=a. If 
 
 sin (a + ft) = sin a cos ft + cos a sin ft, 
 then 
 
 y (p; 
 
 
 
 
 
 (M) 
 
 
 
 N 
 
 
 
 rx 
 
 fa) 
 
 
 * 
 
 v a 
 
 
 
 
 (C) c 
 
 ! 
 
 
 
 X 
 
 Z + 4& 
 
 ivWc 2 +;& 
 
 Va 2 + c 2 
 
 Va 2 + o 2 
 
 Va 2 + c 2 ' vW(<* + *) 2 
 
 \/c 2 +(« + i6) 2 Ve 2 + a 2 ' ' Vc 2 + (a+i'6) 2 
 which is true. 
 
 Analytical verification of the fact that from whatever pair of lines the measure 
 of an imaginary length is obtained its value is the same. 
 
 Y 
 
 (P) 
 
 
 
 
 / 
 
 
 
 // 
 
 
 
 // 
 
 
 
 // 
 
 
 
 / / 
 
 
 
 /"/ 
 
 ib' 
 
 
 f* 
 
 (N) 
 
 I Si' 
 
 // 
 
 b 
 
 
 JS 
 
 
 
 Let the coordinates of a point Preferred to rectangular axes be a + ia', b + ib'. 
 Construct the point Q (a, b) and draw QN parallel to the axis of X to meet the 
 
 ordinate of P in N. 
 
 * 
 
 Then OP i =OQ 2 +PQ 2 -20Q. PQcos OQP 
 
 = (a 2 + b 2 ) - (a' 2 + b' 2 ) - Zi *JoJ+¥ sja' 2 +b' 2 cos OQP. 
 But cos OQP= - cos (PQiV- QOX) 
 
 = - {cos PQN. cos QOX +sin PQiV . sin QOX} 
 
 — _ f ^ a , iV b \ 
 
 \i \/a' 2 + b' 2 \/a 2 +b 2 i *Ja' 2 + b' 2 JaP + bV 
 
 aa' + bb' 
 tjart + b"' '■<JoJ+&' 
 Therefore OP 2 = (a 2 + b 2 ) - (a' 2 + 6' 2 ) + %i (ad + bb') 
 
 = {a + ia') 2 +{b + ib') 2 . 
 
88 The Imaginary in Geometry 
 
 v 
 
 66. To find a real straight line such that the angle between it and 
 a given imaginary straight line has a tangent, which is a purely imaginary 
 quantity. 
 
 P 
 
 (A) E (M " "\ F 
 
 Let AP be the given imaginary straight line and P the real point 
 on it. Take any imaginary point A on the line. Let A' be its con- 
 jugate imaginary point. Then PA' is the conjugate imaginary line of 
 PA and the line AA' is real. 
 
 Join P to the circular points at infinity, CI and D,', to meet A A' in 
 B and B'. Then B and B' are a pair of conjugate imaginary points. 
 
 Since AA', BB', are two pairs of conjugate imaginary paints they 
 have a common pair of real harmonic conjugates E and F (Art. 8). Join 
 E and F to P. Then PE, PF are real lines. 
 
 Since PE, PF are harmonic conjugates of P£l and PD,' they are at 
 right angles (Art. 22). Draw any real line 000' parallel to PF, and 
 therefore perpendicular to PE, to meet PA, PE and PA' in 0, 0, C 
 respectively. Then 0, C are a pair of conjugate imaginary points. 
 
 Also, since (AA'EF) is harmonic, (CO'Ooo ) is also harmonic. 
 
 Therefore is the mean point of the pair of conjugate imaginary 
 
 points and C, and 00, 00' are purely imaginary lengths (Art. 6). 
 
 00 
 t Hence the tangent of the angle 0P0, which is -p^r , is the ratio of a 
 
 purely imaginary quantity to a real quantity. 
 
 It follows from the preceding that two conjugate imaginary lines 
 PA, PA' have always a pair of real bisectors PE, PF, which are at right 
 angles (see Art. 52 (3)). 
 
 Analytical verification. 
 
 Take Pas the origin and let A be a+ib, o + id, and A', a-ib, c-id. 
 
 The equation of PA is x= r-. y. 
 
 The coordinates of the point C where a line y^mx + k meets this line are 
 (n + ib)k (c+id)k 
 
 (c+id) -m(a + ib) ' (c+id) - tn (a+ib) ' 
 
Imaginary Angles 
 
 89 
 
 Therefore 
 
 {(c + id)-m(a+ib)} i ' 
 This expression does not involve i if 
 
 m z 4 ... . t., m-l=0. 
 
 Now the combined equation of PA, PA' is 
 
 {ay - cxf + {yb - xdf = 0, 
 and the equation of the bisectors of the included angles is 
 
 x 2 —y i xy 
 
 c* + d*-a!>-b* = -lac + bd) ' ' 
 Therefore the m's of these lines satisfy the relation 
 
 as 2 +& 2 — c 2 — c 
 
 •« 
 
 .(ii) 
 
 m 2 + - 
 
 m-l=0. 
 
 .(iii) 
 
 ac+bd 
 
 This is the same equation as (i). Hence PC and PC, when CC is* parallel to a 
 bisector of the angle AOA', have as their measures real or purely imaginary quan- 
 tities. GOC is obviously perpendicular to the other bisector from the form of (ii). 
 
 67. Measurement of imaginary angles and evaluation of their 
 sines, cosines and tangents. 
 
 Let OP be any imaginary straight line and the real point on it. 
 
 (p'V 
 
 (p") 
 
 (P r ) 
 
 x 
 
 (pj- 
 
 Ch) 
 
 N 
 
 (a) 
 
 d~\ 
 
 ?0 
 
 (Qj-' 
 
 Let 0Q be its conjugate imaginary line and OX and 0T the bisectors 
 (real) of the angles between these lines (Art. 66). 
 
 Take N any real point on OX at a real distance a from 0. Through 
 N draw a real line NP perpendicular to ON to meet OP in some 
 imaginary point P. 
 
90 The Imaginary in Geometry 
 
 Then as defined in Art. 56 the ratios 
 
 ON NP NP 
 
 OP ' OP ' ON ' 
 
 are respectively the cosine, sine and tangent of the angle XOP. Denote 
 the angle XOP by 0*. 
 
 TV, a 0N ■ a NP i * a NP 
 
 Inen cos 6i = jjp , sin "i = jyp and tan0j = ^-^. 
 
 Now P and Q are a pair of conjugate imaginary points and N is 
 their mean point. Therefore NP is a purely imaginary quantity, i . h. 
 
 NP 
 Let YrTf > which is the ratio of a purely imaginary quantity to a real 
 
 quantity, be denoted by i . m. 
 
 Then a 1 . i.m a . 
 
 cos t>i = . , an Vi = —===. , tan 0i = t. m. 
 
 vl - m 1 vl - m? 
 
 Regard the lines OX, OY as fixed and the lines OP and OQ as being 
 any pair of conjugate imaginary lines through 0, whose bisectors are 
 OX and OY. 
 
 As the line OP moves up to OX and eventually coincides with it, 
 the line OQ will do the same. In the position OX the pair of conjugate 
 imaginary lines coalesce and become the real line OX. Similarly when 
 the line OP coincides with the line OY it also coincides with its con- 
 jugate imaginary line and becomes a real line*. 
 
 Now there is nothing inherent in a real length a and an imaginary . 
 length ih, by which it is possible to tell the relative magnitude of a 
 compared with ih, but it is possible to tell the relative magnitudes of a 
 series of purely imaginary quantities ih, 2ih, 5ih and nih. 
 
 On the line NP take a series pf lengths (all purely imaginary) with 
 values from to i . oo , and let these lengths determine a series of points, 
 P, P', P" , ... on NP. The connectors of these points to are lines of 
 the system of conjugate imaginary lines of which OX and OY are the 
 bisectors. As the imaginary line OP takes up the series of positions 
 OP, OP', OP", OP'", ... the angle U which it makes with OX, will pass 
 
 from a real value 0, when it coincides with OX, to a real value „ when 
 
 it coincides with OY. The right angle XOY may therefore be divided 
 into a series of imaginary angles in the same way in which a right angle 
 
 * If the line OP is given by the equation y = imx, the positions OX and OY correspond 
 to the values m = and m=oo for which values of m. the line is real. 
 
Imaginary Angles 91 
 
 is divided into a series of equal real angles. The values of cos t , sin i} 
 tan { may be evaluated for the imaginary angles into which the right 
 angle XO Y is divided. 
 
 If a real line OP is rotated round through real angles from OX to 
 OY, cos#, where is the angle XOP, passes through all real values 
 from 1 to 0. 
 
 If an imaginary line OP is rotated through imaginary angles — as 
 previously set out— cos 6>» at OX is 1. It increases as m increases, 
 remaining real till m = 1, when it reaches the value oo . This value 
 corresponds to the critical lines through (see Art. 22). It then 
 becomes i . oo (ignoring sign for the present) and as m increases it 
 remains imaginary, decreasing till, when m= oo at the position OY, it is 
 zero. 
 
 Hence, as a real line rotates from OX to OY, the cosine of the angle, 
 which it makes with OX, takes the real values from 1 to and as an 
 imaginary line rotates from OX to OY the cosine of the angle, which it 
 makes with OX, takes the real values from 1 to oo and then the imaginary 
 values from i oo to (ignoring sign for the present). Hence with the 
 extended definition the cosine of an angle can have all values real and 
 purely imaginary. 
 
 Similarly the sine of the real angle made by a real line with OX 
 passes through the real values from to 1. The value of sin#. ; where 
 Qi is the angle made by an imaginary line — as previously described — 
 with OX, increases through imaginary values from to i oo as m in- 
 creases from to 1. As m increases from 1 to oo the value of sin t 
 passes through real values from oo to 1. The cycle of possible real and 
 purely imaginary values is thus complete. 
 
 Similarly for a real line tan passes through the values to oo as 
 it rotates from OX to OY, and for an imaginary line tan t passes through 
 the imaginary values from to i . oo . 
 
 Hence with the extended definition the sine, cosine and tangent of 
 an angle can have all real or purely imaginary values. 
 
 68. When an angle is written 0; it is so written to bring to mind 
 the fact that it is obtained by rotating an imaginary line round and 
 not by rotating a real line round 0. The expression { must not be 
 confused with the expression i . which is the circular measure of a real 
 angle multiplied by the unit of imaginary length. To this expression 
 i . no meaning has — as yet — been attached, and as yet no measure of 
 an imaginary angle 0i has been defined. 
 
92 The Imaginary in Geometry 
 
 T -. 
 
 When the angle 8 t takes values from to ~ it is necessary to find 
 some means of measuring or constructing the different angles. The 
 angle 4 can be constructed from the fact that its tangent is i-, or 
 
 i tan 6, where tan 6 is - . Thus the angle 30°; may be represented by 
 
 constructing a real angle of 30° and regarding the side opposite to the 
 angle as measured in imaginary units. This notation will sometimes be 
 adopted in the following pages. 6 may in this case be termed the sub- 
 sidiary angle of it and 9 t may be written s0, viz., the angle whose 
 subsidiary angle is 6. 
 
 It should be noticed that while the sines and cosines of imaginary 
 angles may be real, the tangent of an imaginary angle — as here defined 
 — is always imaginary, and the tangent of a real angle is real. Hence 
 in dealing with the tangent it is possible to equate real and imaginary 
 parts regarding the tangent of an imaginary angle as imaginary. 
 
 1 1 k/3 
 
 Thus tan 30°j=i -r= , sin 30% =i -75, cos 30%=^, 
 
 tan(45° + 30%)=— ^ = i+^?, 
 
 S in(45° + 30%) = ^ + 4 = ^±-\ 
 co S (45°+30%) = ^-iI = ^-\ 
 
 To find Un-ifl + i^Y 
 
 Let taa(o+ft)=J+i^. 
 
 Then tana+tanft = 1 J3 
 
 1- tan a tan ft 2 2 ' 
 
 T , .. tan ft 
 
 Let # = tana, y = — ~, 
 
 Equating real and imaginary parts 
 
 1 , s /3 
 
 V3 1 
 Solving these, the values #=1, y=> —^ , and x= - 1, y=\/3 are obtained. 
 
Imaginary Angles 
 
 93 
 
 Taking the first solutions 
 
 Therefore 
 
 Similarly 
 
 tana = l therefore a =45°, 
 tanft=-^ „ 0=30%. 
 
 tan(45°+30" i ) = i+i^. 
 
 tan (135° + 60\-)=| + ;>£*. 
 
 To find tan" 1 (a +?'&). 
 
 In a similar manner it is found that 
 
 at + tf-l V(a 2 + 6 2 -l) 2 + 4a 2 
 
 tan a= 5 ± — ^ — , 
 
 2a 2a 
 
 and 
 
 tanff _ a'+6 2 - 1 V{q 2 + (6 + l) 2 } {a 2 +(6- 1) 8 } 
 i 26 26 
 
 It should be noticed, as will be explained hereafter, that s.a+s.fj does not 
 equal s(a + |3). 
 
 69. Relations connecting sines, cosines and tangents of imaginary 
 angles. 
 
 From Art. 67, if tan 6 = - , 
 
 a 
 
 sin 0{ = 
 cos #i = 
 
 ih 
 
 i tan # 
 
 Va 2 -A 2 Vl-tan 2 0' 
 ,a 1 
 
 Va 2 -A 2 Vl-tan 2 0' 
 .A 
 
 tan di = i- = i tan 0, 
 a 
 
 where # has all values from to 
 Let tan = 
 
 7T 
 
 2" 
 
 i-e-y 
 
 ev + e~y 
 
 Then 
 
 , T 1 + tan 6 
 ^ L °ST-tanlr 
 
 Substituting in the above equation in terms of y 
 
 . .ev-e-y ... 
 sin 6i = i : — s = i sinh y 
 
 cos 0* = g — = cosh y 
 
 e/y — g-y 
 tan0i=i-—— i = i tanh y 
 e y + e y I 
 
 •(1) 
 
94 The Imaginary in Geometry 
 
 Hence 
 
 . . . (, T 1 + tanfl 1/ 1T 1+ tan fly 
 
 and 
 
 1 + tan fl 
 1 - tan 
 
 cos 
 
 . , 1 /. T 1 + tan fly . /. _ 1+ tan fl\ 4 , 
 ^=l + ,2 (i L °gi3t^fl) + H^ LOg l^tan^J + 
 
 From equations (1) it follows that fl f is a function of y. Let fl; =f(y)- 
 Then sin/(j/) = i sinh y, ' 
 
 cos f(y) = cosh «/, 
 tan/(y) = i tanh y. 
 Hence sin/(y + z) = i sinh (y + 2) 
 
 = i {sinh y cosh z + cosh y sinh z] 
 = sin/(y) cos/0?) + cosf(y) sin/0). 
 Similar results hold for cos f(y) and tan/(j/). 
 
 Hence, when y is looked upon as the parameter, imaginary angles 
 may be added according to the usual formulae. 
 
 Subsidiary angles. 
 
 Let tan 0; = i tanh y and tan <£; = i tanh z, 
 
 , , T 1 + tan , , _ 1 + tan rf> 
 
 so that y = h Log ., — 5 and z = A- Log . — - — £ . 
 
 * " 6 1 - tan fl ■* s 1 - tan 4> 
 
 Then sin (0* 4- 0,) = sin fl* cos 0; + cos flj sin ^ 
 
 = i (sinh y cosh z + cosh y sinh z) 
 
 = i sinh (j/ + z) 
 
 = sin i|r,-, where y + z = I Log- J . 
 
 ° 1 — tan yjr 
 
 The relation between fl, $ and ijr is consequently given by 
 
 . T 1 + tan -^ 1 + tan fl . T 1 + tan d> 
 
 * Lo gr-^=^ + ^=* Lo gfr^ +^o gr -^. 
 
 rp, f 1 + tan ijr _ 1 + tan fl 1 + tan <f> 
 
 1 — tan if- 1 — tan fl ' 1 — tan <£' 
 
 m, r i t an # + tan <f> 
 
 1 herelore tan -ur = ^ 2— . 
 
 1 + tan fl tan <f> i 
 
Imaginary Angles 95 
 
 This gives the relation connecting the subsidiary angles'. The relation 
 connecting the subsidiary angles can be obtained in a similar manner 
 from the addition formulae for the cosine or tangent. 
 
 Hence if the sine, cosine or tangent of an imaginary angle is required 
 it is possible to proceed in either of two ways. 
 
 Suppose that tan (30°,- + 60%) is required. 
 
 f„\ +„„ /qi\° , ™° \ tan 30°i + tan 60% 
 (a) t m (80 < + 60 0° 1 _ toTO . <tonfl0 - i r 
 
 *^ + tVS ,2 
 
 ,s tan + tan <fr _ tan 30° + tan 60° 
 
 { ) an r - x + tan ^ tan ^ - j + tan 30 c tan 60 o 
 
 1 + 1 V3 ' 
 ' Therefore tan (30°, + 60%) = i tan f = i — 
 
 if 0i + <j>i = ■fi, 
 
 tan + tan rf> 
 
 since tan ilr = - — ■- ^ —, . 
 
 1 + tan a tan <p 
 
 ., , • . sin(0 + <f>) 
 
 therefore simi = r 
 
 V sin 2 (0 + 0) + cos 2 (0 - </>) 
 
 . cos (0 - 0) 
 
 and cos vr = . r/ = . 
 
 V sin 2 (.0 + 0) + cos 2 (0 - <£) 
 
 The advantage of using the subsidiary angle as the parameter lies 
 in the fact that (ignoring sign for the present) while it passes through 
 
 all real values from to ^ , sin and sin 0j pass together through all 
 
 real and purely imaginary values, while cos and cos 6{ and- also tan 
 and tan 0j do the same. There are however advantages, as will be shown 
 later (Art. 70), in taking y or rather iy as the measure of the imaginary 
 angle 0j. 
 
 According to the definition of Art. 56, if 0'* be the angle' OPN, then 
 
 ON 1 PN am 
 
 sin0' i = Td = j T =^, C OB*' 4 =-p0=; 7 — , 
 
96 The Imaginary in Geometry 
 
 ., ON 1 
 and tan ^ i= iW = im" 
 
 Hence 
 
 sin (d { + 6'i) = sin 6 t cos 6'i + cos 6{ sin 6' { 
 
 im im 1 1 . . it 
 
 + ,-_ ,. - =l=sin- 
 
 Vl-m 2 Vl-m 2 Vl-m 2 Vl-m 2 2 
 
 and 
 
 cos {6i + 6'i) = cos 0{ cos 0'j — sin 0* sin 0'* 
 
 1 im im 1 „ w 
 
 : = = cos ; 
 
 Vl— m 2 Vl-m 2 Vl - m 2 Vl — m 2 2 
 
 i 
 
 7T 
 
 This confirms the fact that #< 4- 0'j = ^ . 
 
 The trigonometrical ratios of a purely imaginary angle may be 
 obtained from any triangle, with two sides at right v angles, on which 
 sides the vertices are at distances, the one purely imaginary and the 
 other real. 
 
 Generally, the sines, cosines, and tangents of a complex angle as 
 defined in Arts. 56 and 61 are complex quantities. A complex angle 
 can however be expressed as the sum or difference of a real and a 
 purely imaginary angle, and therefore its trigonometrical functions can 
 be expressed as functions of the trigonometrical functions of real and of 
 purely imaginary angles. 
 
 70. Measure of an imaginary angle. 
 
 In Arts. 56, 60, 61 and G4 the sine, cosine and tangent of an 
 imaginary angle were defined and it was shown that the addition and 
 subtraction formulae, which are true for the sines and cosines of real 
 angles, also hold for the sines and cosines of such imaginary angles. 
 
 No measure of such imaginary angles however was introduced. 
 This leaves the subject in the same position as that, in which the 
 trigonometry of real angles might be supposed to be, before a radian 
 or a degree was defined. 
 
 In Art. 67 it was shown that the sine, cosine and tangent of a 
 purely imaginary angle termed 0i might be constructed by means of 
 a subsidiary angle 0. Still no way of measuring the imaginary angle 
 &i was laid down. 
 
Imaginary Angles 97 
 
 In Art. 69 it was proved that if 
 
 e v _ e -y l + tan 6 
 
 tan = — — - or a = A Log ^ -„ , 
 
 e y + e-v y * 8 1 - tan 6 
 
 ., . ,, .ey-e-y e-y-e* . . , 
 
 then sin 6i = i — = — = — <p — = i smh y, 
 
 cos 8i = — g — = — 2 — = C0S y ' 
 
 , a .ey-e-y e-y-ey ... f 
 
 tan &i = % = —, r = ^ tanh y. 
 
 1 ey + e-y i{e~y + ey} y 
 
 Hence, since the trigonometrical functions of an imaginary angle 8i are 
 functions of y, it was possible to write them as sin f{y), cos f(y), ta.nf(y). 
 
 It was then proved that in this form the y follows the addition and 
 subtraction formulae which hold for the sines and cosines of real angles. 
 
 In this case 
 
 . e-y-e+y e-y + ev ,, . e-w-e» . 
 
 «n/(y) = ^~2i~ > cos /^) = —2~ • tan /^) = i{FM^} ' ■ ■ ■<!) 
 
 Now according to the usual theory, if y is real, 
 
 e w_ e -w tfy + e-iy e w- e -iy 
 
 8111 y=— it - ' cos 2/=-^— • ^2/=-^+^}- -(2) 
 
 And if iy be an imaginary angle 
 
 e~y-e y . e-y + ev . e-y-e y 
 
 by the usual series definitions. 
 
 Comparing (1) and (3) it is seen that they are in agreement if 
 
 f(y) = iy- 
 
 Hence the measure of an imaginary angle may now be defined as 
 being iy, so that 
 
 e-y-e +v . e-y + ev . e - " — e" 
 
 sini2/ = _ __, cosly = ^_, tenty = ____. 
 
 The trigonometrical functions of such an imaginary angle may be 
 
 constructed by finding the real subsidiary angle which is given by 
 
 e y- e -y i + tan 
 
 tan o = - , — -- , where y = * Log ^ — - — ^ . 
 e y + e-y' y 2 6 1 - tan 6 
 
 Then 
 
 itantf . 1 Q 
 
 sin iy = - . = , cos %y = . = , tan ly = % tan 0. 
 
 y Vl-tan 2 6» " VI- tan 2 * 
 
98 The Imaginary in Geometry 
 
 These trigonometrical functions may be graphically constructed by 
 means of Art. 67 and the values of the sine, cosine and tangent of an 
 imaginary angle iy may be obtained by means of the real angle 6. 
 
 For the trigonometrical functions of an imaginary angle so denned 
 the addition and subtraction formulae hold and — as a general rule — the 
 expression iy may be treated as the product of a real and an imaginary 
 quantity. 
 
 It follows that a complex angle a + i/3 has a period 2tt. 
 
 Use of meridional tables to ascertain the subsidiary angle of a 
 given imaginary angle. 
 
 Given an imaginary angle iy the relation to find 6 is 
 
 . x 1 + tan 6 
 
 The tables of meridional parts give v where 
 
 10,800 ,, 10,800 T 1 + tan \ 6 
 -^ = ~^ L °Sl-tan^- . 
 
 Hence if y is given in minutes, the number 2y must be looked out in 
 the tableland the angle 6 corresponding to the value of 2y can be found. 
 One-half of this angle is the required subsidiary angle. 
 
 The values of 6 in the tables are from 0° to 90°. Hence the values 
 of the subsidiary angles are for values from 0° to 45°. 
 
 Summary. , 
 
 These results may be summarised as follows : 
 
 The trigonometrical functions of an imaginary angle { as defined in 
 Art. 67 are 
 
 • /> * m a ! a 
 
 sin Oi = , cos Oi = , tan 0* = irn, 
 
 V 1 - m 3 V 1 - m 2 
 
 i + a t 4. 4. a ev-e-y . , l+tan<? 
 
 where m = tan 6. Let tan 6 = fly + g _, or y = J Log 1 _ teng ■ 
 
 The values of y for corresponding values of 6 can be found from the 
 table of meridional parts. 
 Then 
 
 v = 
 
 . a e-v-ev 
 
 cos d { = — 2 — ' 
 
 tan £_ *-»-«» 
 
 - ulff »- 2t ' 
 
 tan ^ i{e-v + e v\ 
 
 By the series definition 
 
 
 
 
 e~y + ev 
 cos iy = 2 , 
 
 e-~v - ev 
 
 tan %v = .. _, -r 
 
Imaginary Angles 
 
 99 
 
 Hence 6 t = iy, where tan 6 = or w = A Loo: ^. 
 
 * ev + e-y J 2 s 1 - tan 
 
 Therefore 
 
 1 
 
 sin iy = 
 
 Vi- 
 
 m- 
 
 cos %y ■■ 
 
 where 
 
 m = • 
 
 Vl - m : 
 
 , tan ^y = im, 
 
 ' + e-»' 
 
 and m = tan 6, 6 being the real subsidiary angle of 6 t or iy. Hence the 
 
 values of sin iy, cos iy and tan iy can be graphically obtained. If 6 { be 
 
 written s . 0, i.e., the angle whose- subsidiary angle is 6, then s0 is iy, 
 
 , , T l+tan# „ . 1T l+tan'0 
 
 where y - iLog ___ , or rf-.*Log ^^ 
 
 The addition and subtraction theorems have been proved (Art. 69) 
 to hold for y and they therefore hold for iy. 
 
 71. Values of the trigonometrical functions of an imaginary 
 angle. 
 
 The values of the sine, cosine and tangent of iy 6 an imaginary angle 
 have been denned as 
 
 a 
 
 sin iy 9 ■■ 
 
 Vo s -A 2 ' 
 
 costy t = , , taniw9 = i-, where tan# = -, 
 
 *Ja?-h? a a 
 
 7—2 
 
100 The Imaginary in Geometry 
 
 If h>a these may be written 
 
 . . h . —ia . h 
 
 There is an ambiguity of sign in regard to the square root, but it will 
 be convenient to take the above as the definitions. In order likewise 
 to complete the cycle of values it is convenient to take a second line 
 perpendicular to the axis of a; at a distance — a from the origin and to 
 consider intercepts made by the variable line on this line, as well as on 
 the line x — a = 0. 
 
 The following will then be found to be the values of the sine, cosine 
 and tangent of the imaginary angle in the different semi-quadrants : 
 
 e 
 
 sin iy 
 
 cos iy 
 
 tan iy 
 
 iy 
 
 --toQ 
 4 
 
 — i.cc to — i.O 
 
 oo to 1 
 
 ■>-i to —i.O 
 
 2tt — i.oo to 27r 
 
 0to£ 
 4 
 
 i.O to i.oo 
 
 1 to 00 
 
 i.O to i 
 
 to i.oo 
 
 7T , IT 
 
 4 t0 2 
 
 oo to 1 
 
 — i.oo to — i.O 
 
 i to i.oo 
 
 2 +l -°° t0 2 
 
 7T , 37T 
 
 2 t0 T 
 
 1 to X 
 
 i.O to i. oo' 
 
 — i.oo to — i 
 
 7T , 7T 
 
 2 t0 2~ t -°° 
 
 3tt 
 
 —7- tO 7T 
 
 4 
 
 i.oo to i.O 
 
 — oo to — 1 
 
 — i to —i.O 
 
 TT — i.OO to 7T 
 
 7T tO -j- 
 
 4 
 
 — i.O to —i. oo 
 
 — 1 to — 00 
 
 i.O to i 
 
 7T tO 7T + i.OO 
 
 5-ir . Zir 
 
 T t0 T 
 
 — oo to — 1 
 
 i. oo to i.O 
 
 i to i.oo 
 
 3lT . , 37T 
 
 37r x 7tt 
 
 T to T 
 
 — 1 to — 00 
 
 — i.O to —i.oo 
 
 — i.oo to — i 
 
 37T , 37T 
 
 T to T -z.oo 
 
 Hence the sine, cosine and tangent of any angle, real or purely 
 imaginary, can have any value, real or purely imaginary. There is a dis- 
 continuity in the value of iy at the critical lines, where, the value of iy 
 
 is increased by -= . 
 
 In the preceding by the introduction of the line x + a — values of 
 sin#,- and cos^ have been obtained with different signs. Thus both 
 the signs which may be given to VA 2 - a 2 have been taken into account.. 
 
Imaginary Angles 101 
 
 Values of an imaginary angle. 
 
 Let y e ' be the general value of £ Log — — —. and let y e be its principal value. 
 
 1 — E< 1 1 i a 
 
 Then since 
 
 T l + tan0 _, . , l+tan<? 
 
 ^°g i — i „ = 2jrai + log = — t , 
 
 & 1-tand o l-tan0' 
 
 it is seen that iye' = -nir + iy e (1) 
 
 To derive the values of sin 6 t and cos 8i between — and — from those between 
 
 - — and — - , iy e must be increased by n. 
 
 Expressing for shortness the fact that iy$ and iy give the same values of the 
 sine, cosine and tangent of 8 and &', as iyg' =iyg, the following results are arrived at. 
 
 . . , T 1 - tan 8 . T 1 + tan 8 
 
 (a) y_ e = i Log j-^^ = -I Log ^-^ . 
 
 Therefore iy~e=~ tyo (i) 
 
 On reference to the values of the sine, cosine and tangent this is seen to be true. 
 
 , x l-cot0 , _ tan 0+1 _ . , T 1+tand 
 W y. +i ,-i Log r ^^= -i Log t -^-_- 1 = ±Logt - $ Log j-^ 
 
 Therefore We^„= + ^ - »>e • 
 
 There is an ambiguity as to the sign to be taken in ± — . 
 
 Let the first quadrant be that from --to-, the second that from — to — , the 
 
 third that from — to — - and so on. Let m denote the quadrant in which the arm 
 4 4 
 
 of the angle 8 lies. Then if the above result be written in the form 
 
 ^ +}7r = (-l) m+1 f -iy„* (") 
 
 it will be found to be true on reference to the table of values. 
 Hence it follows that 
 
 iy, + „= -(-ir +1 f-^ +i „. 
 
 Hence *%+*= -(-l) m + 1 *- + iy s 
 
 which is in agreement with (1). If n- be added or subtracted the results differ by 
 2jr, hence this result may be stated in the form iy 9+lr =ir+iy 9 ■ 
 
 * This result may also be written in the form 
 
 2m - 1 
 
102 The Imaginary in Geometry 
 
 , , , T l + cot0 iT . ,. T l+tand 
 
 (c) yjir _,= i Log i ___= ± Log H .iLog Trsrtf 
 
 , . 77 , T 1+tanfl 
 
 = ±l 2 + * Los nt^r 
 
 If m be defined as in (6) this formula may be taken in the form 
 
 Wj,_,=(-l)" +1 |+W, ( ii; ) 
 
 On reference to the table of values this is found to be true. 
 
 If 8 be increased by — (iii) becomes iy _ $ = -(-1)™ + 1 — + iy e+ i w - 
 
 This, remembering that iy_ e = — iy e , is (ii). 
 
 ,« , T l+tan(0 + »r) 1T 1 + tantf • 
 
 (<*) ^+^i Lo gl^tan-fc r ) = * Log nta^- 
 
 Hence it would seem that iy B+ =iy B - This on reference to our table of values 
 of sine, cosine and tangent is not true. In fact our geometrical restrictions require 
 
 us to assume that Log , , , „ ~ has its general value and this in fact is y. — iir. 
 
 8 l-tan(0+-rr) 5 "<> 
 
 This result is then in agreement with those already obtained for 
 
 72 . Construction of the siiie, cosine and tangent of a purely imaginary 
 angle by means of the imaginary branch of a circle. 
 
 Let OX and Y be a pair of orthogonal lines. Take a circle centre 
 and radius a. Draw its real branch and also the two imaginary 
 branches* (1, i) and (i, 1) corresponding to the axes OX and OT. 
 
 (1) Through draw any real line OP to make a real angle with 
 OX. Then if this line meets the real branch in P, 
 
 . a PN . . PN „ ON 
 
 tan = tttp , sm«= , cos o = . 
 
 OJ\ a ' a 
 
 (2) Through draw an imaginary line OP' to make an imaginary 
 angle 0; with OX. Let OP' meet the imaginary branch (1, i) in P'. 
 Then, since the measure of the distance of every imaginary point on the 
 curve from is a, 
 
 , P'N' . a P'N' , ON' 
 tan 6i = -qw ' Sln "* = "" ' cos °i = • 
 
 where N' is the foot of the perpendicular from P' on OX. 
 
 * These are respectively the branches for which the coordinates are x, iy, and ix, y. 
 
Imaginary Angles 
 
 103 
 
 (3) Through draw an imaginary line OP" to meet the (i, 1) 
 
 branch in P". Let this line make an angle 6 t with OX. Then as in 
 
 the previous case 
 
 , . P"N" a P"N" a ON" 
 
 tan Vi = - n -^77- » sin 4 = , cos { = , 
 
 Civ a a 
 
 where N" is the foot of the perpendicular from P" on OX. In this case 
 P"iV ' is real and ON" is imaginary. 
 
 
 >M 
 
 Y 
 
 
 N, 
 
 
 
 
 
 
 ^-'V^ 
 
 
 
 \ \, 
 
 B^ 
 
 II -" • ■'' 
 
 ■'' '\Sf / 
 
 
 
 
 1 / \ 
 I / \ 
 
 1 
 1 
 
 /*\yS 
 
 \ / 
 
 
 
 
 1 / \. 
 
 ■ I 
 
 >v ^*^ 
 
 \l 
 
 
 
 
 7 \." 
 
 /A 
 
 '' \ 
 
 \l 
 
 
 
 X' 
 
 i yc 
 
 
 
 N" N lA N' 
 
 X 
 
 
 
 
 
 
 
 ' s "" 
 
 
 "^ X. ^ 
 
 
 
 *//-''' 
 
 
 \ \ \K 
 
 
 
 / ,'* 
 
 
 V N \ V 
 
 
 
 SH 
 
 Y' 
 
 
 GX 
 
 
 
 Hence, if the point P be supposed to move along the real branch 
 
 from A to B, and from B to oo (a critical point) along the (i, 1) 
 
 branch, and then from oo to A along the (1, i) branch, it will describe 
 
 a closed curve in the quadrant OX, OY and the angle, real or purely 
 
 imaginary, which OP makes with the axis OX is such that its tangent is 
 
 PN PN . ON 
 
 . its sine, and its cosine, — , where a is the radius of the circle, 
 
 ON ' & a 
 
 PN the ordinate and ON the abscissa of P. 
 
 In this case the trigonometrical functions are obtained from a right- 
 angled triangle, the hypothenuse of which is a real quantity a, while 
 one side, that measured along the axis of X, takes all real and purely 
 imaginary values. 
 
104 The Imaginary in Geometry 
 
 The values of the sine, cosine and tangent of the purely imaginary 
 angles obtained by rotating the imaginary line through 2ir round for 
 the different semi-quadrants are the same values as those in Art. 71, 
 but they occur in different order. The reason of this is as follows. 
 
 In the preceding the intersections of the line iy — mx with the circle 
 
 a 2 
 
 x* + y 2 = a? are sought for and it is found that «* = •= „. Ifm<l 
 
 J ° 1 — m 2 
 
 the values of x are real. This is the case along the branch A B. If m > 1 
 
 there is no real value of x. In this case the intersections of the line 
 
 with the circle are given by writing the equation of the line in the 
 
 form — y = mix. This is a line in the quadrants 2 and 4 and it meets 
 
 the circle, where y 2 = — — ^-,i.e., foifreal valuesofy, where??i>l. Hence 
 3 m 2 - 1 " 
 
 the line after having met the curve along the branch-if to L meets it 
 
 along the branch M to N. Similarly it again meets the curve from E 
 
 to F and finally from G to H. If the values obtained are interchanged 
 
 in such a way as to take this fact into account they are found to be in 
 
 agreement with those' previously given. 
 
 If, in accordance with the suggestion thrown out in Art. 126, the iy 
 
 axis be taken in the direction OY' and not in the direction OY, the line 
 
 — = m meets the circle in consecutive points, which lie from A to K, 
 
 x r 
 
 to H, F to E, M to N, and from L to A. 
 
 73. Analytical verification of the fact that the definition of an imaginary angle 
 is in agreement with the analytical theory. 
 
 If asr 2 + 2/wcy+by 2 = Q be the equation of a pair of straight lines referred to 
 rectangular axes and 8 be the angle between them, 
 
 tan0= — — (1) 
 
 Consider the pair of conjugate imaginary lines y — imsc=0 and y + imx = 0. Their 
 combined equation is y i +m 2 x i = 0. 
 ' Substituting in (1) it is seen that 
 
 . 2>J-m 2 2im 
 
 tan 6= -^ 5- = = i (2) 
 
 1+m 2 l + m* K ' 
 
 If \&i be taken as the angle between one of them and a bisector of the angle 
 
 which they contain, then 
 
 tan^d{=MK. 
 
 Therefore tan 6 { = z s , 
 
 l+m' 
 
 which is in agreement with (2). 
 
 Hence the angle given by the definition of Art. 67 is the same as that given by 
 the ordinary analytical formula. 
 
Imaginary Eccentric Angles 
 
 105 
 
 74. Eccentric angle of an ellipse. > 
 
 (a) Construct the real branch, the (1, i) branch and the (i, 1) branch. 
 Construct the auxiliary circle and its (1, i) and (i, 1) branches. Let a 
 and b be the semi-major and semi-minor axes of the ellipse so that its 
 
 equation is — + ^ = 1 and that of the auxiliary circle a? + y 2 = a 2 . Take 
 
 P 1; P 2 ,P 3 points on the real branch, the (l,i) branch and the (i, 1) 
 branch respectively of the ellipse and let the ordinates Pi-A^, P 2 iV 2 , 
 P 3 N 3 at these points meet the corresponding branches of the auxiliary 
 circle in P/, P 2 ', P,' respectively. Then, if the angles P/OX, P z '0X, 
 P 3 'OX be lt 2 and 6 3 ; t is real, 6 2 is purely imaginary and < tan -1 V, 
 and d 3 is purely imaginary and >tan -1 r, and the lengths OP/, OP I, 
 OP 3 are real and equal to a. 
 
 Then ONi, 0N 2 , ON 3 are respectively acos6 u acos0 2 , acos# s . 
 Hence from the equation to the ellipse P X N X , P 2 N 2 , P S N 3 are respec- 
 tively b sin 6} , b sin # 2 and b sin 6 3 , where a cos 9 3 and b sin 2 are imaginary. 
 
106 
 
 The Imaginary in Geometry 
 
 Hence a cos 6, b sin 6, where 8 can have all values, real or purely 
 imaginary, are the coordinates of points on the real, the (1, i) and the 
 (i, 1) branches of the curve. 
 
 (6) Construct the real branch, the (a, /3) branch* [see Art. 127] and 
 the (/3, a) branch. Let a' and b' be the semi-conjugate diameters OA and 
 OB of the curve corresponding to these branches so that the equation 
 
 f/So)/' 
 
 
 z'fojS) 
 
 x y % 
 of the curve referred to these diameters as. axes is — r „ + rr = 1- On OB 
 
 a" . o 2 
 
 take two points B' and 2?/ at distances equal to- OA from 0. Describe 
 
 an ellipse through A, B', A', B^ to touch the given ellipse at A. The 
 
 equation of this ellipse is x 2 + y 1 = a" 2 . Construct the imaginary 
 
 branches of this ellipse corresponding to the axes OA and OB. 
 
 Take P 1; P 2 , P, points on the real branch, the («, 0) branch and the 
 
 (ft, a) branch respectively of the ellipse, and let the ordinates P 1 N 1 , 
 
 P^N 2 , P 2 N 3 at these points meet the corresponding branches of the 
 
 ellipse af + y 2 = a' 2 in P/, P ? ', P 3 ' respectively. 0N t and 0N 2 are real 
 
 * This is the branch for which the coordinates are x, iy, the axes of coordinates being 
 the conjugate diameters of the curve, which make angles a and /S with the major axis. 
 
Imaginary Eccentric Angles 107 
 
 and ON 3 is purely imaginary. Through N lt N 2 , and N 3 erect perpen- 
 diculars NiQ x , N«Q 2 , and N 3 Q 3 equal respectively to N^', N 2 P 2 , and 
 N 3 P 3 , the first and third of which are real and the second purely 
 imaginary. Let Q 1 0N 1 , Q 2 0N 2 , and Q 3 ON 3 be U 2 , and 3 respectively. 
 
 Then OQf = ON 2 + N, Q, 2 = ON, 2 + N.P,' 2 = «' 2 . 
 
 Similarly 0Q 2 2 = 0Q 3 2 = a' 2 . 
 
 Therefore 0N lt 0N 2 , 0N S are respectively a'cos^, a'cos# 2 , and 
 a' cos 8 3 . Hence from the equation to the ellipse Pi-JTi, P a N 2 , P 3 N 3 are 
 respectively b sin 0j, & sin 8 2 , and 6 sin # 3 . 
 
 Hence a cos #, 6 sin 8, where 8 can have all values real or 1 purely 
 imaginary, are the coordinates of points on the real, the (a, /3) and the 
 (yS, a) branches of the curve, the axes of coordinates being inclined at an 
 angle /3 — a. If the major axis is taken as the initial line the eccentric 
 angle of a point on the (a, /3) branch is of the form (a + 8i). Hence, if a 
 be constant, points on this branch are obtained by varying #,-, and the 
 different (a, /3) figures are obtained by varying a. 
 
 The locus of Q x , Q 2 , Q 3 , ... is a circle described on A A' as diameter. 
 
 (c) Similarly the coordinates of a point on the hyperbolae 
 
 n~ 
 
 --75 = 1 and ---+f =1 
 a? b 2 a 1 o 2 
 
 may be expressed in the form a cos 8, ib sin 8 and ia cos 6, b sin 6, where 6 
 can have any value, real or purely imaginary. If the axes are inclined 
 at an angle a> the points lie on (a, /8) and (/S, a) branches, where a and 
 j3 give the pair of conjugate diameters, which are inclined at an angle o>. 
 
 Consider generally what is represented by the point 
 a cos (0 + sOi), b sin (6 + s^)- 
 These coordinates may be written 
 
 a cos 6 cos s0 x - a sin 9 sin s0 x , b sin 8 cos s#, + b cos 8 sin s^ . 
 Now, if cos sd-L is real, sin s0 x is imaginary and vice versa. Considering 
 the real and imaginary parts of these coordinates and assuming that 
 cos s0! is real, a point is given by a real length 
 
 Va 2 cos 2 8 + b 2 sin 3 . cos s8 lt 
 
 measured in a direction making an angle tan -1 a with the axis of 
 
 00 a cos a 
 
 x, and by an imaginary length 
 
 Va 2 sin 2 8 + b* cos 2 8 . sin s0 lt 
 
108 The Imaginary in Geometry 
 
 measured in a direction making an angle tan -1 ; — a with the axis 
 
 ° ° asm 8 
 
 of x. a 2 cos 2 8 + b* sin 2 8 and a 2 sin 2 8 + ¥ cos 2 8 are the squares of semi- 
 conjugate diameters of an ellipse of semi-axes a and b, the eccentric 
 angle of the end of a diameter being 8. Their directions are also the 
 direction of a pair of conjugate diameters, for if m and m' be the tangents 
 of the angles which they make with the axis of x, then 
 
 ¥ sin 8 cos 8 ¥ 
 
 mm = — 
 
 a 2 cos 6 sin 6 a 2 
 
 Hence if s8 l = 0, the real branch of an ellipse of semi-axes a and b is 
 given ; if 8 = 0, the (1, i) and (i, 1) branches are given ; if 8 has a con- 
 stant value the (8, $) and (<j>, 0) branches of the ellipse are given, where 
 <f> is the angle which the diameter conjugate to the diameter given by 
 8 makes with the axis of x. 
 
 75. Evaluation of integrals. 
 
 If sin = -j=r , then cos 5 = . v \Jax 2 +'3,hx4-b, 
 
 \/h?-ab *JW — ab 
 
 1 ax+h , d8 -iJa 
 
 ta,n8=—. . and 
 
 *V a *Jax 2 + 2hx + b dx ' *Jax' i +2hx+b'' 
 
 whatever are the values of a, h, b and x, provided a is not zero. These results may 
 of course be applied for the evaluation of integrals. Thus if it is required to 
 evaluate the integral 
 
 \f(\/ax 2 +<Zha:+b)dx, 
 this integral may at once be written as 
 
 f ,(\IW- ah cos 8\ >Jh 2 -ab ... 
 
 f -!-— — —. . cos 8 d8. 
 
 } J \ t-Ja } a 
 
 An integral of the form 
 
 I / (ax 2 + 2 hx + b, x) dx 
 
 becomes at once 
 
 f ,f*Jh*-ah „ -Jh 2 -abs\a6-b\ >Jh?- 
 
 I f I - -T-. cos 8, . 
 
 J J \ i*/a a ] a 
 
 It may be noticed that 
 
 d sin id . . , . , 
 -00 * sin (*<? + -), 
 
 d cos id 
 
 dd ~~ 
 
 <K)- 
 
Imaginary Straight Lines 
 
 109 
 
 ■(l) 
 
 76. Tracing of imaginary straight lines. 
 
 Let the equation of the pair of lines considered be 
 
 y 1 + m 2 « 2 = 0, 
 
 the axes being rectangular. 
 
 (1) Draw a system of straight lines parallel to the axis of y. Any 
 one of these straight lines PNP' will 
 meet the axis of a; in a real point N, 
 and the pair of straight lines in a pair 
 of conjugate imaginary points P and 
 P'. The locus of the points P and P' 
 so obtained is a pair of lines OP and 
 OP', which make angles tan -1 (+ im) 
 with the axis of a-. 
 
 (2) Change the axes of coordi- 
 nates so that x sin a — y cos a = and 
 x sin /S — y cos /3 = are the axes of x 
 and y respectively, the angles a and /3 being real. 
 
 The equation of the pair of lines is then 
 X 2 (sin 2 a. + m 2 cos 2 a) + F 2 (sin 2 £ + m 2 cos 2 /3) 
 
 + 2XF(sin a sin /3 + m? cos a cos /3) •■ 
 Then the equation is 
 
 = 
 
 -6> 
 
 P' 
 
 0. 
 
 Let tan a tan /S = — m 2 . 
 sin 2 a + m 2 cos 2 a 
 
 F 2 + X 
 
 or 
 
 or 
 
 sin 2 /8 + m 2 cos 2 /3 
 F 2 
 
 X 2S 4^=0 
 
 F 2 -X 2 
 
 sin 2/8 
 M-m*M' 
 
 M' - wi 2 i!f 
 iftana = il/ and tan £ = .¥'. 
 Hence if m 2 = 1 the equation 
 becomes F 2 + X 2 = 0. 
 
 The locus of the points of 
 intersection of the given lines 
 with real lines parallel to the new 
 axis of y, can as in the previous 
 case be shown to be a pair of lines OP and OP' which pass through 0, 
 P and P' being conjugate imaginary points. 
 
110 
 
 The Imaginary in Geometry 
 
 If tan a tan /S = — m° the lines y — x tan a — and y — *' tan /3 = are 
 harmonic conjugates of the lines y 2 + m 2 # 2 = 0. They are also a pair of 
 
 conjugate diameters of the ellipse - 4- ^ = 1 if - = in. 
 
 Hence the graphic representation of a pair of conjugate imaginary 
 lines is as follows. The pair of conjugate imaginary lines are the 
 double rays of a real overlapping involution pencil. All pairs of real 
 conjugate rays of the pencil are harmonic conjugates of the pair of given 
 lines. If any pair of these conjugate rays be taken as axes of coordi- 
 nates the graph of the pair of imaginary lines with respect to these 
 axes is a pair of straight lines through their real point. By varying 
 the pair of conjugate rays which are taken as axes all points on the 
 given straight lines can be graphically represented. 
 
 To construct the point of intersection of a real line with an imaginary 
 line. 
 
 Take as axes of coordinates the 
 bisectors of the angles between the 
 imaginary line and its conjugate 
 imaginary line. Let the combined 
 equation of these lines be 
 y* 4- m*a? = 0. 
 
 Let the given real line (I) make 
 an angle /3 with the axis of x. 
 
 Draw through a line, making 
 an angle « with the axis of x, where 
 tan a. tan y3 = — m 2 , to meet I in N. 
 Let QN~=h. 
 
 Then the equation of the lines 
 3/ 2 + m 2 * 2 = referred to ON and a 
 line parallel to I through as axes is 
 
 sin 2o 
 
 Y*-X 
 
 A Y . ■ / sin 2a 
 
 0, or ? =±n/ - — 
 JC V sin 
 
 Therefore 
 
 sin 2/3 
 
 NP=-NP' = -ih x /-^ 
 V si 
 
 sin 2/3" 
 
 sin 2a 
 
 sin 2/3 ' 
 
Imaginary Straight Lines 111 
 
 Hence P and P' are a pair of conjugate imaginary points and for 
 lines parallel to I they lie on the lines OP and OP'. 
 
 The required points P and P' are the points of intersection of I with 
 the graph of the lines, for which the imaginary axis is parallel to I and 
 the real axis is the harmonic conjugate of this line with respect to the 
 pair of imaginary lines. 
 
 If the point A where the Hue I meets the axis of X is fixed and OA =■■ K, it may 
 be easily deduced that, when 8 varies, the locus of N is the ellipse 
 
 A'V 
 
 X ' 2, ____ 
 
 (t) H) 
 
 If the values a^ + ii'j, x 1 tan a + ix 2 tan 8 [see Arts. 129 and 132] are substituted 
 for x and y in y- + m-x' i = 0, it follows at once that 
 
 tan a tan 8 = — m' and — r,= — ; . 
 
 xf tan a 
 
 _ „ x, m _ tan 8 
 
 Therefore — -= ± t = + . 
 
 x 2 tan a ot 
 
 77. To find the imaginary angle, which the line OP' in the preceding makes 
 with the axis of x. 
 
 Let 8-a = a> and P'OX' be 6. 
 Then from the equation of the line 
 
 /sin 2a P'JV sin 8 
 
 /sin 2a _ - 
 
 V sin 2/3 _ 
 
 sin 20 ON sin(<o + 0)' 
 
 /sin 2/3 
 Therefore sin w cot + cos <» = » / ■ n ■ 
 
 V sin za 
 
 Now 5 will generally be a complex angle. Let 6 = 8 1 + s6 2 . Then 
 
 tan fl 1 + tan«fl 2 
 
 tan0=tan(0 1 + *0 2 )^ 
 
 v ' r 1 — tan 0i tan «0 2 
 
 1 — tan 6. tan «0 2 . /sin 2/3 
 
 Therefore sin a> — — -t—\ ^- + cos <b = A / _ — si . 
 
 tan^+tans^ /y sm 2a 
 
 /sin 2/3 . . 
 Equating real and imaginary parts, since ^J ^- 2a is imaginary, 
 
 sin (B + coswtan 6,= . / -. — ~ tan s0 2 , (1) 
 
 V sin 2a 
 
 an d (cosa)-sino>tan5 1 )tansd 2 =tan^ 1 */— — 5- (2) 
 
112 
 
 The Imaginary in Geometry 
 
 . sin 2/3 
 Therefore (sin a + cos a> tan 61) (cos a> - sin a tan e x ) = tan 6^ g . p gn , 
 
 „ v • „ „ sin 2/3 
 or sin ((o + flj cos (a)+5 1 )=sin 9^ cos X - — ^- . 
 
 Therefore sin 2a sin {2a> + 2^} = sin 20! sin 20, 
 
 whence it follows that 6 x = a. 
 . Let #i = a in (1), then 
 
 /sin 2/3 . a 
 smco + cos <b tana= » / -^— — tanw 2 . 
 V sin 2a 
 
 Therefore 
 
 /sin a sin /3 
 V cos a ( 
 
 sin 2a 
 = tani 
 
 COS0 
 
 Therefore im = tan s# 2 ■ 
 
 Hence the angle P'OA = NOA - NOP' 
 
 = a-(a+tan~ 1 iro) 
 
 = - tan - 1 (im)=tan ( — im). 
 
 This is the angle which the line y + imx=0 makes with the axis of x. 
 
 It will be noticed that in the preceding the positive sign was given to 
 in the first equation. The reason for this is set forth in Art. 126. 
 
 /sin 2/3 
 V sin 2a 
 
 78. Critical lines and the circular points at infinity. 
 
 Consider the double points E wAE' of an overlapping involution, 
 which is orthogonal at S. If, SO be the per- 
 pendicular from *S on the base, then 
 
 0E=-0E'=i.S0. 
 
 Hence if P be any point on SE and N 
 the foot of the perpendicular from P on SO, 
 
 SO 1 +0E* = and SN°- + PN* = 0. 
 
 Hence the measure of the distance of any 
 point on one of these lines, SE and SE', 
 from the point S in which they intersect (or 
 from any other point on the lines) is zero. 
 These lines are termed the critical lines of the point (see Art. 22). 
 They are the locus of all points in the plane the measure of whose 
 distances from S is zero. 
 
 Since the measure of the distance between any two points on one of 
 these lines is zero, the definition of the sine and cosine (and hence also 
 
Critical Lines 113 
 
 of the tangent) of the angle between two lines fails, when one of them 
 is a critical line. 
 
 It maybe noticed that even the formula c 2 =a 2 + 6 2 -2a&cosO fails for a real 
 triangle in certain cases. Let A, B be two real 
 points at a finite distance apart. Through A 
 and B draw two parallel lines AC, BO, to make 
 given equal angles with AB. Then AC and BC 
 meet at infinity. 
 
 In the triangle ABC, by the given formula 
 A B* = A C 2 + BC* - ZA C . BC cos C 
 = (BC- A Cf, since cos C= 1. 
 
 But, if BC—AC represents anything, it re- 
 presents the distance BN,. where AN is the perpendicular from A on BC, 
 
 Hence AB*=BN i . 
 
 This obviously is not true unless C lies on AB. Since even the formula quoted 
 fails in certain cases for a real triangle it is not a matter of surprise that it and 
 other formulae should fail for imaginary points, lines, and angles in certain cases. 
 
 Generally a real or an imaginary straight line meets the critical lines 
 of different points in the plane in different pairs of points. But this is 
 not the case with "the line at infinity." On the line at infinity, regarding 
 the region at infinity as a straight line, all involutions of orthogonal lines 
 through different vertices determine the same involution and therefore 
 the critical lines of different points meet the " line at infinity " in the 
 same pair of imaginary points. Hence the critical lines of a point may 
 be regarded as the connector^ of the point in question to this pair of 
 points which have been termed the circular points at infinity, but may 
 be better called the critical points of the plane. (See Art. 22.) 
 
 For the graphic representation of the critical lines of a point see 
 Arts. 76 and 147. 
 
 Hence unless two points are situated on a critical line it follows that, 
 if the measure of the distance between them is zero, they must coincide. 
 
 Analytical. The equation of the pair of bisectors of the angles between the lines 
 
 #2 -f y 2 = is — — y- = ~ . Hence the bisectors of the angles between a pair of critical 
 
 lines are indeterminate. 
 
 It may be noticed that, -if it be assumed that the addition formula holds for the 
 
 tangents of angles involving the angle s — , then whatever angle a may be 
 
 tan o + !j =i, unless tan a + i= 0. 
 H. I. G. .8 
 
114 The Imaginary in Geometry 
 
 The value of y, which corresponds to the value * — of s6 in the formula, 
 
 tan s8 = _ a , is infinite. Heuce a finite addition to y does not generally make 
 
 any difference in the value of y. 
 
 The critical lines through a point have an equation of the same form whatever 
 pair of lines at right angles through the point are taken as axes of coordinates 
 (see Art. 76). Let the coordinates of any point referred to rectangular axes be 
 x and y and those referred to another pair of axes x 1 and y', the two systems of 
 axes being inclined at an angle a. The equations of the critical lines of the origin 
 referred to the first axes are x + iy—O and x — iy=0 and those referred to the second 
 axes are 3/+iy' = and x? — iy'=0. If the second axes are rotated through an 
 angle a by means of the usual substitutions, the first equations are obtained from 
 the second. 
 
 The critical points in a plane and the critical lines generally are considered 
 in Art. 147. 
 
 79. The perpendicular from a given point a? y' on an imaginary line ax + by + c=0. 
 In the same way as when the line and point are real it may be proved that the 
 perpendicular from the point x!y' on the line is 
 
 ax' + by' + c 
 
 Equation of the bisectors of the angles between a pair of conjugate imaginary lines. 
 
 Let the equations of the lines be 
 
 ax + by+c + i(a'x + b'y+c')=0 
 and ax + by + c — i(a'x+b'y + c')—0, 
 
 or shortly L + iL'=0 and L — iL'=0. 
 
 Equating the perpendiculars from any point on the two lines and squaring, the 
 
 equation is found to be 
 
 £ 2 -Z' 2 LL' 
 
 a 2 + 6 2 -(a' 2 + &' 2 ) aa' + bb" 
 
 (ax + by + c) 2 - {a! x + b'y + c') 2 _ (ax + by+c)(a'x + b'y + c') 
 
 or , (a 2 + 6 2 )-(a' 2 + 6' 2 ) ~ aa' + bb' (> 
 
 If c = e' = 0, the combined equation of the lines is 
 
 (ax + byf + (a'x + b'y) 2 = 
 
 and by the usual formula the equation of the bisectors of the angles between these 
 
 lines is f 
 
 x^—y 1 xy 
 
 tf+a'*- W-b't ~ ab + a'b' ' 
 
 If c and c' are made equal to zero in (1) the two equations may be shown to be 
 
 identical. 
 
 80. Systems of lines through a point. v 
 
 From the earlier articles of this chapter it will be seen that a real point contains : 
 (1) an infinite number of real lines determined by real angles measured from a 
 base line drawn through the point : 
 
Imaginary Straight Lines 115 
 
 (2) an infinite number of what may be termed purely imaginary lines determined 
 by angles, which are entirely imaginary, measured from the base line : 
 
 (3) an infinite number of infinite systems of imaginary lines — whose determining 
 angles with reference to the base line are complex angles, i.e., angles which may be 
 expressed as the sum or difference of real and purely imaginary angles. Each infinite 
 system may be obtained by measuring imaginary angles from some real line of (1), 
 or by measuring real angles from some line of (2). 
 
 The system of lines through an imaginary point is of the same nature except for 
 the fact that the point itself is imaginary. 
 
 The above should be compared with the statement as to points on a real line 
 given at the commencement of Art. 51. 
 
 Systems of lines through a real and through an imaginary point considered in 
 reference to the critical lines. Through every real point there passes : 
 
 (1) a pair of lines termed the critical lines of the point. These lines are the 
 connectors of the point to the circular points at infinity : 
 
 (2) an infinite number of pairs of real lines, which are harmonic conjugates of 
 the critical lines and are therefore at right angles, also an infinite number of pairs 
 of imaginary lines, which are harmonic conjugates of the critical lines and therefore 
 at right angles. Such pairs of imaginary lines are not generally pairs of conjugate 
 imaginary lines. 
 
 Since every line through a point has a conjugate in every involution at the 
 point it follows that (2) includes all the lines through the point. 
 
 (3) an infinite number of pairs of conjugate imaginary lines. Each pair of such 
 conjugate imaginary lines has a pair of the real orthogonal lines of (2) for the 
 internal and external bisectors of the angles between them. Those pairs of conjugate 
 imaginary lines, which have the same pair of real orthogonal lines for bisectors, form 
 an involution of which the real bisectors are the double rays. The critical lines 
 being harmonic conjugates of all pairs of real orthogonal lines through the point are 
 a pair of conjugate rays of all such involution pencils. 
 
 Through every imaginary point there passes : 
 
 (1) a pair of critical lines which are the connectors of the point to the circular 
 points at infinity : 
 
 (2) one real line which is its connector to its conjugate imaginary point : 
 
 (3) a line perpendicular to this line, which is real except in so far as it passes 
 through the imaginary point, i.e. this line would be real if the imaginary point were 
 looked upon as the origin : 
 
 (4) an infinite number of pairs of lines which are harmonic conjugates of the 
 critical lines and are therefore at right angles. These pairs of lines divide themselves 
 up into two groups (a) those which would be real if the point were taken as origin 
 and (6) those which would still be imaginary : 
 
 (.5) an infinite number of pairs of lines obtained by joining the point to pairs of 
 conjugate imaginary points. These pairs of lines have the pairs of lines (a) of (4) 
 for bisectors of the angles between them, and those pairs which have the same pair 
 
 8—2 
 
116 The Imaginary in Geometry 
 
 of bisectors form an involution. The critical lines are a pair of conjugate lines of 
 all these involutions. 
 
 Hence, as might be expected, the system of lines through an imaginary point is 
 the same as that through a real point, except for the fact that the point through 
 which they all pass is an imaginary point. . This also follows from the fact that 
 the origin may be regarded as either a real or an imaginary point. The same was 
 shown to be the case for real and imaginary straight lines. 
 
 If a+ib, h+ik be an imaginary point the equation of the real line through it is 
 
 x — a y-h . ., ,. , ,. x-a — ib y — h — ih „ 
 -v— = *—r- and the perpendicular line r h - — £ — =0. 
 
 81. Theorems connected with projection. 
 
 If five points A, B, G, D, E, are situated on an imaginary straight 
 line, other than a critical line, and (ABCB) = {ABCE), then E and D 
 coincide, AC, AD, etc., being the measures of the distances between the 
 points. 
 
 The imaginary line can be rotated round its real point into coincidence 
 with some real line through that point. In this position A, B,G, D, E, 
 will coincide with points A , i? , C , B , E and the anharmonic ratio of 
 any four of the points A, B, G, D, E will be equal to, the anharmonic 
 ratio of the corresponding four of the points A , B , G , D , E . 
 
 Hence since {ABGD) = (ABCE), ' 
 
 (A B C Do) = (A B G E ). 
 Therefore B E must be zero. 
 
 Hence the measure of BE must be zero. 
 
 Therefore unless D, E are on a critical line they must coincide 
 (Art. 78). This result may also be obtained by equating the an- 
 harmonic ratios of the measures of the distances between the points. 
 
 Two projective ranges on an imaginary straight line have a pair of 
 self-corresponding points, real, coincident, or imaginary. 
 
 This follows in a similar way from Art. 21 (6). 
 
 82. (1) A pencil with an imaginary vertex is intersected by all 
 real transversals in ranges which are equi-anharmonic. 
 
 This has been proved in Art. 21. 
 
 (2) A pencil with an imaginary vertex is intersected by all imaginary 
 transversals in ranges which are equi-anharmonic. 
 
 This theorem may be proved as in Arts. [10] and [11] of the Principles 
 of Projective Geometry. The proof there given depends on the fact. 
 
Imaginary Projection 117 
 
 that the sides of a triangle are proportional to the sines of the opposite 
 angles, and this theorem has been shown to be true for an imaginary- 
 triangle, provided that no side is a critical line. 
 
 The theorem may also be deduced from the general case of Menelaus' 
 theorem. (Art. 65) as follows. 
 
 Let S be the vertex of the pencil and let two imaginary transversals 
 meet the rays in A,- B, G, D, 
 
 and A', B', C, D' respectively ,*, 
 
 and intersect at 0. Through S 
 draw any line to mee^ the trans- 
 versals in U and V. Let the ~"A''a 
 ratios of A, B, C, D, A', B', C, D' / ~f ^LJ 
 and of 8 with respect to the /' / / ; ~--: ; u 
 vertices of the triangle OUV 
 be a, b, c, d, a', b', c', d' and s 
 respectively. ^ 
 
 Then by Menelaus' theorem 
 
 aa's=l, bb's=l, cc's = l, etc, 
 
 B' C D' 
 
 Therefore aa' = bb' = cc'= = -. 
 
 Therefore (ABCD) = (A'B'0'D'). 
 
 The point is obviously a self-corresponding point of the ranges. 
 
 Correlatively the pencils formed by projecting a range situated on an 
 imaginary line from two imaginary points are equi-ankarmonic. 
 
 This may be proved in a similar way to the above by the general 
 case of Ceva's theorem (Art. 65). 
 
 (3) If two pencils with imaginary vertices have the connector of their 
 vertices for a self-corresponding ray, all pairs of corresponding rays 
 intersect on a fixed straight line. 
 
 Let s be the line joining the vertices S and S' of the pencils. Let 
 corresponding rays a, a' and b, b' \s 
 
 intersect respectively at A and /f v '-\ 
 
 B. Join AB by the line u. Let ,/ y \ b ^K^ 
 
 d and d' a pair of corresponding / I \ \ s' 
 
 rays meet u in D and D' and let / ; \.--"„-s 
 
 SS' meet u in 0. / X'''''y'* 
 
 Then (ABOD) = (ABOU). -—j.-^-l-^-—- 
 
 Therefore D and D' coincide (Art. 81). 
 
 
118 The Imaginary in Geometry 
 
 Conversely if three pairs of corresponding rays intersect on a straight 
 line, the connector of the vertices is a self-corresponding ray. 
 
 In a similar manner it is possible to prove the correlative theorem, 
 viz., if the point of intersection of the bases of two projective ranges is a 
 self-corresponding point, then the ranges are in plane perspective. 
 
 Conversely if the connectors of three pairs of corresponding points of 
 two projective ranges are concurrent, the point of intersection of the bases 
 is a self-corresponding point. 
 
 If two projective pencils are such that for three pairs of corresponding 
 rays the angles between the rays of one pencil are equal to the corre- 
 sponding angles between the rays of the other pencil, the pencils are said 
 to be equal and the angles between all. pairs of corresponding rays are 
 equal. 
 
 This follows from the preceding. 
 
 83. Real and imaginary correspondence. 
 
 If a pair of corresponding elements of two superposed projective forms are given 
 by a relation 
 
 Axaf + Bx + Gx' + D = 0, 
 
 where x, x 1 determine pairs of corresponding elements, the correspondence is said 
 to! be real if A, B, C, D are real. 
 
 If pairs of corresponding elements are given by a relation 
 
 {A + iA') xx' + {B + iB') x+(G+iG')x' + D+iD' =0, 
 the correspondence is said to be imaginary. 
 
 (1) Any pair of superposed projective pencils with a real vertex, and any pair of 
 superposed projective ranges on a real base, have either two real elements of one which 
 correspond to two real elements of the other or a pair of conjugate imaginary elements 
 of one which correspond to a pair of conjugate imaginary elements of the other. 
 
 Consider the correspondence of two superposed projective ranges on a real base. 
 
 This is given by 
 
 (A+iA')xa/ + (B+iB')x+(G+iC')x'+(D + iD') = 0. 
 
 The real elements are given by 
 
 Axx' + Bx+Cx' + D=0, 
 and A'xx' + B'x+C'x' + D' = 0. 
 
 Therefore (A'B -AB')x+(CA' - C'A)x' + (A'D-AD')=0, (1) 
 
 and (BA'-B'A)x 2 + (BC'-B'C+DA'-iyA)x+(DC'-D'C)=0. 
 
 The roots of this quadratic are either (1) a pair of real quantities or (2) a pair 
 of conjugate imaginary quantities. From (1) these correspond to a pair of real 
 quantities or to a pair of conjugate imaginary quantities. 
 
Imaginary Correspondence 119 
 
 The theorem for a pair of pencils is obtained by joining the points of the range 
 to two real vertices. 
 
 (2) Every pair of superposed projective pencils with a real vertex, and every pair of 
 superposed projective ranges on a real base, have two elements of one which correspond 
 to the conjugate imaginary elements of the ot/ier, either each to its own imaginary con- 
 jugate or each to the imaginary conjugate of the other. 
 
 Write y + iy' for x xaAy — iy' for x'. Then equating real and imaginary parts it 
 is found that 
 
 A (y*+y'*) + (B+C)y+(C'-B')y'+D=0, 
 and A'l V *+y'*) + (B' + C')y+(B-C)y' + B J =0. 
 
 Hence Ky+Li/+M=0, (1) 
 
 where 
 
 K=A'(B+C)-A{B' + C), . L = A'(0'-B') + A (C-B), M=A'D-AD, 
 and y 2 ,(Z 2 + £ 2 )+y{2A'Jf+X(C 2 + C" 2 --B' 3 -5 i! )} + if 2 
 
 + L{D>{C'-B') + D{C-B)} = (2) 
 
 (a) If the roots of (2) are real, viz., y 1 and y 2 , the corresponding values of 
 y' from (1) give a pair of conjugate imaginary points, and two pairs of conjugate 
 imaginary points are corresponding points. 
 
 (6) If the roots of (2) are imaginary, viz., (a + ib) and (a — ib), let the corre- 
 sponding values of y' from (1) be (c+id) and (c — id). 
 Then {a + ib) and (c+id) give as corresponding points 
 
 (a + ib) + i (c+id) and (a+ib) — i(e+id), 
 i.e., (a-d)+i(b + c) and (a + d)+i(b-c); 
 
 also-(a-i6) and (c — id) give as corresponding points 
 
 (a — ib) + i (c — id) and (a — ib)-i(c — id), 
 or (a+d)-i(b-c) and (a-d)-i(b + c). 
 
 Hence the theorem is proved. 
 
 84. The following are immediate consequences of Arts. 81 and 82 and may be 
 proved in the same way that the corresponding theorems for real lines and points 
 are proved in The Principles of Projective Geometry. They hold for all imaginary 
 points and straight lines, excluding critical lines and pairs of points situated on the 
 same critical lines. 
 
 (i) The properties of two triangles in perspective, Art. [13 (as)]. 
 
 (ii) The harmonic property of the quadrangle and quadrilateral, Art. \4o]. 
 
 (iii) Construction of harmonic conjugates depending thereon, Art. [46]. 
 
 (iv) The involution property of the quadrangle and quadrilateral, Art. [56]. 
 
 Also the properties of involution ranges and pencils hold generally for pencils 
 with imaginary vertices and for ranges on imaginary straight lines. Those for real 
 involutions are set forth in Art. 7. 
 
 Formulae connecting the angles of an imaginary pencil. 
 
 Let a, b, c, d be any four concurrent lines real or imaginary (excluding critical 
 
120 
 
 The Imaginary in Geometry 
 
 lines) and let ab denote the angle between the lines a and b. Then as in Art. [11] it 
 may be shown that the anharmonic ratio of the pencil a ; b, c, d is 
 
 sin ac sin ad , , _ 
 
 ~ : —. — ^ = (abed). 
 
 sin be sin bd 
 
 The relations connecting the angles, when the pencil is harmonic, may be proved 
 as in The Principles of Projective Geometry. 
 
 85. Projection of points into the circular points at infinity. 
 
 To project by a real projection any pair of conjugate imaginary 
 points into the circular points at infinity. 
 
 Let the given pair of conjugate imaginary points be the double 
 points of the involution A A', BS, situated on the real line v in the 
 plane <r. Through v draw any plane <r 1; and take any plane a parallel 
 to o"! as plane of projection. On AA', BB', in the plane <r l , describe 
 two semi-circles to intersect at $. They will always intersect since the 
 segments AA' and BB' overlap. Take 8 as centre of projection. Then 
 
Projection into Critical Points 
 
 121 
 
 the line v is projected into the line at infinity in the plane cr' and the 
 involution A A', BB' is projected into the orthogonal involution on this 
 line and its double points are the circular points at infinity. 
 
 To project a pair of real points into the circular points at infinity. 
 
 Let E and F, the two real points which are to be projected into the 
 circular points at infinity, be situated on the real line v in the plane <r. 
 
 Through v draw any real plane <r lt and let CI and CI' be the circular 
 points at infinity in this plane. 
 
 Join FC1 and ECU to meet at S, and FCl' and ECl to meet at S'. 
 
 a' cv 
 
 Then, if a plane a parallel to o-j be taken as plane of projection, and 
 8 or S' for centre of projection, E and F will be projected into CI and 
 CI'. These points are the circular points at infinity in the plane a since 
 the planes o-j and a' are parallel. 
 
 In this projection v is projected into a real line, viz., the line at 
 infinity. 8 and 8' are a pair of conjugate imaginary points. Hence the 
 line SS' is real and its point of intersection with v is a real point and 
 is projected into a real point 0' at infinity. The connector to S of any 
 other real point in the plane a must meet the plane a in an imaginary 
 point. For such connector, since it does not contain 8', must be an 
 imaginary line, and an imaginary line can only contain one real point. 
 The real points on the line <r<r' however remain real. 
 
122 
 
 The Imaginary in Geometry 
 
 If D be any real point on the line aar', the line BO corresponds to a 
 real line in the plane a-', viz., DO'. Hence a pencil of real lines through 
 in the plane a corresponds to a pencil of parallel real lines through 1 in 
 c-'. The real points on lines of one pencil correspond to imaginary points 
 on the corresponding lines of the other pencil with the exception of 
 and 0', which are both real, and the points on aa' which are unaltered. 
 
 Hence, if a pair of real points are projected into the circular points 
 at infinity, all real points in the plane are projected into imaginary points 
 with the exception of one point 0, on the connector of the pair of real 
 points, and points on the line of intersection of the planes. 
 
 Consequently it is imaginary points on a locus in the plane a which 
 are projected into real points on the corresponding locus in the plane a, 
 and it is for such imaginary points in the plane a that theorems are 
 proved, when they are deduced from the properties of real points in the 
 plane a. It is only if and when projective properties of real and of 
 imaginary points have been shown to be identical, that the process of 
 projecting real points into the circular points at infinity is justifiable as 
 a means of obtaining a theorem for real points. 
 
 SES'F is a semi-real square of the second kind. 
 
 86. Projection of a semi-real quadrangle into a semi-real square. 
 
 (a) Let the quadrangle be of the first kind. Let A, A', B, B be the vertices and 
 EFG the diagonal points triangle, which is real. Let 
 AA' and BE meet FG in K and L. Project the pair 
 of conjugate imaginary points, - which are the double 
 points of the involution determined by FG and KL, into 
 the circular points at infinity. Then in the new figure 
 
 (1) the lines FAB', FBA' and FE are parallel, as 
 are the lines OB A, GA'B' and GE ; 
 
 (2) the first three of these lines are perpendicular 
 to the second three lines ; 
 
 (3) the lines EK and EL are at right angles ; 
 
 (4) the lengths BE and EB! are equal, as are the 
 lengths AE and EA'. 
 
 Hence in the new figure the two pairs of conjugate 
 imaginary points A, A' and B, B' are situated on two real 
 lines EK and EL which are at right angles. Also 
 EB = B'E and AE = EA' . Also since the pencil 
 (E.KLFG) is harmonic and the lines EF and EG 
 are at right angles, EF and EG are the bisectors of the 
 angle AEB. But ABG is parallel to EG and therefore by symmetry (or see Arts. 53 
 and 66) EA=EB. 
 
 >G 
 
 *-G 
 
Semi-real Square 123 
 
 Therefore the points A, A', B, B! are the vertices of a semi-real square of the 
 first kind. 
 
 (b) Let the quadrangle be of the second kind and let B and B' be the pair of 
 real vertices. Then FG is the real side of the diagonal points triangle and E is its 
 real vertex. F and G are a pair of conjugate imaginary points. Project these 
 points into the circular points at infinity. 
 
 Then in the new figure 
 
 (1) the lines FAB', FBA' and FE are parallel, as are the lines GBA, GA'B' 
 and GE; 
 
 (2) the lines EK and EL are at right angles ; 
 
 (3) the lengths BE and EBI are equal, as are the lengths AE and EA'. 
 
 Hence in the new figure the pair of conjugate imaginary points A, A' and the 
 pair of real points B and B! are situated on two real lines EK and EL which are 
 at right angles. Also EB=B'E and AE=EA'. Also the line ABG, which is 
 parallel to a critical line EG, meets the real lines EA and EB through E, which are 
 at right angles, in A and B. Therefore EA=i. EB [see Art. 78]. Hence A, A', B, B' 
 is a semi-real square of the second kind. 
 
 Conversely it follows that each of the four sides of a semi-real square of the second 
 kind passes through a critical or circular point. 
 
 Signs of trigonometrical functions. 
 
 In Art. 56 the sine, cosine and tangent of the angle between a real and an 
 imaginary straight line were defined. In Art. 61 these definitions were extended to 
 the case of the angle between a pair of imaginary straight lines. In these definitions, 
 as far as the sine and cosine are concerned, there is an ambiguity of sign in respect 
 to the denominator. This arises from the fact that there is a geometrical ambiguity — 
 with real as well as with imaginary straight lines — in regard to the angle between 
 two straight lines. There are four angles, less than 2jt, between two given straight 
 lines real or imaginary and by changing the sign of the square root in the expressions 
 in Arts. 56 and 61 there are — both for real and for imaginary straight lines — 
 four sets of values of sin 6, cos 9 and tan 6 corresponding to the four angles between 
 the straight lines. 
 
 Certain conventions are laid down in the trigonometry of real lines as to which 
 angles are those whose sine, cosine and tangent are represented by the expressions 
 in Art. 56. It is convenient that the conventions introduced, when imaginary lines 
 are considered, should be such as to make the sum and difference formulae, as used 
 for real angles, apply when some or all of the angles considered are imaginary. The 
 expression s/OP* + OQ 2 -20P.0Q cos d> will in general be complex. It would seem 
 that the correct convention is, in the case of the angle between two imaginary straight 
 lines which has the smaller real part, to take the same value of 
 
 *j0P i +0Q*-20P.0Q cos a, 
 in the expressions for sin 6 and for cos 6. This has been done in Arts. 57 to 64, in 
 which the sum and difference formulae for imaginary angles have been deduced. If 
 different signs are given to this expression in the values of siu 8 and cos 6, the angle 
 
124 The Imaginary in Geometry 
 
 n — 6, positive or negative, is obtained. Generally it is believed that it will be 
 found best to assume — as in the geometry of real lines — that the real part of 
 \/0P 2 + OQ 2 -20P.0Q cos <o is positive, when it exists, and to follow the conventions 
 laid down in Arts. 71 and 72, when it is a purely imaginary quantity. 
 
 If the values of sin6 { , cos6t, tan# 4 are obtained by the method of Art. 72 — 
 in which the hypothenuse of the triangle is always real and positive — and if the 
 positive axis of iy is taken in the negative direction of the real axis of y, it is found 
 that the values in question are the same as those given on page 100 with the sign 
 of i changed. 
 
 EXAMPLES 
 
 (1) Prove that the sum of the measures of the distances of a point on an ellipse 
 from the imaginary foci is equal to the minor axis. 
 
 (2) Prove that the sum of the measures of the distances of a point on a hyper- 
 bola from the imaginary foci is equal to the transverse axis, the length of which is 
 imaginary. 
 
 (3) If two conjugate imaginary lines form a harmonic pencil with two real lines 
 which are at right angles, then the real lines are the bisectors of the angles between 
 the imaginary lines. 
 
 (4) Prove that the ratio of the measures of the distances of a point on an 
 ellipse from an imaginary focus and from the corresponding imaginary directrix is a 
 constant quantity which is purely imaginary. 
 
 (5) Prove that the equation of the chord joining two points on an ellipse 
 determined by complex eccentric angles 6 and (j> is 
 
 _cos^+fsm— = cos-^, 
 
 and that the equation of the tangent at the point is 
 
 -cos 5 + t sin = 1. 
 a o 
 
 (6) Prove that the real and imaginary parts of the eccentric angle of the points 
 
 x 2 v 2 b 
 
 where the line y=mx + c meets the ellipse -^ + jj - 1 = are respectively tan ~ l - - 
 
 and tan" 1 — if c 2 >m 2 a 2 + l 
 
 a' o' ' ma 
 
 (7) Prove analytically that the real part of the angle between two imaginary 
 straight lines through the origin is equal to one or other of the angles between the 
 real bisectors of the angles between the lines and their conjugate imaginary lines. 
 
 (8) Prove under the same circumstances that the imaginary part of the angle 
 between the lines is the difference of the angles which the lines make with the 
 bisectors in question. 
 
CHAPTER IV 
 
 THE GENEKAL CONIC 
 87. Definition of a conic. 
 
 Let S and S' be any two points real or imaginary; a, b, c three 
 lines, real or imaginary, through S, and a', b', c' three lines, real or 
 ■imaginary, through S'. 
 
 Then a, a, b, b', c, c' may be regarded as three pairs of corresponding 
 rays of two projective or equi-anharmonic pencils with vertices at S and 
 S', and the ray of one corresponding to any ray of the other may be 
 constructed. 
 
 Def. 5. The locus of the points of intersection of corresponding rays 
 of two projective pencils is a conic. 
 
 It follows immediately from this definition that (1) a conic is com- 
 pletely determined when five points on it are given, and (2) that a 
 conic may be described through any five given points. 
 
 The definition is illustrated by the figure. In this figure a real 
 conic is determined by five real points S, S', A,,, B , C and the pencils 
 (S.A t B,,G ) and (S'.A B G ) are constructed. P is any imaginary 
 point on the conic It can be constructed by means of the -Poncelet 
 figure corresponding to the diameter OK, which is parallel to the real 
 line AA'BB'CC through the point P, and its conjugate diameter OL. 
 
126 The Imaginary in Geometry 
 
 By definition the pencils {S.A B^G P) and (S'. -4 .B C' I) P) are projective. 
 This is the case if (ABCP) = {A'B'C'P), where A, B, G and A', B', C 
 are the points in which the rays of the pencils meet the real line 
 through P. That this is true may be verified by the figure. If P' be 
 •the conjugate imaginary point of P, then, the point M, where OL meets 
 the real line PP', is the mean point of P and P'. 
 
 If the angles a, /3, y, 8 determine four points on a circle the anharmonic ratio of 
 
 the pencil subtended by these points at the centre is — — ^r — —. : -. — )-= — J- , and that 
 * . sin^-'y) sin — 8)' 
 
 a — t 
 
 a — y 
 
 Bm 2 Sln <-> 
 of the pencil subtended at any point on the circle is 
 
 . p-y . /3-S" 
 8111 2 Sm ~T~ 
 
 Similarly, if a, j3, y, 8 be the eccentric angles of points on the ellipse -g + ^ - 1 =0> 
 the anharmonic ratio of the pencil subtended by these points at any point on the 
 
 . a—y . * 
 sin —zr- 1 sin 
 
 ellipse is 
 
 . /3— y . 0-S" 
 Sln 2 Sln -2~ 
 
 If a, /3, y, 8 be complex angles, this expression still represents the anharmonic 
 ratio of the four imaginary points on the ellipse (see Art. 74). 
 
 88. Through five given points, no three of which are collinear, only 
 one conic can be described. 
 
 In Art. 87 the five points S, S', A, B, C completely determine the 
 correspondence between the pencils whose vertices are S and S'. Hence 
 the conic determined by the pencils with vertices S and >S' is com- 
 pletely and uniquely determined. The only question which arises is 
 whether the same conic is obtained if, of the five points, S, S', A, B, G, 
 points other than S and S' are taken as the vertices of the pencils. 
 That the same conic is obtained may be proved as follows. 
 
 To prove that whichever of the five given points, which determine a 
 conic, are taken as the vertices of the two generating pencils, the same 
 conic is obtained. 
 
 Let A, B,G, D, E be the five given points. Join A to G, B, E and 
 B to G, B, E. Let a pair of corresponding rays of the pencils so deter- 
 mined meet at F. Join GF, and let CF meet BB, AB, BE and AE in 
 K, L, M, N respectively. . Then F is the second self-corresponding 
 point of the ranges GKM and GLN and is a point on the conic deter- 
 mined by the pencils with vertices A and B. 
 
The General Conic 
 
 127 
 
 Hence {GFKM) = {GFLN). 
 
 Consider the pencils, DC, DB, DA and EG, EB, EA whose vertices 
 are D and E. They determine ranges GKL and GMN on the line GF, 
 and the second self-corresponding point of these ranges is the second 
 point in which the conic determined by the pencils, vertices D and E, 
 meets GF. Let F' be the second self-corre- 
 sponding point. 
 
 Then (CF'KL) = (GF'MN) 
 
 which is the same as (GF'KM) = (CF'LN). 
 
 But {GFKM) = (CFLN\ 
 
 Therefore F and F' coincide and the two 
 conies meet the line CF in the same pair of 
 points and are therefore the same conic. 
 
 Converse. 
 
 If A, B, G, D, E, F are six points on a conic the pencils subtended 
 by the other four points at A and B are projective. But the five points 
 A, G, D, E, F determine the conic and B may therefore be regarded as 
 any point on the conic determined by these five points. Hence at any 
 point on a conic four given points on the conic determine a pencil of 
 constant anharmonic ratio. 
 
 Hence it follows that a conic may be looked upon as the locus of a 
 point at which four given points, which are on the conic, subtend a pencil 
 of constant anharmonic ratio. It is sometimes convenient to look upon 
 a conic from this point of view. 
 
 89. Every straight line, in the plane of a conic, meets the conic — 
 unless it breaks up into a pair of straight lines — in two points which may 
 be real or imaginary, and may be coincident. 
 
 If any two points on a conic be joined to four other points on the 
 conic, the pencils so formed determine two superposed projective ranges 
 on every line in the plane. The conic meets each line in the self- 
 corresponding points of these ranges, which are two points, real, 
 imaginary or coincident (Art. 81). 
 
 There is one and only one tangent at every point on a conic, which 
 does not break up into a pair of straight lines. 
 
 Let the conic be determined by five points A, B, G, P and 0, real or 
 imaginary. For every point P' on the curve (P' . ABGP) = (0 . ABCP). 
 
128 
 
 The Imaginary in Geometry 
 
 Let P' be a point adjacent to P. Then (P' . ABGP) = (0 . ABGP). 
 
 Since P'A, P'B, P'O and the anharmonic ratio of the pencil 
 (P' .ABGP) are given, the line PP is a given line and is uniquely 
 determined. This straight line is termed the tangent at P and contains 
 all the points on the curve infinitely, near to P. 
 
 If P' is real there will usually be a real point on the curve infinitely 
 near to P and also an imaginary point infinitely near to P. Both of 
 these points lie on the tangent at P > 
 
 90. Given two tangents to a conic and their points of contact, to 
 construct the tangent at any other point on the conic. 
 
 Let the tangents at B and G 
 meet at A. Take P and Q 
 any two points on the conic 
 and let GP, BQ intersect at M 
 and meet AB and AG in G' 
 and B'. 
 
 Then 
 (B.BPQC) = (G.BPQG). 
 
 Therefore 
 (C'PMG)=(BMQB')=(B'QMB). 
 
 Hence G'B', PQ and GB are concurrent at a point T. Let P and Q 
 coincide. Then G'B', BG and ihe tangent at P are concurrent. From 
 the harmonic property of the quadrangle TG'PB it follows that, if TP 
 and AG meet at R, the range ARB'G is harmonic. 
 
 Hence to construct the tangent at any point P, join BP to meet 
 AG at B' and construct R the harmonic conjugate of A with respect to 
 B 1 , C. Then PR is the tangent at P. 
 
 Four fixed tangents to a conic determine ranges of constant anharmonic 
 ratio on all other tangents to the conic. 
 
 Let AG and AB be any two tangents to the conic. Take points 
 Pi, P 2 , P 3 , P 4 on the conic. 
 
 Let BP lt BP„ BP S , BP, meet AG in 5/, B{, B s ', B 4 ' and let the 
 tangents at P u P 2 , P s , P 4 meet AG in the points R u R it R s , P 4 . Then 
 the ranges AR&'C, AR 2 B S 'G, AR 3 B S 'G, ARJBJC are all harmonic. 
 Hence (RiR i R s R i ) = (B 1 'B2B 3 'B i ') = the anharmonic ratio of the pencil 
 
The General Conic 129 
 
 formed by joining P lt P t , P s> P 4 to any point on the conic = a constant 
 for all positions of B on the conic. 
 
 91. Correlative definition of a conic. 
 
 The correlative definition of a conic is as follows : 
 The envelope of the lines joining pairs of corresponding points of two 
 projective ranges is the correlative of a conic. 
 
 From this it follows by means of the correlative proofs 
 
 (1) that the correlative of a conic can be described to touch five 
 given lines: 
 
 (2) the anharmonic ratio of the range formed by the intersection of 
 four fixed tangents to the correlative of a conic with a variable tangent is 
 constant : 
 
 (3) only one correlative of a conic can be described to touch five 
 given lines, no three of which are concurrent : 
 
 (4) through a given point two tangents real, coincident, or imaginary 
 can be drawn to the correlative of a conic : 
 
 (5) every tangent to the correlative of a conic has one and only one 
 point of contact. 
 
 From Art. 90 and (2) above it follows that the correlative of a conic 
 is a conic. 
 
 92. If, through the vertices of two projective pencils, a conic be 
 described, these pencils determine on the conic ranges of points, which 
 are termed projective ranges on the conic. Such ranges subtend pro- 
 jective pencils at every point on the conic. 
 
 The anharmonic ratio of four points A, B, G, D at any point on the 
 conic is. written (ABGB). Hence the condition that A, B, G, B and 
 A', B', G', D' should constitute two projective ranges on a conic is 
 {ABCD) = (A'B'G'D'). 
 
 All pairs of superposed projective pencils have a pair of self-corre- 
 sponding rays. Hence every pair of projective ranges on a conic have a 
 pair of self-corresponding points. If the points of the two ranges be 
 joined to any point on the conic, the self-corresponding rays of the 
 pencils so formed meet the conic in the self-corresponding points of the 
 projective ranges. Likewise, if through the vertex of an involution 
 pencil a conic be described, the involution pencil determines on the 
 conic a range of points which is termed an involution range on the conic. 
 
 H. i. g. 9 
 
130 The Imaginary in Geometry 
 
 The range in question is such that if the points which constitute it be 
 joined to any point on the conic, the pencil so formed is an involution 
 pencil. 
 
 It follows that, if three pairs of points A, A' ; B, B' ; G, C are such 
 that the anharmonic ratio of the pencil formed by joining four of them, 
 not constituting two pairs, to a point on the conic, is equal to the 
 anharmonic ratio of the pencil formed by joining the other points of the 
 pairs to a point on the conic, then A, A' ; B, B' ; 0, C" are pairs of con- 
 jugate points of an involution on the conic. 
 
 Since every involution pencil has a pair of double rays, every 
 involution range on a conic has a pair of double points. 
 
 The correlative theorems and definitions, which hold for a real conic, 
 are obviously true for the conic in general. 
 
 There can be no double point on a conic, which does not break up into 
 two straight lines. For if possible let there be such a point P. Join P 
 to any point Q on the conic. Then the line PQ meets the conic in 
 three points, which is impossible unless the conic breaks up into a pair 
 of straight lines. 
 
 93. It is now possible to prove for a conic in general the important 
 theorems of projective geometry, which are proved for the real branch 
 of a real conic in the Principles of Projective Geometry. The proofs 
 are in most cases similar although in some cases they admit of simplifi- 
 cation in the case of the general conic. They are shortly set forth in 
 the following articles. The correlative theorem may in each case be 
 proved by the correlative method. 
 
 Involution property of a conic. 
 
 A system of lines through any point S, real or imaginary, determines 
 on any conic in their plane pairs of conju- , 
 gate points of an involution. 
 
 Let S be the point and SA A', SBB', 
 SCC' any three chords through it. Join 
 AC' to meet SBB' in 0. 
 
 Then 
 
 (BB'CA) = (C . BB'GA) = (BB'SO) 
 and 
 
 (B'BG'A') = (A.B'BC'A') 
 
 = (B'BOS) = (BB'SO). 
 
The General Conic 
 
 131 
 
 Therefore (BB'GA) = (BBC A'). 
 
 Hence (Art. 92) A A', BB' , CC are pairs of conjugate points of an 
 involution on the conic. 
 
 If SAA', SBB' are regarded as fixed it is seen that SCC is an'' 
 chord through S. ( 
 
 Conversely if A A', BB ', CC are three pairs of conjugate points of an 
 involution on a conic, then AA', BB', CC are concurrent. 
 
 Similarly the correlative theorem and its converse may be proved. 
 
 of a conic be drawn 
 
 94. Pole and polar. 
 
 If through a point S chords SAA', SBB', SCC 
 and L, M, N, ... are the harmonic con- 
 jugates of S with respect to A A', BB, 
 CC, ..., then the locus of L, M, N is a 
 straight line, which is termed' the polar 
 ofS. 
 
 Join A to A', B, C, A, B, ..., and 
 join A' to A, B, C, A', B,.... Then, 
 since AA', BB', CC form an involu- 
 tion, the pencils (A . AA'BB'CC) and 
 {A'. A' A BB CC) are projective. But 
 A A' is a self-corresponding ray of the 
 pencils and therefore (Art. 49(3)) the 
 pencils are in plane perspective and 
 pairs of corresponding rays intersect on a fixed straight line s. 
 
 Since s, in the figure, passes through U and U', etc., the points 
 L, M, N in which s meets the chords through S are, by the harmonic 
 property of the quadrangle, the harmonic conjugates of S with respect 
 to A, A'; B, B; C, C, .... Hence the locus of these harmonic conju- 
 gates is the straight line s. 
 
 The points where s meets the conic, viz. E and F, are the double 
 points of the involution AA' , BB, CC, .... They are the points of 
 contact of the tangents from S to the conic and form a harmonic range 
 on the conic with any pair of conjugate points 'A, A' ; B, B '; etc. 
 
 The correlative theorem for the construction of the pole of a given 
 line can be proved by the correlative method, which is given for the 
 case of a circle in Art. [76]. 
 
 9—2 
 
132 The Imaginary in Geometry 
 
 Every real point in the plane of a real conic has a real polar with respect to the 
 conic. 
 
 A real conic is a conic which is met by every real line in its plane in u, pair of 
 points, real, coincident, or conjugate imaginary. A conic with a real branch is always 
 real as is also a conic with a real equation. The latter includes the former 
 (Art. 106). 
 
 Through a real point S draw real chords SAA' and SBB' to meet the conic in 
 real points (figure, Art. 94) or in pairs of conjugate imaginary points. These form 
 the real or semi-real quadrangle AA'B'B, the diagonal points triangle of which has a 
 real side UU'. The line UU' is the polar of S. By Art. 94 it is the locus of 
 harmonic conjugates of S with respect to the points of intersection with the conic of 
 all chords through S. 
 
 From the preceding it follows that the polar of a real point with respect to a 
 conic, which has no real branch, such as the ellipse —^ + j^ + 1 = 0, is real. 
 
 95. A conic determines on every straight line in its plane an 
 involution, the double points of which are the points of intersection of 
 the conic with the line. 
 
 Let s be any straight line in the plane of the conic. On this 
 straight line take points P, Q, R, .... These points have each a polar 
 with respect to the conic. Let these polars meet the line s in P', Q', 
 R', . . . respectively. Then P, P' ; Q, Q' ; R, R' ; ... are harmonic con- 
 jugates, of the points E and F in which the line s meets the conic. 
 Therefore PP ', QQ', RR', . . . form an involution of which the points of 
 intersection of s with the conic are the double points. If the conic and 
 line are real the involution is real. 
 
 ^Correlatively, a conic determines at every point in its plane an involu- 
 tion pencil, the double rays of which are the tangents from the point to 
 the conic. This involution pencil is real if the conic and point are real. 
 
 Two points P and P' are said to be conjugate points with respect to a conic 
 if (PP'QQ')= — 1, where PP' meets the conic in Q and Q'. The correlative definition 
 holds for conjugate lines. 
 
 96. From Arts. 93 — 95 it follows that the properties of pole and 
 polar hold not only for the real branch of a conic but for the conic in 
 general. From the harmonic property of a point and its polar it follows 
 that every conic is in harmonic perspective with itself, any point and its 
 polar being the centre and axis of perspective. When the point is real, 
 the polar is real, if the conic is real. In this case the real branch of the 
 
The General Conic 133 
 
 conic corresponds to itself and points on imaginary branches correspond 
 to points on imaginary branches. 
 
 If the conic is real and an imaginary point be taken as centre of 
 perspective and its imaginary polar as axis of perspective, real points on 
 the curve correspond as a general rule to imaginary points. 
 
 97. Pascal's theorem. 
 
 The three points of intersection of the three pairs of opposite sides of 
 a hexagon inscribed in a conic are collinear. 
 
 Let A, B, C and A', B', C be any six points on a conic. If A, A'. ; 
 B, B' ; C, C are looked upon as pairs of corresponding points, they 
 determine two projective ranges on the conic. 
 
 Consider the pencils 
 
 (A.'. ABC) and (A.A'KC). 
 They have a self-corresponding ray A A' and are therefore in perspective. 
 Hence the lines -which join any pair of 
 corresponding points of the ranges to A' and 
 A intersect on a line s, which passes through 
 K and L in the figure. This line meets the 
 conic in the pair of self-corresponding points 
 U and V of the projective ranges on the 
 conic. 
 
 Similarly consider the pencils 
 
 (B'.ABC) and (B.A'B'C). 
 They have a self-corresponding ray BB' and are therefore in perspective. 
 As in the previous case pairs of corresponding rays of the pencils 
 intersect on a line s', which passes through K and M. This line 
 s' meets the conic in the pair of self-corresponding points U and V of 
 the projective ranges on the conic. Hence s' must coincide with s. 
 Therefore K, L, M are collinear. But these are the points of inter- 
 section of the pairs of opposite sides of the hexagon AB'GA'BC. Hence 
 the theorem is proved. 
 
 98. Briarichon's theorem. 
 
 The correlative theorem, Brianchon's theorem, may be proved by 
 the correlative method. The theorem is to the effect that, the three 
 connectors of the three pairs of opposite vertices of a hexagon circum- 
 scribed to a conic are concurrent. 
 
134 The Imaginary in Geometry 
 
 The converse of Pascal's theorem and also of Brianchon's theorem 
 hold as for the real branch of a conic (see Art. [100]). 
 
 In the proof of Pascal's theorem, if A, A'; B, B"; G, C be three pairs 
 of conjugate imaginary points the Pascal line is real. For in the figure 
 K, L, M are all points of intersection of pairs of conjugate imaginary 
 lines and are therefore real. 
 
 If A, A'; B,B' are pairs of conjugate imaginary points and G and C 
 are real, then /the Pascal line is imaginary. For K is real while L and 
 M are imaginary. The real lines through L and M are the polars of 
 AA' . GC and BB' . CC. Hence no other real lines can pass through L 
 and M and therefore the Pascal line is imaginary, K being the real 
 point on it. 
 
 Similarly if A, A' are conjugatejmaginary points and B, B', G, C are 
 real, the Pascal line is imaginary, M being the real point on it. 
 
 The condition that three imaginary points should lie on a real conic 
 is that the hexagon formed by these points and their conjugate imaginary 
 points (as opposite vertices) should be a Pascal .hexagon. This condition 
 is clearly satisfied if the real lines through the imaginary points are 
 concurrent, for in this case the Pascal line is the polar of their point of 
 intersection. Hence three conjugate imaginary points, the real lines 
 through which are concurrent, determine a real conic. 
 
 99. Self-corresponding elements. 
 
 Determination of the self-corresponding elements of two superposed 
 projective pencils or ranges. 
 
 Let a, b, c and a', V , c' be corresponding rays of two superposed 
 pencils, vertex S. Describe a conic through S to meet the rays of the 
 pencils in A, B, G, and A', F, G'. 
 
 Let the Pascal line of AFGA'BG' meet the conic in L and M. 
 Then SL and SM are the self-corresponding rays of the two pencils. 
 
 The self-corresponding points of two projective ranges can be obtained 
 by the correlative method or can be deduced from the preceding. 
 
 If tivo conies, real or imaginary, intersect in two points, they also 
 intersect in two other points, real or imaginary (see page [272]). 
 
 Let S and S' be the given points of intersection of the conies (1) 
 and (2). Take any three points A , B , C on the conic (1). Join these 
 points to S and S'. Then the conic (1) may be regarded as generated 
 by the projective pencils (S.A B„C ...) and (S' . A B G„ . . .). 
 
The General Conic 
 
 135 
 
 Let the rays of the first pencil meet the' conic (2) in A, B, G, and let 
 the rays of the second meet the conic (2) in A', B', C. ' Then the self- 
 
 The figure represents the case in which the conies are real and intersect 
 in two real points (given) and in a pair of conjugate imaginary points. 
 
 corresponding points of the ranges ABC and A'B'C are the points of 
 intersection of the Pascal line of AC'BA'CB' with the conic (2). Let 
 these points be L and M. Then SL, S'L and SM, S'M are pairs of 
 corresponding rays of the pencils, which generate the conic (1), and 
 therefore the conic (1) passes through L and M, which are the two other 
 points of, intersection of the conies. 
 
 100. Desargues' theorem. 
 
 Every transversal is cut by a system of conies through four fixed 
 points, in pairs of conjugate points of an involution. 
 
 Let any conic of the system of conies through the four fixed points 
 Q, R, S, T meet any transversal s in P and P', and let opposite sides of 
 the quadrangle meet s in A, A' and B, B' as in the figure. 
 
136 
 
 The Imaginary in Geometry 
 
 Then (PP'AB)±(Q.PP'AB) = (PP'TR), 
 
 and (P'PA'B) = (S . P'PA'B') = (P'PRT) = (PP'TR). 
 
 In the figure the transversal s, which is real, meets the 
 conic through Q, R, S, T, which are real, in a pair of 
 conjugate imaginary points. Beal pairs of conjugate 
 points of the involution are situated on the same side 
 > of C and imaginary conjugates on opposite sides of G, 
 where C is the centre of the involution. ' 
 
 Therefore {PP'AB) = (P'PA'B'). 
 
 Therefore P, P' are a pair of conjugate points of the involution deter- 
 mined (Art. 84) by the quadrangle QRST on s, i.e. of a given involution. 
 
 Correlatively the pairs of tangents from any point to a system, of 
 conies, touching four fixed lines, are pairs of conjugate rays of an involu- 
 tion pencil. 
 
 This may be proved by the correlative method. 
 
 The converse theorems as stated in Art. [101] hold for the general 
 conic. 
 
 One and only one conic, real or imaginary, can be described through 
 two given points A and B to determine pairs of .conjugates of given 
 involutions on three given straight lines c, d, e. If the points A and B 
 are real, and likewise the involutions on c, d, e, then the conic is real. 
 
 This can be proved by the method employed in Art. [114(/)]. 
 
 101. Carnot's theorem. 
 
 If the points determined by a conic on the sides BC, CA, AB of any 
 triangle be A„ A 2 ; B„B,\ C„ C„ then 
 
 BA, BAj CBi OR, AG, ACj _ 
 
 CA, ' CA, * AB, - AB, ' BC, ' BC, 
 
The General Conic 
 
 137 
 
 Let the lines A,B, and A 2 B 2 meet the side BA in G s and G t . 
 C„ C 2 ; A, B; and C s , (7 4 are pairs of con- 
 jugate points of an involution (Art. 84). 
 Therefore 
 
 AC, AG 1 _AC S AG, 
 BC, ' BC 2 ~ BGl " BC 4 " 
 But by Menelaus' theorem 
 
 AC 3 _ _J 
 
 £C 3 
 
 Then 
 
 and 
 
 AC, 
 BC, 
 
 CB,BA,' 
 AB, CA\ 
 1 
 
 Therefore 
 
 CB 2 BA 2 
 
 AB 2 GA 2 
 BA, BA 2 CB, CB, 
 CA, " CA 2 " AB, ' AB 2 
 
 AC, 
 
 = 1. 
 
 AC, 
 
 ' BC, " .BCs 
 
 Correlatively if the tangents from the vertices A, B, C of a triangle 
 meet the opposite sides in three pairs of points A,, A 2 ; B,, B 2 ; C 1} C 2 , then 
 BA, BA 2 CB, CB 2 AC, AC 2 _ 
 CA, CA 2 AB, AB 2 BC, BC 2 ~ 
 
 This may be proved by the correlative method. 
 
 The converse theorems as stated in Art. [99] are all true. 
 
 102. , If two conies, real or imaginary, intersect in one point they 
 also intersect in three other points. 
 
 Let the conies S and S' intersect 
 at A. 
 
 Let B be any other point on S. 
 
 Draw any chord through B meet- 
 ing S' in Q and Q'. 
 
 Let AQ, AQ' meet S again in R 
 and R'. 
 
 Then the pencils B (Q), B(R) 
 have a (1, 2) correspondence. For 
 BQ determines BR and BR', and 
 either BR or BR' uniquely determines 
 BQ. Hence there are 2 + 1 or 3 
 positions in which BQ and BR coin- 
 cide (Art. [143]). For such positions of BR, Q and R coincide. But 
 
138 
 
 The Imaginary in Geometry 
 
 Q and R can only coincide at a point of intersection of the conies 8 and 
 S'. Hence in addition to A the conies S and 8' intersect in three 
 points. 
 
 103. The locus of the common 
 conjugates of points on a fixed line 
 with respect to two conies is a 
 conic. 
 
 Let P be any point on the 
 fixed line I and let L and L' be 
 the poles of I with respect to the 
 given conies. The polars of P with 
 respect to the conies are two 
 straight lines LQ and L'Q, which 
 pass through L and L' and meet 
 at a point Q, which is the common 
 conjugate of P with respect to the 
 conies. As P moves along I, LQ 
 and L'Q describe two pencils 
 through L and L' which are each 
 equi-anharmonic with the range 
 described by P and are therefore 
 projective with each other. 
 
 Hence the locus of Q is a conic, 
 through L and L', and this conic 
 is the locus of common conjugates 
 of points on I with respect to the 
 two conies. 
 
 The envelope of the common 
 conjugates of lines through ■ a fixed 
 point with respect to two conies is 
 a conic. 
 
 Let p be any line through the 
 fixed point L, and let I and V be 
 the polars of L with respect to the 
 given conies. The poles of p with 
 respect to the conies are two points 
 Iq and I'q which lie on I and V 
 and have for their connector q a 
 line - which is the common conju- 
 gate of p with respect to the 
 conies. As p rotates round L, Iq 
 and I'q describe two ranges on I and 
 I' which are each equi-anharmonic 
 with the pencil described by p and 
 are therefore projective with each 
 other. 
 
 Hence the envelope of q is a 
 conic which touches I and V , and 
 this conic is the envelope of com- 
 mon conjugates of lines through L 
 
 with respect to the tw6 conies. 
 104. Every pair of conies have a common self-conjugate triangle. 
 
 The loci of the common conjugates 
 of points on any two lines a and b with 
 respect to the conies (1) and (2), are 
 two conies. 
 
 The common conjugate of the point 
 ab must be on both of these conies, 
 which therefore intersect in one point. 
 Therefore, by Art. 102, they intersect 
 in three other points. Let P be one 
 of these points. Let P a and P b be the 
 
The General Conic 139 
 
 common conjugates of P on the lines a and b. Then P a Pb is a common 
 polar of P with respect to the two conies. Let P a Pb meet the conies in 
 A, A' and B, B'. The pair of common harmonic conjugates of A, A' 
 and B, B' together with P form a common self-conjugate triangle of the 
 conies. 
 
 It follows from the preceding that the locus of common conjugates 
 of points on any straight line with respect to two conies passes through 
 the vertices of their common self-conjugate triangle. 
 
 105. Every pair of conies intersect in four points. 
 
 Consider the locus of the common conjugates, with respect to the 
 conies, of points on a line a which passes through one of the vertices F 
 of their common self-conjugate triangle. The poles of the line a with 
 
 F 
 
 a 1 / 
 
 regard to the conies are two points A and A' which lie on EG, the side 
 of the common self-conjugate triangle opposite to F. The polars of P 
 any point on a pass through A and A' and meet at a point P'. 
 
 Since A, A' and also the points E, G, F are on the conic which is 
 the locus of P', this conic must break up into a pair of lines, one of 
 which is EA'AG and the other FP'. 
 
 The polars of P' with respect to the conies pass through P. There- 
 fore, if FP' be a, the locus of P for different positions of P' on a' is the 
 line a. Hence the lines a and a form an involution pencil. If FG is 
 taken as a, then EF is a', so that FG and FE are conjugate, elements of 
 the involution. Consider the two double rays of this involution. For 
 
140 The Imaginary in Geometry 
 
 these P and P' lie on the same straight line. Therefore on these lines 
 the two conies determine the same involution. Hence these lines meet 
 the conies in the same pair of points. 
 
 Similarly there are a pair of lines through E and a pair of lines 
 through G, which meet the conies in the same points. These six lines 
 intersect in four points which are common to the two conies. Hence 
 the conies intersect in four points. 
 
 By the correlative method it may be proved that every pair of conies 
 have four common tangents! 
 
 Since the conies considered in Art. 103 intersect in four points ijb 
 follows by Desargues' theorem that the locus of common conjugates of 
 points on a straight line is (1) the locus of common conjugates of all 
 conies through the four points of intersection of this conic, (2) that the 
 locus passes through the vertices of the common self-conjugate triangle,' 
 (3) that it is also the locus of the poles of the line with respect to conies of 
 the system, (4) that it passes through the harmonic conjugates of the 
 points, where the sides of the common inscribed quadrangle of the conies 
 meet the given line, with respect to the vertices, and also through the double 
 points of the involution determined by the quadrangle on the given line. 
 
 No pair of conies real or imaginary can have more than one common 
 self-conjugate triangle if they intersect in four distinct points. 
 
 This is proved in the same way as the corresponding theorem in 
 Art. [124]. 
 
 Hence the diagonal points triangle of the common inscribed quad- 
 rangle of two conies and the diagonal triangle of their common circum- 
 scribed quadrilateral coincide. 
 
 EXAMPLES 
 
 (1) Prove that the imaginary tangents from an internal point to an ellipse are 
 equally inclined to the focal distances of the point. • 
 
 (2) Prove that a pair of imaginary tangents from a real point to a conic subtend 
 equal angles at a focus. 
 
 (3) Prove that an imaginary tangent to a conic intercepts on two fixed real 
 tangents a length which subtends a real constant angle at a real focus. 
 
 (4) In Art. 94 if A A' and BB! are pairs of conjugate imaginary points and UU' 
 meets SAA' and SBB' in A , B , prove that U and D' divide A B in the ratio 
 
 (internally and externally), where K=-j-^j, and K'= £„, , M and M' being 
 
 the points in which OAA' and OBB' are met by diameters conjugate to diameters 
 parallel to these lines. 
 
CHAPTER V 
 
 THE IMAGINARY CONIC 
 
 106. Distinction between real and imaginary conies. 
 
 Hitherto the conic considered has been the general conic. A real 
 conic may be defined as a curve which is met by every real straight line 
 in its plane in a pair of points, real, coincident or conjugate imaginary. 
 
 In view of the correspondence between real points and purely 
 imaginary points this definition may also be stated as follows. A real 
 conic is a curve which is met by every purely imaginary straight line in 
 its plane in a pair of points, purely imaginary, coincident or conjugate 
 imaginary. 
 
 It follows that, if a real conic has a real point on it, it has a real 
 branch. For, if through the real point a real straight line be drawn, it 
 must meet the conic in a second real point. 
 
 If a conic be described from data depending on imaginary points 
 and quantities, another conic can be described in a similar way depending 
 on the conjugate imaginary points and quantities. The second conic is 
 termed the conjugate imaginary conic of the first. If a conic and its 
 conjugate imaginary conic coincide, the conic in question must be a 
 real conic. 
 
 When only the real rays of the generating pencils of the conic are 
 taken into account, it is proved in the Principles of Projective Geometry 
 that a conic determines on every real line in its plane a real involution. 
 When real, the double points of this involution are the points of inter- 
 section of the line with the conic. From Art. 95 it is seen that, when 
 these double points are imaginary, they are also the points of intersection 
 of the line and conic. Being the double points of a real involution they 
 are a pair of conjugate imaginary points. Hence all conies considered 
 in the Principles of Projective Geometry comply with the definition of a 
 real conic. If a conic of this nature can be described through five 
 .points or to satisfy other conditions, which determine the conic, the 
 conic satisfying these conditions must be a real conic. 
 
 If a real conic passes through an imaginary point, it also passes 
 through the conjugate imaginary point. 
 
142 The Imaginary in Geometry 
 
 For let J. be a point on the conic. Draw the real line I through A. 
 This real line will meet the conic in the conjugate imaginary point of A, 
 which is consequently on the conic. 
 
 A conic must be real if it passes through 
 
 (a) 5 real points, 
 
 (b) 3 real points and a pair of conjugate imaginary points, 
 
 (c) 1 real point and 2 pairs of conjugate imaginary points, or 
 
 (d) 3 pairs of conjugate imaginary points. 
 
 In Art. [149] real conies were described to comply with conditions 
 (a), (b) and (c). Therefore a conic determined by these conditions is 
 real. The fact that the conic is real in case (d) may be proved as 
 follows. Let A, A'; B, B'\ G, C, be three pairs of conjugate imaginary 
 points on the conic. Then 
 
 (G . AA'BB') = (C . AA'BB') = (C . A'AB'B). 
 
 But if (C. A A' BB) = K + iK', then (C . A' AB'B) = K-iK'. 
 
 .-. (K + iK') = (K-iK'), .-. K' = and (G. AA'BB') is real. 
 
 Hence the conic is real. (See Art. 23.) 
 
 This result may also be obtained as follows. The four points A, A', 
 B, B', together with the point C, determine a conic. 
 
 The four points A', A, B', B ? together with the point C', determine 
 a second conic. This second conic is the con- 
 jugate imaginary conic of the first., But the 
 six points A, A', B, E, C, G' all lie on a 
 conic. Hence the conic and its conjugate 
 imaginary conic coincide, and therefore the conic 
 is real. 
 
 The conies (a) which can be described through two 
 pairs of conjugate imaginary points to touch a given real- <a> 
 line and (b) correlatively the conies which can be described 
 to touch two pairs of conjugate imaginary lines and to 
 pass through a real point, are real. 
 
 (a) Let A A' and BE be the real sides of a 
 quadrangle AA'BB, and let A B', A'B be a pair of con- 
 jugate imaginary sides of the quadrangle. Let the 
 given real line I meet these pairs of the sides in X, X' 
 and Y, 7' A conic through A, B, A', B' which touches I will touch I at one of the 
 double points of the involution determined by X, X', Y, Y'. These double points 
 
The Imaginary Conic 143 
 
 are the common harmonic conjugates of A", X' and Y, ¥' Since one pair of 
 these are conjugate imaginary points their common harmonic conjugates — which 
 are the double points of the involution — are real (Art. 8). Hence each of the two 
 conies through A, A', B, B', which touches I, passes through a real point and 
 therefore is real. 
 
 (6) This is proved by the correlative method. 
 
 Of the conies, which pass through one real point, a pair of conjugate imaginary 
 points, and touch two real straight lines, one pair is real and one pair is a pair 
 conjugate imaginary conies. 
 
 Let K be the real point and A, A' a pair of conjugate imaginary points through 
 which the conies are described, and I and m 
 the two real straight lines which they touch. 
 
 Then the lines KA and KA' (a! and a") 
 are a pair of conjiigate imaginary lines. 
 Hence the points in which they meet the 
 real lines I and m are pairs of conjugate 
 imaginary points, i.e. in the figure L, L" 
 and if', M" are pairs of conjugate imaginary 
 points. Hence the points A, U, M' ', are 
 the conjugate imaginary points of A', L", 
 M", respectively. 
 
 Hence the double points F' and E' of 
 the involution determined by K, A , M', L' \ / 
 
 are the conjugate imaginary points of the 
 double-points F", E" of the involution determined by K, A', M", L". 
 
 Here the lines F'F" and E'E" are real and they meet I and m in real points. 
 Hence the two conies corresponding to these chords of contact are real. Art. [106 (&)]. 
 
 The lines F'E" and E'F" are conjugate imaginary lines and they meet I and m 
 in two pairs of imaginary points, the first pair being conjugate imaginary points of 
 the latter pair. Hence the two conies corresponding to these chords of contact are 
 a pair of conjugate imaginary conies. 
 
 Of the conies which pass through a real point, a pair of conjugate imaginary 
 points, and touch a pair of conjugate imaginary lines, one pair is real and one pair 
 is a pair of conjugate imaginary conies. 
 
 In the pro'of of the last theorem let I and m be a pair of conjugate imaginary 
 lines. Then M' and L" are a pair of conjugate imaginary points as are also L' and 
 M". Hence the points A, M', L', are the conjugate imaginary points of A', L"; M". 
 Therefore the result follows as before. 
 
 The correlative theorems hold. 
 
 Since, all circles pass through the circular points at infinity, a circle which passes 
 through one real pbint and a pair of conjugate imaginary points, or through two 
 pairs of conjugate imaginary points, is real. < 
 
144 The Imaginary in Geometry 
 
 Analytical. 
 
 Analytically a real conic is a conic whose equation does not explicitly contain " i". 
 The points of intersection of suoh a conic with a real straight line are given by a 
 quadratic equation, which does not explicitly involve "i", and whose roots are there- 
 fore real, coincident or conjugate imaginary. This definition includes as real conies 
 
 such curves as those given by the equations sc 2 +y 2 + a 2 = and -| + j^ + 1 = 0, although 
 
 there are no real points on the curves in question. 
 
 107. Every imaginary conic contains one real or semi-real 
 quadrangle. 
 
 Let S be the given conic and 8' its conjugate imaginary conic. Let 
 any real line a meet S in P and P', and S' in Q and Q'. Then Q and 
 Q' are the conjugate imaginary points of P and P', and P, P' and Q, 
 Q' determine a real involution on a (Art. 8). Similarly the conies 
 determine real involutions on any other three real lines b, c and d. 
 
 (Si-'" ---. B 
 
 Aj' \C 
 
 
 / 
 / 
 
 / 
 
 i 
 i 
 
 i 
 
 \ 
 \ 
 
 / 
 / 
 
 / 
 
 / 
 
 
 
 (a) 
 
 P\ 
 
 Q\ 
 
 E /P* 
 
 ;'<2' 
 
 E' 
 
 --'D-- 
 
 Let E and E' be a pair of real conjugates of the involution on a. 
 Describe a conic 2 through E and E' to determine conjugates of the 
 given involutions on b, c and d. This conic is real (Art. 99). 
 
 Let F and F' be a second pair of real conjugates of the involution 
 on a. Describe a second conic X' through F and F' to determine 
 conjugates of the given involutions on 6, c and d. This conic also is 
 real. 
 
 The conies S and 2' being real intersect in the vertices of a real or 
 semi-real quadrangle ABCD, and determine pairs of conjugates of the 
 involutions on a, b, c, d. 
 
The Imaginary Conic 145 
 
 Describe a conic through P and A, B, C, B. It will pass through 
 P'. Hence through P and P' there are two conies, viz. S and this new 
 conic, which both pass through a pair of conjugates of the involutions 
 on b, c and d. Hence these conies coincide and S passes through A, B, 
 G, D. Therefore S contains a real or semi-real quadrangle. 
 
 The following is a particular case of the preceding. 
 
 If a conic be generated by pencils with real or with conjugate imaginary points for 
 •vertices, it contains another pair of real or conjugate imaginary points. 
 
 (a) Let the vertices of the projective pencils be real points Si and S 2 . 
 determine on any real straight line s two projective 
 ranges in which a pair of real, or conjugate imaginary, 
 points, A, A', correspond to a pair of real, or con- 
 jugate imaginary, points, A r and A{ (Art. 83 (1)). 
 
 Let A and A' be real. Then in the figure P and 
 Q are real. 
 
 Let A .and A' be a pair of conjugate imaginary 
 points. Then Si A and Si A' are conjugate imaginary 
 lines as are S 2 Ai and 52 -4/. Hence P and Q are a 
 pair of conjugate imaginary points. 
 
 -(&) Let the vertices Si and S 2 be a pair of conjugate imaginary points. Let A 
 and A' be a pair of points which correspond to their conjugate imaginary points A x 
 and A{ (Art. 83 (2)). Then the pair of conjugate imaginary lines SiA and S^A X 
 meet in a real point P, and the pair of conjugate imaginary lines SiA' and S 2 Ai meet 
 in a real point Q. Hence P and Q are real. < 
 
 Let A and A' be a pair of points each of which corresponds to the conjugate 
 imaginary of the other, so that A and Ai, and A' and A x are pairs of conjugate 
 imaginary points (Art. 83 (2)). Then SiA and S 2 A{ are conjugate imaginary lines as 
 are also SiA' and S 2 Ai. Hence P and Q are a pair of conjugate imaginary points. 
 
 Analytical. 
 
 The most general form of the equation of a conic is 
 
 S+iS'=0, 
 where S=0 and <S" = are the general forms of the equation of a real conic. 
 
 Every imaginary conic whose equation is of the form S+iS'=0 has a conjugate 
 imaginary conic whose equation is S—iS'=0 associated with it. If the first conic 
 pass through an imaginary point, the second conic passes through the conjugate 
 imaginary point. The points of intersection of these conies, through which the 
 real conies S=0 and S'=0 pass, consist in general of either 
 ,(a) Four real points. 
 
 (6) Two pairs of conjugate imaginary points, 
 (c) Two real points and a pair of conjugate imaginary points. 
 Hence every imaginary conic contains four points, which constitute either a real 
 quadrangle or a semi-real quadrangle of the first or second kind. 
 
 H. i. G. 10 
 
146 The Imaginary in Geometry 
 
 Every imaginary conic has a real or semi-real circumscribed quadri- 
 lateral. 
 
 This may be proved by the correlative method to that employed to 
 prove the first theorem of this article. 
 
 108. JYo imaginary conic can have more than one self-conjugate 
 triangle real or semi-real. 
 
 Let ABC and A'B'C be two self-conjugate triangles of the conic. 
 
 (a) Let ABC and A'B'C be real. Join A A' to meet BC and BC 
 in L and K. Then A', K and A, L are pairs of conjugate points with 
 respect to the conic, and the conic therefore meets AA' in a pair of 
 points which are real or conjugate imaginary. Similarly it meets BB' 
 and GC in pairs of points which are real or conjugate imaginary. 
 Hence the conic is real. 
 
 (6). Let ABC and A'B'C be semi-real. Let A and A' be the real 
 vertices. Then as before the conic meets AA' in a pair of real or con- 
 jugate imaginary points. Produce BC and EC to meet at _D. Then 
 K, D and B, C are pairs of conjugate points with respect to the conic,; 
 also B', C is a pair of conjugate imaginary points. Hence the conic 
 meets B'C in a pair of real points (Art. 8). Similarly it meets BC in a 
 pair of real points. Hence the conic is real. 
 
 (c) If ABC is real and A'B'C is semi-real, it may be proved as in 
 case (6) that the conic must be real. 
 
 109. Construction of the self- conjugate triangle — real or semi- 
 real — of an imaginary conic. 
 
 Let ABCD be the real or semi-real inscribed quadrangle of the conic 
 and let abed be the real or semi-real circumscribed quadrilateral. Then 
 
Real Self-Conjugate Triangle 147 
 
 the diagonal points triangle of ABGD, and the diagonal triangle of abed, 
 are both real or semi-real self-conjugate triangles of the conic. Hence 
 they coincide in a real or semi-real triangle EFG. Therefore the 
 sides of the quadrilateral abed intersect in pairs on the sides of the 
 triangle EFG. Also by the properties of pole and polar the points of 
 contact of a, b, c, d are collinear in pairs with the vertices E, F, G of 
 the triangle EFG. 
 
 Let ABGD be real or semi-real of the first kind. 
 
 Then the conic is in real harmonic perspective with any vertex of 
 EFG for centre and the opposite side for axis of harmonic perspective. 
 
 Hence A, B, G, D are all real or are conjugate imaginary in pairs, 
 and a, b, c, d „ „ „ „ „ • 
 
 If A, B, G, D are real and a, b, c, d are imaginary the conic is real. 
 This is also the case if A, Bj G, D are imaginary and a, b, c, d are real 
 (Art. 106). 
 
 Therefore, if the inscribed quadrangle is real or semi-real of the first 
 kind, the bircumscribed quadrilateral is real or semi-real of the first kind. 
 
 Let ABGD be semi-real of the second kind. 
 
 Then the triangle EFG is semi-real. Hence the quadrilateral abed 
 has a semi-real diagonal triangle, and therefore it must be semi-real of 
 the second kind. 
 
 Therefore the inscribed quadrangle and the circumscribed quadrilateral 
 of an imaginary conic are always of the same kind, real, semi-real of the 
 first kind, or semi-real of the second land. 
 
 The conic cannot pass through any real points other than A,B,G, D, 
 or through any pairs of conjugate imaginary points other than A, B, 
 G, D, for if it did so it would be a real conic. 
 
 Likewise the conic cannot touch any real lines other than a, b, c, d, 
 or any pairs of conjugate imaginary lines other than a, b, c, d, for if it 
 did so it would be a real conic. 
 
 Hence the tangents at A, B, G, D and the points of contact of a, b, 
 c, d are all imaginary. The tangents at A, B, G, D intersect in pairs 
 of imaginary points in the sides of the triangle EFG and the points of 
 contact of a, b, c, d are collinear in pairs with E, F, and G. 
 
 It should be noticed that the real or semi-real quadrangle, and 
 quadrilateral of an imaginary conic are the common inscribed quadrangle 
 and the common circumscribed quadrilateral of the conic and its conju- 
 gate imaginary conic. 
 
 10—2 
 
148 The Imaginary in Geometry 
 
 110. Geometrical origin of an imaginary conic. 
 
 Any imaginary conic can be described as the conic which circumscribes 
 a real or semi-real quadrangle and touches either a real line or an 
 imaginary line. In the latter case the quadrangle is semi-real of the 
 second kind and the real point on the imaginary line touched by the conic 
 lies on a side of the diagonal points triangle of the quadrangle. 
 
 Correlatively : 
 
 Any imaginary conic can be described as the conic which is inscribed 
 in a real or semi-real quadrilateral and passes through either a real 
 point, or an imaginary point. In the latter case the quadrilateral is 
 semi-real of the second kind and the real line through the imaginary 
 point, through which the conic is described, passes through a vertex of the 
 diagonal triangle of the quadrilateral. 
 
 These results follow from the last article. 
 
 Through the vertices of a real or semi-real quadrangle two conies can 
 be described to touch a real line. If these conies are imaginary they touch 
 the real line at a pair of conjugate imaginary points and are therefore 
 conjugate imaginary conies. 
 
 This result follows from Desargues' theorem. 
 
 Conies circumscribing a real or semi-real quadrangle so as to touch a 
 given imaginary line. 
 
 Two conies can be described through the vertices of a real or semi- 
 real quadrangle ABGD to touch an imaginary line I. The points of 
 contact are the double points of the involution determined by the sides 
 of the quadrangle A BCD on I. Let these points be E and F. Take 
 I' the conjugate imaginary line of I. A conic through A, B, C, D may 
 be described to touch this line at either of two points E' and F'. The 
 points E, F, E', F' are two pairs of conjugate imaginary points. Hence 
 the conic through A, B, C, D and E will be the conjugate imaginary 
 conic of one of the conies through A,B,C,D and E' or F'. Similarly the 
 conic through- .4, B, C, D and F will be the conjugate of the other one 
 of these conies. 
 
 Similar results hold for the correlative construction of an imaginary 
 conic. 
 
 111. Real conjugate points with respect to an imaginary conic. 
 
 Every real point in the plane of an imaginary conic has one real 
 conjugate point with respect to the conic. 
 
Real Conjugate Points 149 
 
 Let A be any real point. Describe a system of real conies through 
 the vertices of the real or semi-real quadrangle contained by the given 
 imaginary conic. Let the polars of A with respect to two of these real 
 conies meet at A'. Then A and A' are real and are the double points 
 of the involution determined on AA' by the conies described through 
 the vertices of the quadrangle. Hence by Desargues' theorem they are 
 conjugate points with respect to all conies — including the given imaginary 
 conic — which pass through the vertices of the quadrangle. 
 
 The point A' is the one real point on the imaginary polar of A. It 
 
 can be constructed as the point of intersection of the polars of A with 
 
 respect to the given imaginary conic and its conjugate imaginary conic. 
 i 
 Correlatively : 
 
 Every real line in the plane of an imaginary conic has one real 
 conjugate line with respect to this conic. 
 
 It follows from the above that the points of intersection of the real 
 line A A' with the imaginary conic are a pair of conjugate points of a 
 real involution. This involution is that determined on the line by the 
 real or semi-real quadrangle of the conic. 
 
 The locus of the real conjugates of real points on a given real straight 
 line with respect to an imaginary conic, is a conic (a) which passes 
 through the vertices of the diagonal points triangle of the real or semi-real 
 inscribed, quadrangle of the conic, (b) which meets the sides of the quadrangle 
 in six points which are the harmonic conjugates of the points, where the 
 sides meet the given line, with respect to the vertices, and (c) which meets 
 the given line in the double points of the involution determined on the line 
 by the quadrangle. (Eleven points locus.) 
 
 Describe two real conies through the vertices of the real or semi-real 
 inscribed quadrangle of the imaginary conic. The real conjugates with 
 respect to. the imaginary conic of real points on the given line are the 
 common conjugates of these same points with respect to the two real 
 conies. Hence the result follows from Art. [117 (2)]. 
 
 In the proof in question two imaginary conies could have been 
 substituted for the two real conies. Hence it follows that the Ipcus— 
 which is the eleven points locus — is not only the locus — as proved in Art. 
 [117 (2)] — of the real poles of the given line with respect to real conies of 
 the system, but also that the imaginary points- on the locus are the poles 
 of the real line with respect to imaginary conies of the system. 
 
150 The Imaginary in Geometry 
 
 Cremona transformation. 
 
 ■ The fact that every real point in the plane of an imaginary conic has one and 
 only one real conjugate point with respect to the conic affords a method of 
 deducing geometrical theorems from each other. The points are said to be derived 
 from each other by what is termed the Cremona transformation. 
 
 If A' be the real conjugate of A, the relationship between A and A' is reciprocal. 
 If A moves along a straight line a, the conjugate point A' describes a conic a', 
 which passes through three fixed points E, F, G, which are the vertices of the real 
 or semi-real self-conjugate triangle of the imaginary conic. Conversely if A describes 
 a conic through E, F, G the conjugate point A' describes a straight line. If a 
 corresponds to a' and b to V, the point ab corresponds to the point a'V. To a 
 straight line through any one of the points E, F, G corresponds a straight line 
 through the same point together with the opposite side ofi the triangle EFG 
 (Art. [105]). The anharmonic ratio of any four points on a is equal to the 
 anharmonic ratio of the four corresponding points on a' (Art. [117 (2)]). Hence, if six 
 points on a form an involution, the corresponding six points on a' likewise form an 
 involution. 
 
 It will be convenient to term conies which pass through the same three points, ' 
 three point conies, and conies which pass through the same four points, four point 
 conies. 
 
 ^_ The method may be illustrated as follows. Consider Desargues' theorem and 
 let a', b', d be three four point conies. Take three of the points of intersection of 
 these conies as E, F,. G. Then to a', V, d correspond three straight lines a, b, a 
 which meet at a point, since a', V, d intersect in a fourth point. Hence since 
 a', V, d determine an involution on any transversal s 1 , the corresponding points, 
 viz. the points of intersection of a, 5, c with the conic s, form an involution. 
 Hence the theorem which corresponds to Desargues' theorem is to the effect that 
 three concurrent straight lines determine an involution on a conic. 
 
 From the involution property of the quadrilateral it follows that if four three 
 point conies intersect in pairs in A, A', B, B, C, C" the connectors of these points to 
 any one of the three points of intersection of the conies form an involution pencil. 
 
 From Brianchon's theorem it follows that if six three point conies be described 
 to touch a given straight line, three of the three point conies, which can be drawn 
 through pairs of points of intersection of these six conies, meet at a point. 
 
 If a pair of the points E, F, G are the critical points the results obtained by this 
 method are similaT to those obtained by inversion. 
 
 Generally in this method (1) to a conic corresponds a quartic curve through 
 E, F, G, having nodes at these points, (2) to a conic through one of the points 
 E, F, G corresponds a cubic curve through E, F, G having a node at one of these 
 points, (3) to a conic through two of the points E, F, G corresponds a conic 
 passing through two of the vertices of the triangle, (4) to a conic through the 
 three points E, F, G corresponds a straight line. 
 
The Harmonic Locus 151 
 
 112. Harmonic locus of two imaginary conies. 
 
 As in Art. [133] it may be proved that the envelope of a chord which 
 is cut by two imaginary conies in pairs of points, which are , harmonic 
 conjugates, is a conic, which touches the eight tangents, which can be drawn 
 to the two conies at their four points of intersection. This conic is 
 generally imaginary. 
 
 The correlative and also the converse theorems can also be proved 
 as in Art. [133]. 
 
 The envelope of chords cut harmonically by a conic and its conjugate 
 imaginary curve is a real conic, which touches the eight tangents which 
 can be drawn to these curves at their four points of intersection. 
 
 In this case the four points of intersection form a real or semi-real 
 quadrangle A BCD. 
 
 (1) Let A, B, C, D be real. The tangents to the two conies at these 
 points, being" tangents to conjugate imaginary curves at the same real 
 points on the curves, are pairs of conjugate imaginary lines. Thus, if a 
 and a' be the tangents to the two conies at A, they are conjugate 
 imaginary lines. Hence the harmonic locus of the conies touches four 
 pairs of conjugate imaginary lines and is therefore real. (Art. 106.) 
 
 (2) Let A and B be real and G and D a pair of conjugate imaginary 
 points. '-The tangents a and a' at A and the tangents b and b' at B to 
 the two conies are two pairs of conjugate imaginary lines. Let c and c' 
 be the tangents at G, and d and d' the tangents at D. Then as C is an 
 imaginary point the lines c and c' are not conjugate imaginary lines. 
 Consider however c and d'. They are tangents to conjugate imaginary 
 curves at conjugate imaginary points and are therefore conjugate 
 imaginary lines. Similarly c' and d are conjugate imaginary lines. 
 Hence in this case also the harmonic locus touches four pairs of con- 
 jugate imaginary lines and is therefore a real conic. (Art. 106.) ' 
 
 (3) Let A and B and also G and B be pairs of conjugate imaginary 
 points. Then as in the last case a and b', a' and b, c and d', and c' and 
 d, are pairs of conjugate imaginary lines and as they are tangents to the 
 harmonic locus it is a real conic. (Art. 106.) 
 
 Hence the harmonic locus of two conjugate imaginary conies is a 
 real conic. For its equation see Art. 114. 
 
 Similarly the correlative locus is a real conic. 
 
152 The Imaginary in Geometry 
 
 113. Analytical. 
 
 Let S=0, S'—O, be the equations of two real conies, 2=0, 2' = their tangential 
 equations, and *=0 the tangential equation of the envelope of lines cut harmoni- 
 cally by the conies. 
 
 Consider the conic S+iS' = 0. Its tangential equation is 2-2' + i'*=0. Here 
 the conic contains a real or semi-real quadrangte given by S=0 and <S" = 0, and has 
 likewise a real or semi-real circumscribed quadrilateral whose sides are given by 
 
 2-2'=0, *=0. 
 
 The fact that *=0 is one of these equations, shows that the real or conjugate 
 imaginary tangents are tangents to the harmonic locus of S=0, and *S" = 0. 
 
 Consider the conic 2+t'2'=0. Its ordinary equation is AS- A'S' + iF=0, where 
 F=0 is the locus of points from which tangents to 2 = and 2'=0 form a harmonic 
 pencil. • 
 
 Hence the conic has a real or semi-real circumscribed quadrilateral the sides of 
 which are given by 2 = and 2'=0 and has likewise a real or semi-real inscribed 
 quadrangle whose vertices are given by 
 
 AS-A'S'=0, F=0. 
 The locus of points the tangents from which to the conies 2 = and 2'=0 form 
 a harmonic pencil passes through these points. 
 
 Let the equation of the conic be 
 
 S+iS' = Q (a) 
 
 Consider the two real conies 
 
 £=0, 
 
 and S' = 0, \ 
 
 when these conies intersect in four real points or two pairs of conjugate imaginary 
 points. The conies then have a real self-conjugate triangle. Hence S and jS" may 
 be written in the form : ' 
 
 S = au 2 + bv 2 + cw 2 , 
 
 S' = a'u 2 + b!'v 2 + c'w 2 , 
 where u, o and w are linear functions of x and y. 
 Hence the equation (as) may be written 
 
 (a + las') u 2 + (6 + ib') v 2 + (c + id) w 2 = 0, 
 or replacing u, v and w by x, y, z, as 
 
 (a + ia!) x 2 + (b + ib') y 2 + (c + ic') z 2 = (1) 
 
 The vertices of the inscribed quadrangle of this conic are given as the points of 
 intersection of the real conies, 
 
 ax 2 + by 2 +C2 2 =0, (2) 
 
 a'x 2 + b'y 2 + c'z 2 = (3) 
 
 ' The tangential equation of (1) is 
 
 I 2 m 2 n 2 
 
 as + ios- b+tb c + ic v ' 
 
 v a' + a' b' + b 2 N ' c 2 + c 2 v ' 
 
Anharmonic Loci 153 
 
 Hence the circumscribed quadrilateral is given hy 
 
 1 \m^+°^o, W 
 
 a? + a' 2 b 2 + b' 2 c 2 + c' 2 
 j / ^ 2 7/ ™ 2 , n 2 r. m\ 
 
 and **+& +V F+F +,f ?+F-° ( ± (0 
 
 (i) The values of x 2 , y 2 and z 2 obtained from (2) and (3) are the same as 
 
 the values of -=- — ^, T , — ,,,, -= — T . obtained from (6) and (7). Hence, if the 
 a 2 +a 2 ' b 2 + b 2 c 2 + e 2 w 
 
 values of x, y, z are real, the values of I, m, n will be real Hence, if the inscribed 
 
 quadrangle is real, the circumscribed quadrilateral is real, and if the inscribed 
 
 quadrangle is semi-real of the -first kind the circumscribed quadrilateral is also 
 
 semi-real of the first kind (cf. Art. 109). 
 
 (ii) Consider the polar reciprocal of (2) with regard to the conic 
 
 \Ja 2 + a' 2 x 2 + iJW+V 2 y 2 + ^J+d 2 z 2 = 0, (8) 
 
 which is always real. 
 
 The line lx + my+nz = touches (2) if 
 
 ? + £ + - = » (9) 
 
 a b c 
 
 The polar of x', y', z 1 with respect to (8) is 
 
 \/a 2 + a' 2 xx' + \JW+V 2 yy 1 + «Jc 2 + c' 2 zz' = 0. 
 Substituting in (9) the locus of x'y'z 1 is 
 
 r' 2 i/ 2 z' 2 
 
 (cfi+a' 2 ) — +(6 2 + b' 2 ) \- + (c 2 + d 2 ) - = 0. 
 v 'a b c 
 
 The x, y, z equation of (6) is 
 
 <a 2 + a' 2 ) -+(b 2 + b' 2 )V + (c 2 + d 2 ) -=0. 
 V 'a b x c 
 
 Hence the polar reciprocal of (2) with respect to (8) is (6). 
 
 Similarly „ „ (3) „ „ (8) is (7). 
 
 Hence the polars of the vertices of the quadrangle with respect to (8) are sides 
 
 of the quadrilateral. This confirms (i). 
 
 (iii) The polar reciprocal of (1) with respect to (8) is 
 
 a 2 + a' 2 . , b 2 +b' 2 „ , c 2 + c' 2 2 . 
 a + ia b+ib " c+ic 
 
 or (a - ia') x 2 + (b - iV) y 2 + (c - id) z 2 = 0. 
 
 This is the conjugate imaginary curve of (1). Hence the polpr reciprocal of (1) 
 wit\i respect to (8) is its conjugate imaginary curve. 
 
 114. Harmonic and anharmonic loci of two conies. 
 
 The equation of the envelope of a line, which is cut in constant anharmonic 
 ratio by two conies S=0 and *S" = 0, can be easily obtained as follows : 
 
 All conies of the system S+\S' = are cut by a transversal lx+my + ns=0 
 in an involution two pairs of conjugate points of which A, A' and B, B are given as 
 
154 The Imaginary in Geometry 
 
 the points of intersection of the line lx + my + nz=0 with the conies S=0 and 
 S' = Q. 
 
 The double points of this involution are given by the values of X, viz. X x and X 2 , 
 'which are obtained from the tangential equation of <S+X»S"=0, i.e. from 
 
 '2 + X*+X 2 2'=0, 
 $ 
 
 and XiX 2 = |;. 
 
 But by example (6), Art. 27, if (AA'BB') = n, 
 Therefore the equatioia of the anharmonic locus is 
 
 1 A+/»Y_(>»+>«) 1 
 
 -G3 
 
 22'. 
 
 The harmonic locus is * = 0, for l+/i=0 in this case. 
 The equation of the common points is * 2 = 422', for fi = in this case. 
 Correlatively the equation of the locus of points, the tangents from which form 
 a pencil of constant anharmonic ratio fi, is 
 
 The corresponding harmonic locus is F=0, for l+fi = in this case. 
 The equation of the common tangents is F 2 = 4,AA'SS', for fi = in this case. 
 In the preceding 1 
 
 * = 2 (6c' + cb' - %ff) P = la-P (suppose), 
 and F= 2(BC' + OB' - 2FF') x^-sA jx 2 (suppose). 
 
 Consider the two conies 
 
 S+iS' = 0, and S-iS' = (1) 
 
 The tangential equations of these conies are 
 
 2-2' + i'*=0 and 2-2'-i'* = 0. 
 The equations * = and ^=0 of the preceding become in this case 
 
 2(2 + 2') = 0, (2) 
 
 and 2(AS+A'S'-F+& 1 ) = 0, 
 
 where *! = is the x, y, z equation of * = 0. 
 
 Hence the equations of the anharmonic loci of the conies (1) are respectively 
 
 (2 + 27=(-}^) 2 {(2-2') 2 + n '.....(3) 
 
 and. (A5+A'^"- J F+* 1 )2=^l±^Y{(A-e') 2 +(e-A') 2 }(* 2 +/S"2) (4) 
 
Anharmonic Loci 155 
 
 If n—O (3) becomes 422' = * 2 , which is the equation previously obtained for the 
 points of intersection of the conies. 
 
 The x, y, z equation of 2 (2 + 2') = is 2(aS+A'S' + F)=0. 
 
 Consider the two conies 
 
 2+i2'=0 and 2-i'2' = 0. 
 The x, y, z equations of these conies are 
 
 AS-A'S' + iF=0 and AS-A'S'-iF=0. 
 Their *=0 and F=Q equations become respectively 
 A i 2 + A' 2 2'-AA't- + F i 
 
 (A-e')*+(e-A') 2 • 
 
 2(SA+S'A')=0, 
 where Fi = is -the tangential equation of F=0. 
 Hence their anharmonic loci are 
 
 (A 2 2 + A' 2 2'-AA'* + i?' 1 ) 2 = (^ X Y{(A-e') 2 + (e-A') 2 } 2 (2 2 + 2' 3 ) ) (1) 
 
 (SA + S'Ay=(j^\ 2 {(AS- A'Sif+F"} (2) 
 
 If fi = equation (2) gives the same equation of the common tangents as that 
 previously obtained. 
 
 Hence it follows that the anharmonic loci of two conjugate imaginary conies are 
 real conies. 
 
 115. Anharmonic loci of two conjugate imaginary conies. 
 
 These may also be obtained as follows : 
 
 Consider the conies S=0 and S'=0. 
 
 The points of intersection of the line lx + my+nz=0 with the first of these are 
 given by 
 
 (cl 2 +an?-2gln)x 2 + 2 (clm—fln-gmn + hn 2 )xy + (em 2 + bn 2 — '2fmn)y i = (1) 
 
 with a similar equation (2) for the points of intersection of the line with the second 
 conic. 
 
 But if X be one of the anharmonic ratios of the points given by 
 a 1 x 2 + 2h 1 x+b 1 =0 and a 1 'x 2 + 2h 1 'x+b 1 ' = 0, 
 by example (3) Art. [14] 
 
 (UA' ~ afii ~ «i'6i) 2 = 4 (j±£) * (A, 2 - o,6,) (A,' 2 - <*,'&,'). 
 
 Now, if the values from (1) and (2) are substituted in this equation and 2, 2', * 
 are the tangential equations of the conies and their harmonic locus, this becomes at 
 once 
 
 * 2=4 (S) 222 ' (3) 
 
156 The Imaginary in Geometry 
 
 Consider the conies S+iS'—O and S-iS' = 0. 
 
 Their tangential equations are 
 
 2-2'+i*=0 
 
 and 2-2'-i*=0, 
 
 and their harmonic locus is 
 
 2 + 2' = 0. 
 
 Hence from (3) their anharmonic locus is the real curve 
 
 (2 + 2') 2 =(^) 2 {(2-2') 2 + * 2 }. 
 
 116. The locus of real points at which a pair of conjugate imaginary 
 points A and A' and a pair of imaginary points B and C subtend a 
 pencil, whose anharmonic ratio is real, is a cubic curve. 
 
 Let B and C be the conjugate imaginary points of B and C. Let 
 P be a real point on the locus. 
 
 ' Then if (P . A A' BO) = K + iK' , (P . A'AB'C) = K- iK'. < 
 
 Therefore since K' = 0, 
 
 (P . AA'BG) = (P . A'AB'C). 
 
 Hence the pencil subtended at P by the three pairs of points is an 
 involution pencil and the required locus is the cubic obtained in Art. 
 [142 (3)], which passes through the six points A, A', B, B', C, C. 
 
 (1) If the chords A A', BB', CC are concurrent at the six points 
 A, A', B, B', C.C lie on a conic and the cubic breaks up into this conic 
 and the polar of with respect to the conic. (See Art. [98].) 
 
 (2) In the general case the construction for certain points on the 
 cubic is given in Art. [142 (3)]. 
 
 In case (1) the points subtend at a pencil whose anharmonic ratio 
 may be 0, 1, or x , and in case (2) the same applies to points on the lines 
 AA', BB, CC in all cases, since two of the four rays coincide. 
 
 In the proof of the involution cubic given in Art. [143 (3)] the three 
 pairs of points A, A' ; B, B' ; G, C are assumed to be real. From the 
 preceding it follows that this need not be the case. Any of the pairs 
 of points may be pairs of conjugate imaginary points. In case (1) the 
 six points, whether real or conjugate imaginary, subtend an involution 
 pencil at real points on the polar of 0. 
 
The Cubic Locus 
 
 157 
 
 This result may be proved analytically as follows : 
 
 Take A A' for axis of x and let the coordinates of A and' A' be ia! and — ia'. 
 Let the coordinates of B and C be b + ib', c+ic' and k + ik 1 , l+il' respectively. 
 
 Then the connectors of the points A, A', B, C to any point P {x, y) meet the 
 axis of x in points given by 
 
 ( y-e)Z-CJ7 (y-c)X + c'X {y-l)X 1 -l'X{ Ay-l)X{ + l'X , 
 
 where X=by- ex, X' = 6'y — c'a;, X x = 
 
 -£s?, X{ = Ky — J!x. 
 
 The condition that the anharmonic ratio of a range should be real is given in 
 Art. 24. The relation in this case can be reduced to the form 
 
 1 ( 2 ,_ C )S +C '! (y_Z)» + f2 
 
 (y-c)X-e'X' {jj-l)X x -VX{ 
 a' 2 X 2 + X' 2 Zj 2 + AY 2 
 
 = 0. 
 
 In this determinant, if y is made equal to zero, the result is zero. Hence y is a 
 factor and the curve is a cubic. 
 
 117. To find the equation of the conic, points on which subtend a pencil of 
 anharmonic ratio K+iK' at any fowr given points. 
 
 Let the coordinates of the four given points A, B, C, D be x x y x ; x 2 y 2 ; x^; 
 x t y t respectively. 
 
 The connector of the point x^yi to the point P (xy) meets the axis of # in a 
 point X,, at a distance *#— ^,from the origin. 
 
 Substituting this and the similar values in the usual expression for au anharmonic 
 ratio and equating the result to K+iK', the equation of the conic is found to be 
 
 --K+iK'. 
 
 If the four points A, B, C, D are real this result is at once obtained from the 
 fact that 
 
 x , y 
 
 1 
 
 
 x y 1 
 
 x\ y\ 
 
 1 
 
 
 *i yi i 
 
 #3 ys 
 
 1 
 
 
 Xi y 4 1 
 
 x y 
 
 1 
 
 x y 1 
 
 *i #2 
 
 1 
 
 
 x 2 y 2 1 
 
 «s ys 
 
 1 
 
 
 xi yi 1 
 
 iAPC= 
 
 AP.CP 
 
 area APC = 
 
 AP.CP 
 
 1 
 
 1 
 
 i 
 
 *1 
 
 2/i 
 
 i 
 
 «s 
 
 yi 
 
 i 
 
 If A, B and (7, D are pairs of conjugate imaginary points, while x andy are real, 
 the expression on the left-hand side is real. 
 
158 The Imaginary in Geometry 
 
 118. Theorem. The anharmonic ratio of the four points of 
 intersection of the conies S = and S' = at any point of the conic 
 
 S— XjS'=0 is —" — - : -=? — =-, where K lt K 2 , K 3 are the roots of the 
 
 jti-2 A. JUL 2 -Kg 
 
 discriminating cubic of S— \S' = and the quantities K x , K it K s and X 
 are interchanged according to the order in which the points of intersection 
 of the conic are taken in the anharmonic ratio. 
 
 Let S - K.S' -0,5- K 2 S' = 0, S-K 3 S' = determine respectively 
 the pairs of lines P^, P S P, ; P.P,, P^P, ; P^, P 2 P S . Let be any 
 point in the plane and let S — \S' = be the conic through 0, P lt P 2 , 
 P 3 , P t . Then the equations of the polars of with respect to the three 
 pairs of lines and this conic are of the form P — K 1 P' = 0, P — K 2 P' — 0, 
 P — K S P' = 0, P — \P' = 0, the last being the equation of the tangent 
 at to the conic. 
 
 Now the polars of two fixed points with respect to four conies through 
 four fixed points form two projective pencils. Hence the pencils formed 
 by the polars of any two points with respect to the same four conies of 
 the system have the same anharmonic ratios. But the polars of P 2 
 with respect to the three pairs of lines and the conic considered are 
 P^Pz, P,P 3 , P.Pi and P X P^ the tangent at P, to the conic S-\8' = 0. 
 But (P x . PzPzPtPJ = (0 . P^PsPiP,) by the anharmonic property of 
 the conic. Hence (0 . PzPsPtP]) equals the anharmonic ratio of the 
 polars of 0. That is to say it is equal to 
 
 K 2 — X K 2 — fl) 
 
 The author is indebted to Mr S. G. Soal for this proof. The result 
 may also be proved analytically for the imaginary conic by means of 
 Art. 117. 
 
 119. The theorem proved in the last article affords an easy method of 
 distinguishing between real arid imaginary conies which pass through the four 
 vertices of a real or semi-real quadrangle. 
 
 Consider the conic S — A/S"=0 and let the four points of intersection of the conies 
 £=0 and S' = be real or two pairs of conjugate imaginary points. In this case 
 2Ti , K%, K 3 are all real. 
 
 (i) Hence if the conic is real the anharmonic ratio of four real points or of two 
 pairs of conjugate imaginary points on the curve is real : if the conic is imaginary 
 the anharmonic ratio of the four real points on the curve or of the two pairs of 
 conjugate imaginary points on the curve is imaginary. 
 
 Let the four points of intersection of the conies £=0 and (S' = be two real 
 points C and D and a pair of conjugate imaginary points A and A'. In this case 
 
Anharmonic Ratios 159 
 
 .STi and A~ 2 , two of the roots of the discriminating cubic, are a pair of conjugate 
 imaginary quantities a + ia', and a -id, while the third root Z 3 is real. 
 Hence 
 
 (AA'CB) = a + ia 'rf : ?+-**=* = K+-iK' (suppose). 
 a — ta-Ks a-iu'-\ v rr ' 
 
 Let X be real, then 
 
 (A'ACD)=^ m ,=K-iK l , .-.Kt+K'^l. 
 
 Let X be •' i," then (A'ACD) + K-iK' and the relation K i + E' i = l does not hold. 
 
 (ii) Hence if the conic is real an anharmonic ratio of two real points (C and D) 
 and a pair of conjugate imaginary points {A, A') on the curve is K+iK' where 
 K 2 + K' 2 = \ ; if the conic is imaginary the anharmonic ratio of the two real points 
 (C, -D) and the two conjugate imaginary points (A and A') on the curve is K+iK' 
 where ^ 2 + /r 2 =t=l. 
 
 The only case in which the anharmonic ratio of the points of intersection of S=0 
 and <S'=0 at any point of the imaginary conic S+iS'=0 can be real is when two of 
 the roots K it K 2 , K s are equal, in which case the anharmonic ratio is 1, 0, or oo . 
 
 If a conic passes through a real or semi-real quadrangle and through a real 
 point, the curve is real, for the equation to determine \does npt involve "i." 
 
 If a conic passes through a real or semi-real quadrangle and through an imaginary 
 point, the conic is real or imaginary according as it passes or does not pass through 
 the conjugate imaginary point. The condition necessary and sufficient is that the 
 hexagon formed by the six points is a Pascal hexagon. (See Art. 98.) 
 
 A real conic can always be described through three pairs of conjugate imaginary 
 points the real connectors of which are concurrent, for the Pascal line of the hexagon 
 so formed is the polar of the point of intersection of the connectors. 
 
 (iii) Generally the anharmonic ratio of a pair o£ conjugate imaginary points 
 (A, A') and of two imaginary points (B, C) on a real conic is imaginary. If, how- 
 ever, the real chords through A, B, C are concurrent at a point the anharmonic 
 ratio is real. 
 
 Let B' and C" be imaginary conjugates of B and C. Then if 
 
 (C'.AA'B0)=K+iK, {C. A'ABC')=K-iK'. 
 Since B and C" are on the conic 
 
 (AA'BC) = K+iK' and (A'AB'C') = K-iK'. 
 
 If A A', BB', CC are concurrent the six points form an involution on the conic 
 a,nd (AA'BC) = (A'ABC). 
 
 . • . K + UC =K-iK'. .. K'= 0. 
 
 (iv) Generally the anharmonic ratio of four imaginary points A, B, C D on a 
 real conic is imaginary. If however the real chords through the points are con- 
 current at a point the anharmonic ratio is real. 
 
 Let A', B, C, J) 1 be the conjugate imaginary points of A, B, C, D. They are on 
 the curve. 
 
160 The Imaginary in Geometry 
 
 Hence if (ABCD) = K+iK', {A'B'G'D) = K-iK'. 
 
 But {ABCD) = (A'B'C'iy). 
 
 Hence K+iE' = K-iIC, -. A r '=0. 
 
 In case (iii) when the chords are concurrent the four points -subtend a pencil of 
 an anharmonic ratio 0, 1, or oo at 0, and a pencil of real anharmonic ratio at any 
 real point on the polar of 0. The latter follows from the fact that the six points 
 
 A, A', B, B', 0, C" subtend an involution at such a point (Art. 116). 
 
 In case (iv) when the chords are concurrent the four points subtend a pencil of 
 real anharmonic ratio at and also a pencil of real anharmonic ratio at any real 
 point on the polar of 0. The latter follows from the fact that the six points A, A', 
 
 B, B', C, C subtend an involution at such a point (Art. 116). 
 
 (v) Generally at any point on a real conic through the points the anharmonic 
 ratio of 
 
 (1) Three real points and an imaginary point ; 
 
 (2) Two real points and two imaginary points ; 
 
 (3) One real point and three imaginary points ; 
 
 (4) One real point, a pair of conjugate imaginary points and an imaginary 
 
 point 
 is imaginary. 
 
 In case (3) if the real lines through the three imaginary points (A, B, C) are 
 concurrent at and meet the conic again in A', B', C", their conjugate imaginary 
 points, join to D to meet the conic again in a real point D 1 . If D and D are 
 distinct the anharmonic ratio is imaginary, but if D and D' coincide, then 
 
 {ABCD) = {A , EC'D). 
 .-. K+iK'=K-iK'. : A"=0. 
 Hence the anharmonic ratio is real. 
 
 Similarly the anharmonic ratio in case (4) may under certain circumstances be 
 real. 
 
 120. Construction of conies from real data. 
 
 If a conic is given by the fact that it either passes through real 
 points or touches real straight lines, it is easily possible to distinguish 
 between the cases when this conic is real and when it is imaginary. 
 Using the notation of Art. [144] and denoting a real conic by 1 and an 
 imaginary conic by 1. i, the results are as follows : 
 
 (1) [:•:] =1, (4) [:///] = 4 or 4. i, 
 
 (2) [::/]=2or2.i, (5) [■////] = 2 or 2. i, 
 
 (3) [•///] = 4 or 4. i, (6) [/////]= 1. 
 
 In case (2) the conic is real or imaginary according as the double points 
 of the involution determined by the four points on the line are real or 
 imaginary. 
 
Conies satisfying Certain Conditions 161 
 
 If in any position of the line the double points are real and the line 
 be rotated round a fixed point on it, till it passes through one of the 
 four fixed points, A, the double points in the new position coincide with 
 A and the two conies are coincident. After the line has passed through 
 A, the double points become imaginary and the two conies are a pair of 
 conjugate imaginary conies. 
 
 In case (3) the four conies are either all real or are two pairs of 
 conjugate imaginary conies. By Art. [113 (E)] the points of contact of 
 the conies with the two straight lines are the intersections of these lines 
 with the connectors of two pairs of double points of two involutions 
 situated on real lines. If the four double points are real the inter- 
 sections of their connectors with the given lines are real and therefore 
 the four conies are real. If one pair of these double points (which form 
 a ,real or semi-real quadrangle) or both pairs, are pairs of conjugate 
 imaginary points, the points of contact of the conies with the two lines 
 are imaginary and the conies are two pairs of conjugate imaginary conies. 
 
 This may be proved as follows. Let the given lines be p and p' and 
 the two pairs of double points conjugate imaginary points A, A' and B, B. 
 Then AB and A'B" are a pair of conjugate imaginary lines and they 
 meet p and p in two pairs of conjugate imaginary points. Hence one 
 pair of the given conies are a pair of conjugate imaginary conies. 
 Similarly, since AB' and BA' are a pair of conjugate imaginary lines, 
 the other pair of conies are a pair of conjugate imaginary conies. 
 
 If A and A' be a pair of real points, say D and E, and B and B' be 
 a pair of conjugate imaginary points, the lines DB, T)B' are a pair of 
 conjugate imaginary lines as are also the lines EB and EB': Hence the 
 same result follows in this case. 
 
 121. In the statement Art. 120 a pair of conjugate imaginary 
 points may be substituted for a pair of real points, or a pair of conjugate 
 imaginary lines for a pair of real lines, in the determining elements 
 (see Art. 106), except (a) that in (2), if the four points are two pairs 
 of conjugate imaginary points, the conies must be real (Art. 19, case 11) 
 with a similar alteration in (5) ; and (6) that in (3), if the three points 
 are real and the tangents are conjugate imaginary lines, the conies are 
 real as the double points of the involutions are real, and, if of the three 
 points two are conjugate imaginary, then a pair of the conies are real 
 and a pair are conjugate imaginary conies, with similar alterations 
 in (4). [See Art. 106.] 
 
 H. I. G. 11 
 
162 The Imaginary in Geometry 
 
 122. Foci of an imaginary conic. 
 
 If the equation of the conic be 
 
 ax 2 + Ihxy + by 2 + 2gx + 2fy + c = 0, 
 it can be easily proved that the coordinates of the foci (a, #) are given by the 
 equations 
 
 n s =/3 s = ( _&)^. and a^=h~, 
 
 ft w 
 
 after the origin has been transferred to the point ~, -^. 
 
 It can also be proved that, if T be the sum of the measures N of the distances of 
 any point on the curve from a pair of conjugate foci, 
 
 y 2 = 2{+ x /(a-6) 2 +4A 2 -(a + 6)}^r 2 ' 
 where the + sign is to be taken according to the pair of conjugate foci considered. 
 
 Hence the sum of the measures of any point on an imaginary conic from a pair of 
 conjugate foci is constant. 
 
 123. Every imaginary conic contains a real or semi-real quadrangle which by a 
 real projection (Art. 86) may be projected into a square ABOD, real, semi-real of 
 the first kind, or semi-real of the second kind. Take the real diagonals of this 
 square, OA and OB, as axes of x and y respectively. 
 
 Then in the three cases the coordinates of the vertices are respectively : 
 ABC D 
 
 1st case a, 0, a - a, , 0, - a ' 
 
 2nd case ia, 0, ia - ia, 0, - ia 
 
 3rd case a, 0, ia —a,0 0, ia 
 
 Hence if X be one of the anharmonic ratios, which any point on the curve 
 subtends at these points, the equations of the conic in the three cases are from 
 Art. 117, respectively: 
 
 (1) x*+y 2 -a? = 2xy— --, 
 
 (2) #2+jr 2 +a 2 =2^£±i, 
 
 (3) -# 2 +i/ 2 +a 2 = 2iOT^±^. Cf. Art. 137. 
 
 A— 1 " 
 
 124. Foci of the conies into which a pair of conjugate imaginary conies 
 can be projected. 
 
 In cases (1) and (2) of the last article the conic and its conjugate imaginary 
 conic have their axes in the same real directions and. have a common real centre 
 (Art. 86). It can be proved geometrically or analytically .that the eight foci of the 
 two conies are four pairs of conjugate imaginary points^ which lie four on each of the 
 two real axes. 
 
 If fij and Q 2 be the critical points and the tangents from Qj to the two conies 
 are a u ai, and 6^ 6/, and from Q 2 to the two conies a 2 > <h' and 6 2 > &s'» then the 
 
Conies with Double Contact 163 
 
 pairs of tangents which are conjugate imaginary lines may be taken as a x and 6 2 , 
 a{ and b 2 ', ch and b h a{ and b{. Consider the quadrilateral a x a{b2b{. The points 
 ai& 2 and a{b£ are real points, also a x a{ gives the conjugate imaginary point to 
 6 2 62'- Hence the two diagonals of the quadrilateral, which are at a finite distance, 
 are real. They may be easily shown to be at right angles, and their point of inter- 
 section is the centre of the two conies. There is a second quadrilateral Ogdfe'&i&i' 
 with similar properties. 
 
 125. Real conies having double contact at a pair of conjugate imaginary 
 points. 
 
 Consider the real conies S=0 and S-\a 2 =0, where a is a linear expression in 
 x and y. 
 
 If the line a=0 meets the conic S=0 in a pair of conjugate imaginary points the 
 conies have double contact at these points. 
 
 Take a line through the centre parallel to a = as axis of y and its conjugate 
 diameter with respect to S=0 for axis of x. The equations of the two conies will 
 then be of the form 
 
 ax* + by* -1=0, 
 
 and aa? + by i -\-k{x-lf=Q. 
 
 These conies can be graphed for the pair of conjugate diameters, which are the 
 axes of x and y, and it will be found that their imaginary branches touch. 
 
 The properties of conies having double contact at real points, which were proved 
 in Art. [130], are true also for conies having double contact at pairs of conjugate 
 imaginary points. 
 
 In Art. [134] it was proved that the harmonic locus of a conic and a pair of 
 points was related to the conic in such a way that if a simple quadrilateral was 
 circumscribed to the conic and inscribed in the harmonic locus, an infinite number 
 of quadrilaterals could be similarly described. 
 
 If the pair of points are on the conic, the harmonic locus of these points and the 
 conic has double contact with the conic at these points. Hence given a real conic 
 and a real line, another conic can be described to have double contact with the first 
 conie at the points where it is met by the line, which also possesses the property 
 that an infinite number of simple quadrilaterals can be circumscribed to the first 
 conic and inscribed in the latter. 
 
 This should be compared with the property of conies having double contact 
 proved in Art. [131]. The conies in question possess the remarkable property that 
 not only do the sides of the inscribed quadrilateral meet in two points on the chord 
 of contact, but the connectors of their points of contact with the other conic also 
 pass through these same points. 
 
 The properties of conies having double contact at a pair of conjugate imaginary 
 points may be obtained at once from the fact that, in a real plane perspective, two 
 such conies may be made to correspond to a pair of concentric circles. 
 
 11—2 
 
CHAPTER VI 
 
 TRACING OF CONICS AND STRAIGHT LINES 
 
 126. Graphic representation of the imaginary. 
 
 Although in the preceding pages of this book the term " graph of an 
 imaginary point" has been used, and in certain places — especially in 
 Ghapter II — diagrams of imaginary branches of curves have been given, 
 such use of " graphs " and figures is not essential, and these have been 
 used rather as a convenient way of expressing what was meant or of 
 explaining and illustrating results, than because their use was essential. 
 
 The graphical representation of the imaginary must always, .from the 
 nature of the original hypothesis, be difficult and defective. Still some 
 representation is felt to be better than none — as long as the limitations 
 of the method employed are clearly understood. In this chapter, for this 
 reason, the representation of imaginary and complex points by means of 
 what were termed in Chapter II Poncelet figures will be more fully 
 considered. 
 
 In Chapter I, for convenience the graph of an imaginary point on a 
 fixed straight line at a distance il from a fixed point 
 on the line was defined as being the point at a 
 distance I from the point 0. Hence, to obtain a 
 representation of a point P, whose position is deter- 
 mined by an imaginary.length ix measured parallel 
 to a real axis of x and a real length y measured 
 parallel to a real axis of y, it is natural to measure 
 a length ON along OX equal to x' and a length 
 NP' equal to y' and parallel to Y, and thereby to obtain a point P', 
 which represents P graphically. 
 
 A point Q, referred to the same axes of coordinates and determined 
 by coordinates x", iy", may be represented in the same way, remem- 
 bering that in this case distances parallel to the axis of y are imaginary 
 and those parallel to the axis of x are real. This is possible, but 
 there are certain advantages in modifying somewhat this system of 
 representation. 
 
 In this modification — the advantages of which have already been 
 apparent in Arts. 72 and 77 — with the usual system of real axes, the 
 
> Systems of Axes 163 
 
 positive imaginary axis of x is taken along OX and the negative 
 imaginary axis' of x along OX' , but the positive imaginary axis of 'y 
 
 is taken along OT' and the negative imaginary axis of y along OY. 
 This system of coordinates will be adopted generally in this chapter. 
 
 Representation of imaginary straight lines through the origin. 
 
 30 • 
 
 Consider the straight line ax + iby = 0. Here — — ='-. Hence the 
 
 ° —ty a' 
 
 line may be represented by OP in the figure. But the equation of the 
 
 line may be written iax — by = 0. Hence — = - . Therefore in this case 
 
 also the line is represented in the figure by OP. Similarly the line 
 ax — iby = has the same graph in whichever way its equation is 
 written. 
 
 t 
 127. Tracing of real conies. 
 
 If imaginary and complex points are taken into account in tracing 
 a conic, x t + ix 2 and y x + iy 2 may be substituted in the equation of the, 
 curve for * and y,- where x u x^,y x , y 2 are real. If real and imaginary 
 parts of the resulting equation are equated to zero, two equations 
 connecting the four quantities x u x it y x , y 2 are obtained. Arbitrary 
 values may be given to any two of these four quantities and the -values 
 of the remaining two may be obtained from the equations. Hence it 
 follows that the equation represents graphically not a* single Curve but 
 a system of curves. 
 
166 The Imaginary in Geometry 
 
 (a) To trace the curve 
 
 a? 
 
 r 
 
 = 1. 
 
 All points on the curve are obtained by giving to x the value x t + ix 2 , 
 where X,. and x 2 can have all real values from — oo to + oo . No two 
 points thus obtained can coincide, for their x coordinates will in all 
 cases be different. 
 
 
 
 Y 
 
 y''AU) 
 
 "n^ ***^ 
 
 
 ,'" ,* 
 
 N >N --- 
 
 
 -**'■ 
 
 \ yr 
 
 " B 
 
 >v ' / 
 
 M 
 
 
 a\ 
 
 *' K 
 
 
 
 )\ x 
 
 / \^ 
 
 
 — vrr -^~l 
 
 
 
 Y' 
 
 Let # 2 be zero. 
 
 (1) Give a?j all values from — a to + a. Real values of y are then 
 obtained* and the part of the curve given by the continuous line is the 
 corresponding branch. This may be termed the (1, 1) branch. 
 
 (2) Give #i all values from — oo to — a and from + a to + oo . 
 Conjugate imaginary values of y are obtained and the dotted part of the 
 figure marked (1, i) is the corresponding branch. This is the real part 
 
 of the curve — — iz = 1. For this part of the curve the coordinates are 
 a? o a 
 
 x and iy. It may be termed the (1, i) branch. 
 Let «! be zero. 
 
 (3) There are no points in this case, the coordinates of which are 
 
The Ellipse 167 
 
 of the form iy, but such a branch may exist. When it exists, it may be 
 termed the (i, i) branch. 
 
 (4) Give x 2 all values from — qo to + oo . The corresponding values 
 
 of y are real and the corresponding branch of the curve is that marked 
 
 {i, 1) in the figure. For this part of the curve the coordinates are ix 
 
 V s x* 
 and y. It is the real part of the curve j- = 1. It may be termed 
 
 the (i, 1) branch. 
 
 The above may be called the parent curve, which consists of a real 
 or (1, 1) branch and a purely imaginary or (i, i) branch, together with a 
 (1, i) branch and an (i, 1) branch. 
 
 It would be possible to determine further points on the curve by 
 giving a constant value to x 2 . In this case it would be necessary to 
 find the corresponding values of y for all real values of x x in the equation 
 
 fa + wj' f , 
 a? ^b* ' 
 
 x % I] 2 
 
 where x 2 is constant. This is the same conic as— ; + tj — 1=0 with its 
 
 a 2 o 2 
 
 centre displaced along the axis of x through a negative distance ix a . 
 
 Such a curve however is an imaginary conic, and conjugate imaginary 
 
 points are not as a general rule situated on the locus. Hence it is 
 
 more satisfactory to find the further imaginary points on the curve by 
 
 another method. 
 
 Two conjugate imaginary points x 1 + ix 2 , y± + iy 2 and x 1 — ix 2 , y± — iy. z 
 
 lie on the real straight line = - — — ; Their mean point is the 
 
 x% y 2 
 
 real point x 1} y±, which is on this straight line. The equation of the 
 
 tangent at either of the points, where the line joining the origin to the 
 
 point x lt y x meets the curve, is of the form 
 
 a 3 + 6 s 
 
 This is parallel to the line joining the conjugate imaginary poirits if 
 
 ^l^ + ii|? = 0. This relation holds if the pair of conjugate imaginary 
 
 points lie on the curve (see Art. 130). Hence the real line joining the 
 pair of conjugate imaginary points and the connector of their mean 
 point to the origin are parallel to a pair of "conjugate diameters of the 
 
168 
 
 The Imaginary in Geometry 
 
 conic. If these conjugate diameters are taken as axes of coordinates, 
 the equation of the conic is 
 
 ^ + ^-1 = 
 
 where a' and b' are the lengths of the corresponding semi-conjugate 
 diameters. 
 
 If this curve be traced for real and for purely imaginary values of 
 
 a? ifi 
 the coordinates, as the curve —, ■ + %- — 1 = was traced, certain new 
 
 a" o 2 
 
 branches of the curve are found. Thus : 
 
 (1) the (1,1) branch is found as before, 
 
 (2) the (1, i) branch is replaced by another hyperbola, which touches 
 the real branch at the ends of the diameter, which is the axis of x, 
 
 (3) the (i, i) branch in this case, as before, does not exist, 
 
 (4) the (i, 1) branch is replaced by another hyperbola, which touches 
 the real branch at the ends of the diameter, which is the axis oi,y. 
 
 By taking different pairs of conjugate diameters for axes'of coordinates 
 all points on the conic are obtained and no points except those on the 
 real and the purely imaginary branch occur more than once. These 
 branches may be called Poncelet figures. They are also, for reasons 
 
The Hyperbola 169 
 
 given hereafter, termed (a, /3) branches of the curve, where a and y8 are 
 the angles which the pair of conjugate diameters, which determine 
 them, make with the major or transverse axis of the conic. 
 
 (b) To trace the curve 
 
 a 2 6 2 
 It will be noticed that this equation may be written 
 
 2l 4. (M a _ i 
 a 8 " 1 " 6 2 
 
 Hence the same graph is obtained as in the previous case, provided 
 
 the axis of y be changed from real to imaginary and vice versa. 
 
 Thus in this case : 
 
 (1) gives an ellipse, as in the previous figure, which is the (1, i) 
 branch, 
 
 (2) gives a real hyperbola as in the previous figure, which is the 
 (1,1) branch, 
 
 (3) this branch, which is the (i, 1) branch, again does not exist, 
 
 (4) gives the same curve as before, but it is a purely imaginary 
 (i, i) curve. 
 
 (c) To trace the curve 
 This may be written 
 
 a 2 "*" 6 2 
 
 {ixf f_ 
 a? + b* 
 
 In this case : 
 
 (1) gives an ellipse of the form (i, 1), 
 
 (2) gives a hyperbola of the form (i, i), 
 
 (3) does not exist, but if it did it would be of the form (1, i), 
 
 (4) gives a hyperbola of the form (1, 1), i.e. a real curve. 
 
 In cases (b) and (c) the (a, /3) branches may be obtained in the same 
 way as for; the curve — - + j- 2 — 1 = 0. 
 
 (d) To trace the curve 
 
 a 2 o 2 
 
170 
 
 The Imaginary in Geometry 
 
 This equation may be written 
 
 {ixf (iyY 
 a 2 b 2 
 
 -1=0. 
 
 a' o° 
 
 Hence : 
 
 (1) gives an ellipse of the form (i, i), 
 
 (2) gives a hyperbola of the form (i, 1), 
 
 (3) does not exist, but if it did it would be of the form (1, 1), 
 
 (4) gives a hyperbola of the form (1, i). 
 
 It follows that all central conies, when traced for all real, imaginary 
 and complex points on the curve, have figures of the same type, and 
 that a real ellipse, an imaginary ellipse and a hyperbola differ from each 
 other only in so far that the (1, 1), (i, i), (1, i) and (i, 1) branches are 
 interchanged and that different branches of the parent curve develope 
 into the (a, /3) branches. Thus in the case of a real ellipse the (a, /3) 
 branches are hyperbolae, and in the case of a real hyperbola the («, 0) 
 branches are ellipses. 
 
 
 Y 
 
 
 
 
 S^J) 
 
 \ 
 X 
 
 
 X 
 
 / 
 1 
 / 
 
 / 
 
 
 
 
 ," 
 
 V 
 
 
 (e) To trace the curve 
 
 y 2 — 4>ax = 0. 
 
 In this case it will be more convenient to treat y as the independent 
 variable and to assume that it is of the form y 1 + iy 2 , where y x and y 2 
 may have any real values. 
 
The Parabola 
 
 171 
 
 Let 2/ 2 be zero. 
 
 (1) If y t be given all real values from — 00 to + 00 , the correspond- 
 ing values of x are real and positive, and the real parabola marked with 
 a continuous line in the figure is obtained. 
 
 Let y, be zero. 
 
 (2) If y 2 be given all real values from — 00 to + 00 „ the correspond- 
 ing values of x are real and negative, and the parabola marked with a 
 dotted line in the figure is the curve obtained. This is a parabola equal 
 to the real parabola but with its axis turned in the opposite direction. 
 This is the (1, i) branch. • 
 
 The branches (1) and (2), which are the (1, 1) and (1, i) branches, 
 constitute the parent curve. 
 
 The (a, /3) loci are obtained by taking any diameter and the tangent 
 at the point, where it meets the real branch, as axes of coordinates. 
 They consist of parabolas with axes parallel to the chief axis of| the 
 parabola and all touch the real branch. They are obtained by tracing 
 the equation of the parabola in the form y* — ka'x = 0. 
 
 128. Special case of the real circle. 
 
 The locus represented by % 2 +y 2 + a?=0 is according ,to definition a real conic, 
 although it has no real branch. 
 
 It has an (1, i) branch, the circle in the figure, and also an (i, 1) branch and a 
 (1, t) branch, both of which are rectangular hyperbolae. 
 
 ' ' ; x 
 
 
 
 
 U,i)y 
 
 \ 
 
 § 
 
 Y 
 
 -■' 
 
 / 
 
 f 
 
 T 
 
 p" 
 
 " \ 
 
 
 
 ^ — — 
 
 .-■P\ 1 
 
 P' X 
 
 /v 
 
 f 
 
 
 """"--,_ 
 
 \ 
 
172 
 
 The Imaginary in Geometry 
 
 Any real line through the centre meets the curve in a pair of conjugate 
 imaginary points, which are the double points of a real involution. Pairs of conju- 
 gate points are, if real, on different sides of and are such that OP .0P"= —a 2 . 
 If the distance of one point from is a purely imaginary quantity, the distance of 
 the other is purely imaginary and the points are on the same side of 0. Pairs of 
 conjugate points of this involution are inverse points with respect to the curve. 
 
 If P be taken such that the distance OP is purely imaginary, the polar of P is 
 the line TP'T' in the figure. 
 
 If P be taken such that the distance OP is , real, the polar of P is. the line 
 QP"Q' in. the figure. 
 
 Since the coordinate geometry of purely imaginary points is the same as that of 
 real points, the properties of purely imaginary points on- the curve are the same as 
 those of real points on the real circle. , ' , * 
 
 Special case of the real conic. 
 
 The locus represented by ,'-g +^4-1 = is /according to definition a real conic. 
 
 Its properties may be deduced from those of the : real ellipse '—$ + jj 
 
 ■1 = in the 
 
 same way that the properties' of the curve ^ 2 -)-y 2 -t-a 2 = 6ah be deduced from those 
 of the circle a; 2 +y 2 - a 2 =0. This conic has a director circle the equation of which 
 is # 2 +y 2 + a 2 +6 2 =0 and likewise a pair, of real foci situated on the minor axis. It 
 may be shown that the sum of the measures of the distances of these points from 
 any point on the curve is equal to 2ib; The corresponding directrices are real. ' 
 
 129. The fact that there is ,only one tangent at any point to a" conic may be 
 verified as follows. 
 
 Let y=f{x) be the equation of a curve so that /(#i) is the value of y corre- 
 sponding to a value x x of x. Then, if h be a small increment to %i, 
 
 The connector of the points given by x x and stii+h, disregarding powers of h 
 above the first, is 
 
 X Y 1 
 
 *i /(*i) 1 =0, 
 
 X Y 1 
 
 «, /(*,)' 1 =0. 
 
 i /'ta) o 
 
 As this does not involve h, the equation of the connector of the points does not 
 depend on the nature of the increment given to x lt i.e. the same result is obtained 
 whether the increment is real or imaginary. 
 
 Geometrically the result seems to depend on the fact that infinitely small and 
 infinitely large quantities may be regarded in the limit as either real or imaginary. 
 
Poncelet Figures 1 73 
 
 130. To find the condition that the point x 1 + ix 2 , y x + iy 2 , which is 
 
 x 2 y 2 
 on the curve — , +T5=1, should lie on that branch of the curve whose 
 a 2 o 2 J 
 
 axes are the conjugate diameters, which are inclined at angles ot and /3 to 
 
 the axis major of the conic. 
 
 If x and iy are the coordinates of the point referred to the conjugate 
 diameters, given by a and /3, as axes, x is equivalent to x cos a along the 
 axis of x, and a; sin a along the axis of y, while iy is equivalent to 
 iy cos /3 along the axis of x, and iy sin /3 along the axis of y. 
 
 Hence the coordinates of the point x, iy, referred to the principal 
 axes of the conic are 
 
 x x + ix 2 = x cos a + iy cos /3, y x + iy 2 = x sin a + iy sin /3. 
 Therefore 
 
 X} = x cos a, yi=x sin a, x 2 = y cos /3, y 2 = y sin /3. 
 
 Therefore ^ = tan a, ^ = tan /3. 
 
 X\ x 2 
 
 Hence the coordinates of a point on the (a, /3) branch are of the form 
 
 «j + ix 2 , a?! tan a + ix 2 tan /3. 
 If the point ^ + ix z , y y + iy 2 is on the conic 
 
 -l^l + y^l =1 and *2f +^ = 0. 
 
 a 2 o 2 a 2 o 2 
 
 Therefore M« = - - t = tan a tan /3. 
 
 If points, whose coordinates are x x + ix 2 , x x tan a + ix 2 tan /3 and 
 Xi+ix 2 , x{ tan a + iae a ' tan yS, are on the conic, they lie on the same 
 branch of it. For different values of x lt x 2 , a and y3, the point having 
 coordinates of this form may be any point. Hence by substituting 
 these expressions in the equation of a central conic the points on the 
 different (a, /S) figures are obtained. 
 
 It will be noticed that, for a point to be on the curve, 
 
 6 2 
 tan a tan # = , 
 
 Cb 
 
 and that if a and /3 are connected by this relation, there are an infinite 
 number of points on the corresponding branch of the curve. 
 
 6 2 . ■ 
 If the relation tan a tan y8 = is not satisfied, there are no points 
 
 on the curve. Thus, corresponding to a diameter given by a, there are 
 
174 The Imaginary in Geometry 
 
 only imaginary points on the curve when the conjugate diameter is 
 associated with it. The planes or figures obtained by associating 
 together the values of a and /8, which determine conjugate diameters of 
 the conic, will be termed the nest of the conic. In this respect the real 
 conic is essentially different from the imaginary conic for which there 
 is no nest. The general use of coordinates of the form x x + ix 2 and 
 x l tan a+ix 2 tan /9 is considered in the following Articles. 
 
 131. General conclusions. 
 
 In the parent branch of a real conic, it has been shown that when 
 the graph is formed for the axes of the conic, there are or may be four 
 parts, viz. 
 - (1) The real or (1, 1) branch. 
 
 (2) The purely imaginary or (i, i) branch. 
 
 (3) The (1, i) branch which is an Argand diagram, the real axis 
 being the axis of x. 
 
 (4) The (i, 1) branch which is an Argand diagram, the real axis 
 being the axis of y. 
 
 The (1, 1) branch represents the curve as usually considered. 
 
 The (i, i) branch is the curve, which' the equation represents on the 
 assumption that the squares of all lines are negative. With this as- 
 sumption it is as real as the real branch and it has identical properties. 
 
 The branches (1, i) and (i, 1) differ in one respect from the Argand 
 diagram in common use. In an Argand diagram it is usual to assume 
 that, if the real variable and also the imaginary variable are infinite, 
 the point at infinity is obtained. This is not the case in the present 
 instance. There are at infinity at any rate two points — if not four — 
 one given by an infinite positive value of the real variable and an 
 infinite positive value of the imaginary variable, the other by an infinite 
 positive value of the real variable and an infinite negative value of the 
 imaginary variable. These two points are the circular points at infinity 
 which from a certain point of view are more distant from the origin 
 than other points on the line joining them,. which is the line at infinity. 
 An Argand diagram is of course a graphic representation of the 
 quantities which make up a complex variable. 
 
 If the conic be graphed for a pair of conjugate diameters making 
 angles a and /3 respectively with the axis major of the conic, it has been 
 shown that there are again four branches, viz. 
 
Poncelet Figures 175 
 
 (1) The rear or (1, 1) branch. 
 
 (2) The purely imaginary or (i, i) branch. 
 
 (3) The (1, i) branch, in which the axis of x is the real axis. 
 
 (4) The (i, 1) branch, in which the axis of y is the real" axis. 
 
 The branches (1, 1) and (i, i) are the same graphically as those of 
 the parent graph, but the branches (3) and (4) are different and vary 
 with different values of a and /3. They may be regarded as Argand 
 diagrams in which the axes of coordinates are not at right angles. 
 
 Whatever value is given to a there is always one corresponding 
 value of /3, and. for every value of B there is one value of a. The graph 
 gives the point of intersection with the curve of real lines parallel to 
 the B axis. Hence with a series of such figures the points of intersec- 
 tion of all real lines with the conic are given graphically. Also since 
 one real line passes through every imaginary point, all points on the 
 curve are graphically represented. 
 
 132. General representation of points by Poncelet or (a, B) 
 figures. 
 
 The preceding renders it possible to conceive the points which exist 
 in a plane, when the values of the determining coordinates of a point 
 are or may be complex. 
 
 Through any point, which may be taken as origin, draw two rect- 
 angular axes. There are with respect to these rectangular axes four 
 systems of points, viz. 
 
 (1) Those whose coordinates are real or (1, 1) points. 
 
 (2) „ „ „ purely imaginary or (i, i) points. 
 
 (3) „ „ „ of the form (1, i). 
 
 (4) „ „ , „ of the form (i, 1). 
 
 Take any real line through the origin making an angle a with the 
 axis of x. With this may be associated any other real line through the 
 origin making an angle B with the axis of x. Let lengths real and 
 purely imaginary be measured along or parallel to these, lines, and let 
 such lengths be regarded as the coordinates of a point. 
 
 Then, as in the preceding case, there are four systems of points, viz. 
 
 (1) Those whose coordinates are real or (1, 1) points. 
 
 (2) „ „ „ purely imaginary or (i, i) points. 
 
 (3) „ „ „ of the form (1, i). 
 
 (4) „ „ „ of the form (i, 1). 
 
176 The Imaginary in Geometry 
 
 The (1,1) and (i, i) points are the same as in the preceding case. The 
 (l,i) and (i, 1) points are different. By varying a and ft the coordinates 
 of all complex points in the plane may be thrown into this form. 
 
 Any value of ft may be associated with any value of a. Hence since 
 
 the graphs of a real conic are such that tan a tan ft = — -r , where 
 
 an? + by" — 1 = is the equation of the curve, it follows that the graphs 
 of a real conic lie in a limited number of the (a, ft) planes, and in each 
 plane in which the graph exists there are an infinite number of points. 
 
 The same is true for a pair of conjugate imaginary lines when they 
 are graphed with respect to their real point as origin. (See Art. 76.) 
 
 The coordinates of a point as set forth in the preceding have been 
 termed the principal coordinates of the point (Art. 8). To find the 
 points on a given line which are in an (a, ft) plane, it is only necessary 
 to substitute in the equation the values x x + ix 2 , x x tana + ix 2 tan ft, and 
 equating real-and imaginary points to find the corresponding real values 
 of Xi and x 2 . It is sometimes more convenient to substitute x t + iy 2 tan y 
 and x x tan a. + iy^ for x and y. 
 
 In order to obtain graphically the points of intersection of two 
 curves, it is not necessary that the o and ft of the two graphs should be 
 the same. It is only necessary that the /3's should have the same value, 
 when the two origins are real with respect to each other. Thus in 
 Art. 49 when the points of intersection of a conic, having a real branch, 
 with a pair of conjugate imaginary lines were obtained, the direction of 
 ft was .obtained from geometrical considerations. The corresponding 
 values of a for the two graphs were deduced and the points of inter- 
 section of the curves were obtained. This is always possible when the 
 two origins are real with reference to each" other. If the origins are 
 purely imaginary points with reference to each other it is necessary for 
 the real axis to have the same direction in both graphs. Generally in 
 the case of the intersections of an imaginary line with a conic— having 
 a real branch — the points of intersection tie in two different (a, ft) 
 figures, but, if the two values of ft are equal, they lie in the same (a, ft) 
 figure. Such a case arises, if the line is the connector of a pair of points 
 on the curve which lie in the same (a, ft) figure. 
 
 133. Change of origin in the case of graphs. 
 
 (a) A curve may be graphed as already set forth for (a, ft) planes 
 with respect to some centre of symmetry such as the centre of a conic 
 
Poncelet Figures 177 
 
 or the real point on an imaginary line. Then, if the origin be transferred 
 to any real point, the graph will be the same but the aVill be changed 
 and will not be constant for points on the same branch. This arises 
 from the fact that the real coordinate of each point on the branch 
 considered has to be combined with the real coordinates of the new 
 origin. Hence the direction of the real coordinate is changed and 
 the a becomes different for different points on the same (a, /3) figure of 
 the original origin. . The /3 however is unaltered. 
 
 (b) A similar process may be employed, when the new origin is a 
 purely imaginary point. In this case however the a remains unaltered 
 for points in the same (a, /3) plane, while the /3 is changed. 
 
 (c) If the new origin is a complex point with respect to the original 
 axes, its position may be determined by means of its principal coordinates. 
 In this case a new real length and a new imaginary length have to 
 be combined with the a, and /3 lengths of the graphs of each point so 
 that the a and /3 of each point of the graph are changed. 
 
 From the above it is seen how the vector method of treating 
 imaginary coordinates becomes possible in the case of the general 
 theory. 
 
 134. Poncelet or (a, £) figures. 
 
 In each case xi+ix 2 and x\ tana + ixi tan are substituted for x and y in the 
 equation of the locus, and the real and imaginary parts of the resulting equation are 
 equated to zero. 
 
 (a) Real straight line. 
 
 (1) Origin on the line. 
 
 Let the equation of the line be ax+by=0. The resulting equations are 
 
 x 1 (a+6tana)=0, x 2 {a + bta,n^) = 0. 
 For these to be satisfied by values of x 1 and x 2 other than zero, it is found that 
 
 tana=tan j3= --=_. 
 
 Hence it follows that the real and imaginary axes must coincide with the real 
 line, and therefore the real line gives the full graphic representation of the line (cf. 
 Art. 1). 
 
 (2) Origin any real point. 
 
 Let the equation of the line be ax+by + c=0. The resulting equations are 
 Xx (a + b tana) + c=0, X2(a+bta,n/3)=0. 
 
 Hence tan 0= — t . Substituting this value in the first equation, it follows that 
 Xx (tan a - tan /3) + t = 0. 
 h. I. G. ' 12 
 
178 The Imaginary in Geometry 
 
 Hence, if tanj3= — r , there are for each value of a an infinite number of values 
 of x 2 tut one definite value of x x . Hence the points all lie in the plane (1, 1) and 
 the planes la, tan -1 ( - t)) • This is geometrically consistent with the previous 
 result. 
 
 (6) Real conic. 
 
 (1) Origin the centre. 
 
 Let the equation of the conic be ax 2 + %hxy + by 2 = l. The resulting equations are 
 a-, 2 {a + 2k tan a + b tan 2 a} - x 2 2 {a + 2A tan |3 + 6 tan 2 /9} = 1 
 and x l x i {a + h (tan a + tan /3) + b tan a tan |3} = 0. 
 
 Hence, if a + A(tana + tan/3)+6tanatan/3=0, there are an infinite number of 
 points in the plane (a, 0). That is, there are an infinite number of points in the 
 planes of the nest of the conic but in no other (a, /3) planes. 
 
 (2) Origin any real point. 
 
 Let the equation of the conic be f(x, y)=0. The resulting equations are 
 
 f(x u 3' 1 tana)-.»5 ! 2 (a + 2/jtan/3 + &tan 2 /3) = 0, (1) 
 
 and x 2 [x 1 {a+h(ta,na+ta.np) + bta,iiata,np} + (g+ftajip)] = (2) 
 
 If #2 = the real points on the curve are obtained. 
 If # 2 % the equation (2) is identically satisfied, if 
 
 <7+/tan/^=0 and a + /i(tan a+tan/3) + 6tanatan/3 = 0. 
 Hence the conic has an infinite number of points in one (a, /3) plane of its nest, 
 viz. the plane for which /3 = tan -1 ( _ 2 J . This plane has its /3 parallel to the polar 
 
 of the origin and its a is the line joining the origin to the centre of the conic. For 
 no other plane of the nest are there points at a finite distance. For all planes 
 outside the nest of the conic 
 
 ff+/tan/3 
 
 Xi=- 
 
 a+h (tan a+tan 0) + b tan a tan ' 
 /(#!, Xi tan a) 
 
 2 a+2Atan0+6tan 2 |3' 
 
 Hence for given values of a and /3, there is one value of x u and if the value of x 2 2 
 is positive, two values of x 2 , which differ only in sign. Hence the corresponding 
 points are a pair of conjugate imaginary points. 
 
 If the value of # 2 2 is negative there are no points on the branch considered. 
 These results are geometrically consistent with those obtained in Art. 127. 
 
 (c) An imaginary straight line. 
 
 (1) Origin the real point on the line. 
 
 If the equation of an imaginary straight line is combined with the equation of 
 its conjugate imaginary straight line an equation of the second degree is obtained, 
 which does not involve the imaginary explicitly. 
 
Poncelet Figures 
 
 179 
 
 The equation may be graphed with respect to the real point on the lines, as if it 
 represented a real conic. 
 
 Let the equation of the lines be ax 2 + 2hxy +6 i y 2 =0. 
 
 Branches are found to exist which correspond to values of a and |3, which satisfy 
 the relation 
 
 a+A(tana+tan/3) + 6tan n tan/3=0. 
 
 That is, branches, which are graphically pairs of straight lines, exist in the planes 
 for which a and give lines, which are real pairs of harmonic conjugates of the pair 
 of conjugate imaginary straight lines. These planes are said to form the nest of the 
 straight lines. No points exist in other planes than those of the nest (Arts. 76 
 and 77). 
 
 (2) Origin any real point. 
 Let the equation of the line be 
 
 ax + by + c + i(a'x + b'y+c') = 0. 
 
 Then x t (ct + fttana) — x 2 {a' + b' tan|3) + c =0, (1) 
 
 and x 2 (a+b tan (3)+.% (a' + b' tana) +c' = (2) 
 
 From (1) and (2) 
 
 Xi {ac' — a'c+(b(/ — cb') tan a} — .v- 2 {a'c' + ac+ {b'd + be) tan j3} = 0. 
 
 a'c'+ac 
 
 Hence, if tan a- 
 
 and tan/3 = — 
 
 b'c'+bc 
 
 there are an infinite number of 
 
 cb' — be' 
 points in the corresponding (a, 0) plane. 
 
 Generally, however there is only one point in an (a, 0) plane, viz. that given by 
 the equations (1) and (2). As a particular case there is only one real point on an 
 imaginary line, i.e. the one point in the (1, 1) plane. If a line joins or contains two 
 points in an (a, 0) plane, that plane is termed the plane of the line, and it contains 
 an infinite number of points in that plane. The relationship of these results to 
 those given in (1) is obvious geometrically. 
 
 135. Table of graphic representation by means of (a, 0) planes. 
 
 In the following, real and purely imaginary points, including the origin, are 
 omitted. 
 
 Nature of locus 
 Real straight line. 
 (1) Origin on line 
 
 Its equation 
 ax + by = 
 
 (2) Origin any point ax+by + c^O 
 
 Planes in which there are points 
 
 The one plane 
 
 (tan- (- j), tan- (-»)), 
 
 in which there are an infinite number of 
 points. 
 
 The planes (a, tan -1 1 -r)), where a 
 
 can have any value. There are an in- 
 finite number of points in each of these 
 planes. 
 
 12—2 
 
180 
 
 The Imaginary in Geometry 
 
 Nature of locus Its equation Planes in which there are points 
 
 Imaginary straight line — together with its conjugate imaginary line. 
 (1) Origin the real ax 2 +2hxy+by 2 =0 An infinite number of points in all (a, /3) 
 point on the line planes, for which a and /3 are harmonic 
 
 conjugates of the lines. These planes 
 form the nest of the lines. * 
 
 (2) Origin any real ax i + 2ha:y + by 2 
 point + 2gx+2fy + c = 0, 
 
 where A = 
 
 Real conic. 
 
 (1) Origin the centre . aaP + tyuey 
 
 + by i + c = 6 
 
 (2) Origin any real ax 2 + 2hmy + by 2 
 point +2gx + 2fy + c=0 
 
 Imaginary conic. 
 (1) Origin the centre S+iS' = Q 
 
 (2) Origin any real S+iS' = 
 
 point 
 
 Imaginary conic — special case. 
 Origin the centre S+iS' = 
 
 An infinite number of points in the (a, ^) 
 plane, the a axis of which passes through ■ 
 the real point on the line, the /3 being 
 its harmonic conjugate with respect to 
 the lines. One point on all other (a, /3) 
 planes except those of the nest, in which 
 there is no point at a finite distance. 
 
 An infinite number of points in all (a, /3) 
 
 ^ planes, where a, /3 determine conjugate 
 
 diameters, i.e. in the nest of the conic. 
 
 An infinite number of points in the (a, 0) 
 plane, the a of which passes through the 
 centre, the /3 being parallel to the con- 
 jugate diameter. Two points in all 
 other (a, /3) planes except those of the 
 nest, in which there is no point at a 
 finite distance. 
 
 Not more than four points in any (a, 0} 
 plane. If there are any points in a*n (a, /3) 
 or (/3, a) plane, the sum of the points in 
 these two planes is four (Art. 144). 
 
 Same result as when the centre is the 
 origin (Art. 144). 
 
 If <S" = are a pair of lines which are 
 conjugate diameters of S=0, there are 
 an infinite number of points in the (a, /3) 
 plane, whose axes are the ' conjugate 
 diameters S'=0 (Art. 145). 
 
 It follows that, as a straight line has always an infinite number of points in 
 some (a, /3) plane, the most general straight line can be found by joining two points 
 in some (a, $) plane. A conic does not always contain an infinite number of points 
 — or more than four points — in any (a, 0) plane. Hence the most general form of a 
 conic cannot be obtained by describing a conic through five points in the same 
 (a, |3) plane. The conic so obtained is the special case alluded to above. 
 
Poncelet Figures 181 
 
 136. Intersections of an imaginary line and a real conic. 
 
 An imaginary line must, by Art. 134, lie in one of the (a, 8) planes of the centre 
 of the conic. 
 
 (1) The line may lie in one of the planes forming the nest of the conic, in which 
 case it has a continuous graph in that plane. This graph may or may not meet the 
 graph of the conic in that plane. If it meets it, the line intersects the conic in two 
 points in the (a, 8) plane in question. If the two graphs do not intersect, the line 
 meets the conic in two points in different (a, 8) planes. 
 
 Let the equations of the line and conic referred to the axes of their common 
 
 (a, 8) plane be 
 
 x iy , , x 2 y 2 ■ 
 
 7 +-2 = l and _+f- = i, 
 I m a 2 b 2 
 
 V 2 a' 2 
 and let K=—. - -=- and L = 1 + K. 
 
 m 2 I 2 
 
 Then for one of the points of intersection 
 
 _o^vA£_o^ . _ a'V JL V 2 
 X ~ m K ,IK' iy ~ I K + mK~ 
 (a) If L is positive this point and also the other point of intersection, obtained 
 by changing the sign of JL, are in the common (a, 8) plane. 
 
 (6) If L is negative the coordinates of the point may be written as 
 
 a' 2 .a'b' , — r d 2 (V ,—f\ .a'b'* , — T ( V 1 \ 
 
 IK mK a lK\d J mK \ a sj - L) 
 
 liy = Mx, y = M'x be the equations of the pair of lines which are the (a, /3) axes 
 
 of this point, M=— \J ' — L and M'= — - , ■ , so that MM'= — ^ and the lines in 
 a a ij — L & 
 
 question are a pair of conjugate diameters of the ellipse. 
 
 The coordinates of the other point of intersection are obtained by changing the 
 
 sign of V — L. Hence these two points are not in the same (a, 8) plane. 
 
 (2) The line may lie in one of the planes which do not form part of the nest of 
 the conic. It cannot then intersect the conic in two points in the same (a, /3) plane, 
 for it only contains one point in planes other than the plane in which it is situated. 
 The two points of intersection may be constructed by means of Art. 49. 
 
 The tangent to the (a, 8) branch of a curve at any point on that branch lies in 
 the plane in question. Hence in the first of the cases dealt with in this article the 
 pole of the line is in the (a, 8) plane considered. Even in the second and third cases 
 
 this is also true. 
 
 i 
 
 Intersections of straight lines and conies. 
 
 A real straight line intersects 
 
 (a) A real straight line in the (1, 1) plane. 
 
 (6) An imaginary straight line in the (a, 8) plane, in which 8 is parallel to the 
 real line and a is the harmonic conjugate of 8 with respect to the imaginary line and 
 its conjugate imaginary line. 
 
182 The Imaginary in Geometry 
 
 (c) A real conic in the (t, 1) plane or in the (a, ft) plane, in which ft is parallel 
 to the real line, and a is the conjugate diameter of this line with respect to the conic. 
 
 (d) An imaginary conic in two points in different (a, ft) planes, which may he 
 determined as follows. Obtain the equation of the pair of imaginary lines joining 
 the centre of the conic — or any real point — to the points of intersection of the line 
 with the conic. This can be done at once analytically. Then by (6) above the 
 points of intersection of the real line with these lines can be obtained. 
 
 An imaginary straight line intersects 
 
 (e) A real straight line in the (a, ft) plane, in which ft is parallel to the real line 
 and a 'is the harmonic conjugate of ft with respect to the imaginary line and its 
 conjugate imaginary line. 
 
 (/) An imaginary straight line in the (a, ft) plane, in which ft is parallel to one 
 of two of the sides of the diagonal triangle of the quadrilateral determined by the 
 imaginary lines and their conjugate imaginary lines. These sides are the axes of 
 perspective of the two involution pencils, of which the imaginary straight lines are 
 the double rays. The points of intersection of the imaginary lines are on these real 
 axes of perspective and the corresponding a's are obtained by joining the real points 
 on the imaginary lines to the mean points of the opposite vertices of the quadri- 
 lateral, which are situated on the axes of perspective. 
 
 (g) A real conic in two points generally in different (a, ft) planes. The real lines 
 on which these points are situated may be constructed as explained in Art. 49, and 
 the corresponding values of the a's may be obtained as in (/) or from the equations 
 of the line and conic (Art. 136). 
 
 (h) An imaginary conic in two imaginary points. These may be found as 
 follows. Express the equations of the conic and straight line in the forms 
 
 , X % - ,w + /z. ^m. -l=0 an <* U+iA^—.-, + (B+iB)—!L--\=*0. 
 (a + ia'f (b+iVf v ' a + ia' K ' b.+ib' 
 
 Let the units of length along the axes be a + ia' and b+ib'. Then the equations of 
 the conic and straight line are respectively 
 
 # 2 +# 2 -l = and (Ax + By-l) + i(A'x+B'y)=Q. 
 
 The points of intersection of these can be found by (g). 
 
 137. Properties of (a, ft) figures. 
 
 Every real point is in the (1, 1) plane and every purely imaginary point is in the 
 (i, i) plane. Every real point is also in all the (a, ft) planes, the a axes of whieh pass 
 through the point in question, and every purely imaginary point is in all the (a, ft) 
 planes, the ft axes of which pass through the point in question. Thus all points are 
 in the (a, ft) planes, and the (1, 1) and.(i, i) planes may be regarded as a regrouping 
 of the real and purely imaginary points or as two planes of reference. From the 
 latter point of view, given a definite origin, the planes (1,' 1) and (i, i) may be 
 regarded as superposed. Then if any line a be drawn in the (1, 1) plane through 
 the origin and any line ft be drawn in the (i, i) plane through the origin, the plane 
 (a, ft) is the plane determined by this pair of intersecting straight lines. 
 
Poncelet Figures 
 
 183 
 
 The following are some properties of (a, £) planes considered in relation to the 
 (1,1) and (i, i) planes. 
 
 Every complex point may be -defined 
 by a real and a purely imaginary co- 
 ordinate in some (a, /3) plane. This con- 
 struction is unique. The point in question 
 is said to be situated in this (a, /3) plane. 
 
 Through every complex point in an 
 (a, /3) plane an infinite number of straight 
 lines may be drawn in that plane and 
 one straight line in every other (a, f$) 
 plane, including the (1, 1) and (i, i) planes. 
 ■ [See (b) below.] 
 
 Every real point contains an infinite 
 number of straight lines in the (1, 1) 
 plane and also an infinite number of 
 straight lines in all the (a, 0) planes, the 
 a axes of which pass through the point 
 in question. The lines in the (1, 1) plane 
 are real and the lines in the (a, |9) planes 
 (with the exception of the axis) are 
 imaginary. 
 
 A purely imaginary point has similar 
 properties. , 
 
 Every complex line may be defined by 
 a real and a purely imaginary intercept 
 on the axes of an (a, /3) plane. This 
 construction is unique. The line in ques- 
 tion is said to be situated in this (a, /3) 
 plane. 
 
 On every complex straight hne there 
 are an infinite number of points in its 
 (a, /3) plane and in every other (a, /3) 
 plane, including the (1, 1) and (i,i) planes, 
 there is one point on the straight line. 
 [See Art. 134 (c) (2).] 
 
 Every real line contains an infinite 
 number of points in the (1, 1) plane and 
 also an infinite number of points in all 
 the (a, /3) planes, the j3 axes of which are 
 parallel to the given line. The points in 
 the (1,1) plane are real and the points 
 in the (a, /3) planes (with the exception 
 of points in which the axis of a meets 
 the line) are imaginary. 
 
 A purely imaginary line has similar 
 properties. 
 
 (a) The line ax + by + c+i (a'x + b'y + c')=0 lies in the (a, |3) plane for which 
 
 a = tan~ 
 
 and /3=tan~ 
 
 — ca — ac 
 
 be'-b'c ^ b'c' + bc ' 
 
 This follows from the fact that the line contains the real point and the purely 
 imaginary point given by the equations 
 
 and 
 
 x y 
 
 
 1 
 
 
 \bc\ lea 
 
 a b 
 
 
 1 V d | | e' a' 
 
 
 a' V 
 
 
 ix iy 
 
 
 1 
 
 
 I V c \ \c a' 
 
 a! 
 
 b'\ 
 
 1 — b d | 1 d — a 
 
 
 — a - 
 
 -b \ 
 
 The connectors of these points to the origin give the o and ;8 axes of the plane in 
 which the line is situated. 
 
 The fact that every imaginary straight line is in an (a, /3) plane for a given origin 
 may also be proved as follows. 
 
 Let the equation of the line be 
 
 Ax+By+i(A'x+B'y+C')=0 (1) 
 
 referred to rectangular axes through the origyr. 
 
184 The Imaginary in Geometry 
 
 Let a line in the (a, /3) plane meet the axes of that plane in, points at distances 
 a and ib from the origin. Referred to the axes of % and y the coordinates of these 
 points are a cos a, a sin a and ib cos /3, ib sin /3. 
 
 Hence the equation of the line joining these points is 
 
 ^•(asin a-ib sin/3) + y (-acosa+i&cos/3) + ia&sin(0-<i)=O (2) 
 
 Comparing coefficients in (1) and (2), it follows that 
 
 tana=--5, tan /3= — -=7, 
 n d 
 
 C'-jA^ + B 2 -C'\/A' 2 + B' 2 
 
 °"~ AB'-BA' ' ,AB'~BA' ' 
 
 Hence the equation of the imaginary line (1) can be uniquely expressed in the 
 
 plane (tan -1 ( - n ) , tan -1 ( - -.=-, J J in the form 
 
 X , • iY = ' C 
 ^A^+B 2 'V A' 2 +B' 2 ~ AB'-BA ' 
 The axes of this plane are Ax + By=0 and A'x+B'y=0. The condition that 
 the line (1) should lie in a plane of the nest of the conic a'xP + b'y 2 +c' = is that 
 
 AA ' 1 * a a ' 
 
 (6) To find the equation of the line through a given imaginary point which lies in 
 a given (a, /3) plane. 
 
 Let the coordinates of the point be Xi + ix^, x t tan a+i# 3 tan (3, and let the plane 
 be the (a, 0) plane. » 
 
 The equation of the line may be written in the form 
 
 y — x 1 tan a — ix 2 tan /3 = (to + im!) (x —x x — ix%). 1 
 
 If this be the same line as 
 
 ax+by + c+i(a'x + b'y+c') = Q, 
 then by (a) 
 
 c c 
 
 ta,nd = m — m'—, and tan/3' = m+»i' — 
 
 a c 
 
 Therefore 
 
 tan a' — m_ in' _ c _ (tan a — tan a) x x _ „ . 
 
 — to' — tan j3' - to — c' ~ (tan /3' — tan /3) x% ~ *■ " '' 
 
 Hence 
 
 J ff 2 tan / 3' + taiia' - K (tan a - tan jf) 1+I{ 2 ' 
 and 
 
 . ,_Ktaaff-ii/asxa_ (tana'— tan a) #, tan ft — i (tan & - tan /3) # 2 t an a' 
 K — i (tana' — taua)#i — i (tan/^ — tanj3)# 2 
 
 If the plane (a', /J") is taken to coincide with the plane (a, j3), then the only 
 relation connecting to and to' is 
 
 (tan a — m) (tan — in) = — to' 2 . 
 This relation is satisfied by the to4oto' of every line in the plane (a, /3). 
 
The Imaginary Conic 185 
 
 (c) To find the point on a given 'imaginary line, which is in a given {a, 8) plane, 
 any real point being the origin. 
 
 Let OX and OF be the axes determining the (a, 8) plane, in which it is required 
 to find the point on the given line. Let 0' be the 
 real point on the given line, and through 0' draw 
 OY' parallel to OF. Let OX' be the harmonic 
 conjugate of O Y' with respect to the line and its 
 conjugate imaginary line. ' Let O'P be the graph of 
 the line for the axes OX' and O'Y'. Let OX meet 
 OX' in N. Draw NP parallel to OY to meet the 
 graph of the line in P. Then P is the required 
 point. For its coordinates are the real length ON 
 measured parallel to OX and the purely imaginary 
 length NP measured parallel to OY. 
 
 Hence it follows that the points on an imaginary line in the (a, 8) planes with 
 any fixed real origin, obtained by keeping 8 constant and varying a, are all on the 
 graph of the line with its real point as origin for the plane, for which B has the 
 given value and a is the harmonic conjugate of 8 with respect to the given line and 
 its conjugate imaginary line. 
 
 138. The imaginary conic. 
 
 The general equation of an imaginary conic, which may be written 
 shortly as S + iS' = 0, where $ = and S' = are the general equations 
 of two real conies, may be treated in either of two ways. 
 
 (1) Transformations of the axes of coordinates may be limited to 
 such as are real. 
 
 (2) Transformations of coordinates may be allowed, which involve 
 the imaginary. 
 
 . In case (2), as a general rule certain points will be changed from 
 real to imaginary and vice versa. Transformations of type (1) will be 
 considered in the first instance. 
 
 Transformations involving a real change of coordinates. 
 
 Such transformations depend on the fact that every conic has a real 
 
 or semi-real inscribed quadrangle. One pair of opposite sides of such a 
 
 quadrangle are real. Hence the equation of every conic can be obtained 
 
 in the form 
 
 S+2ih'xy = 0, (1) 
 
 the axes of coordinates being a pair of real sides of the real or semi-real 
 quadrangle which are inclined at an angle to. 
 
 If the bisectors of the angles between the real sides be taken as the 
 
186 The Imaginary in Geometry 
 
 axes of coordinates, the coordinates are rectangular and the equation 
 may be written , 
 
 S + 2iA'(3/ ! -mV) = (2) 
 
 If a, d, b, b' are the intercepts made by the conic on the pair of real 
 sides of the quadrangle, (1) may be written in the form 
 
 bb'x 2 + ady 2 - bb' (a + a')x- ad (b + b')y + aa'bb' + 2 (h + ih').xy = 0, 
 
 , (3) 
 
 where a, d, b, V may be real, or purely imaginary and conjugate in pairs. 
 Generally the form of the equation of a conic depends on whether 
 the inscribed quadrangle is real, semi-real of the first kind or semi-real 
 of the second kind. 
 
 If the conic has a real self-conjugate triangle, the equation of the 
 conic referred to this triangle as triangle of reference can be expressed 
 in the form 
 
 ax 2 + by 1 + cz* + i (aV + b'y' + c'z 2 ) = (4) 
 
 139. Forms of the equation of a conic into which an imaginary conic can 
 be projected by a real projection. 
 
 (A) (1) Let the conic have a real inscribed quadrangle. 
 
 Project the vertices into a real square, the diagonals of which are 2a. 
 Take the diagonals for axes of coordinates. 
 The equation of any conic through the vertices is 
 
 x 2 + y 2 + 2hxy = a 2 . 
 
 (2) Let the conic have a semi-real inscribed quadrangle of the first kind. 
 Project the vertices into a semi-real square of the first kind, the diagonals of 
 
 which are 2ia. 
 
 Take the diagonals as axes of coordinates. 
 The equation of any conic through the vertices is 
 % 2 +y 2 + 2hahj = - a 2 . 
 
 (3) Let the conic have a semi-real inscribed quadrangle of the second kind. 
 Project the vertices into a semi-real square of the second kind, the diagonals of 
 
 which are respectively 2a and 2i'a. 
 
 Take the diagonal of length 2a as axis of .v and the other diagonal as axis of y. 
 The equation of any conic through the vertices is 
 
 x 2 —y 2 + 2hxy=a 2 . 
 In the above h is or may be a complex quantity. 
 
 (B) (1) Let the diagonal triangle of the inscribed real or semi-real quadrangle 
 be real. 
 
 Project one side to infinity and the other pair of sides into orthogonal lines. 
 
The Imaginary Conic 187 
 
 Take these lines as axes of coordinates. 
 Then the equation of the conic is of the form 
 
 y 2 
 
 ^=1 or (A + iA')a;* + (B + iB')f = l. 
 
 {a + ia'f {b+ibj 
 
 (2) Let the diagonal triangle of the inscribed semi-real quadrangle be semi-real. 
 
 Project the pair of conjugate imaginary vertices into the critical points. 
 
 Take rectangular axes through the centre of the conic, which is real. 
 
 The equation of the conic is then of the form 
 
 (sc+iyf (x-iyf _ 
 {a+ia!f + {b+ib'f ' 
 
 140. Transformation of axes involving the imaginary. 
 
 Let the equation of the conic be 
 
 (a + ia') x 2 + 2 (k+ih') xy + (b+ib')y 2 +2 (g + ig 1 ) x+2 (f+if) y+(e + ic') = 
 
 or a l x 2 + 2h 1 xy + b 1 y 2 + 2g 1 x+2f 1 y + c l =Q, 
 
 the axes being inclined at an angle a>. 
 
 J? f 
 If the origin be transferred to the point -~ , -=^ and the bisector of the angles 
 
 between the axes of coordinates be displaced through an angle 
 
 , , 2Ai — (ai+6,)coS(B 
 
 \ tan -1 — . , . V , 
 
 i («! — &i) sm<» 
 
 the new axes being rectangular, ( the equation of the conic is 
 
 X 2 Y 2 1 
 
 where J t ~ x ^ am , a 
 
 ""A^sin 2 *)' 
 
 C\ a x + &,— 2^008 0) 
 
 and 
 
 , C? (a, + &i cos fi> - 2h x ) 2 + (a x - 5Q 2 sin 2 o> 
 
 with the usual -notation. 
 
 The equation (1) is of the form 
 
 X 2 _ Y 2 
 
 [a+ia'f^ {b + ib'f ' 
 or (A+iA')X 2 +(B+iB')Y 2 =l. 
 
 141. Nature of an imaginary conic. 
 
 A conic may be imaginary either by nature or by displacement. 
 Such a conic as ^ + W* - + £ = 1, where *, is constant, is according to 
 the definition of Art. 106 imaginary. If however the origin is moved 
 
188 The Imaginary in Geometry 
 
 through a distance ix 2 the equation becomes — + ^ = 1. Hence such a 
 
 curve may be regarded as imaginary by displacement. If the constants 
 which determine the nature of the curve, such as the semi-axes, the 
 latus rectum or the angle between a pair of straight lines, involve " i," 
 the curve is said to be imaginary by nature. This is the case when the 
 quadratic equation for the axes of the conic involves the imaginary 
 explicitly or when its roots are imaginary. In the latter case the 
 equation of the curve is of the form 
 
 (a + ibf + (a-iby~ ' 
 Thus with the notation of Art. 140, (1) if «/& and J p are real and J c 
 is positive, the conic is imaginary by displacement, (2) if Jh and J p are 
 real and J c is negative, the conic has a pair of conjugate imaginary 
 axes, (3) if Jh or J v is imaginary or complex, the conic generally is 
 imaginary by nature. 
 
 142. Tracing of an imaginary conic. 
 
 The author is not acquainted with any satisfactory way of represent- 
 ing graphically the points on an imaginary conic in the general case. 
 There are three possible ways of proceeding, but none of these seems po 
 lead to satisfactory results. 
 
 (a) It is possible to transform the equation of the conic as explained 
 in Art. 140 into the form 
 
 of s tf _ 
 (a+ia'f + (b + ib , y~ 
 
 Lengths measured along the axis of x may then be regarded as multiples 
 real or imaginary of a 4- ia, and those measured along the axis of y as 
 multiples real or imaginary of b+ib'. This method however does not 
 seem to lead to satisfactory results. 
 
 (b) If as in the case of a real conic the points in (a, /8) planes are 
 sought, it is found that as a general rule (see Art. 143) there may be 
 four such points in any plane, but generally not more than four points. 
 Hence this method does not lend itself to graphic representation. 
 
 (c) It is possible to obtain a graphic representation by substituting 
 for the systems of real parallel lines in an (a, /3) figure systems of lines 
 satisfying certain conditions, but the graphs obtained in this way are 
 of a complicated nature. 
 
The Imaginary Conic 189 
 
 Thus, if the equation of the conic be 
 
 ax 2 + by 2 + c + i (a'x 2 + b'y 2 ) = 0, 
 
 it is possible to construct the harmonic locus of ax 2 + by 2 + c=0 and 
 a'x 2 + b'y 2 = 0, which is a real curve. The tangents to this curve inter- 
 sect the conic and its conjugate imaginary conic in conjugate imaginary 
 points, the locus of whose mean points is 
 
 (ab' + b'd) {(a* 2 + by 2 ) 2 + (a'x 2 + by) 2 } -he {fb 2 a' + x 2 a 2 b') 2 
 
 - 4c (ab' - ba'fxy (a'x 2 + b'y 2 ) = 0. 
 
 This curve would therefore intersect the tangents to the harmonic locus 
 in points, which would give the real parts of the coordinates of the 
 points on the conic, and a graph corfld be obtained by measuring off the 
 appropriate imaginary lengths along the tangents. The same method 
 could doubtless be employed, substituting any anharmonic locus for the 
 harmonic locus. 
 
 The method (a) groups in the same figure points, whose coordinates 
 referred to the same axes through the centre are of the form k(a + id), 
 l(b + ib'), where k and I are real or purely imaginary quantities. 
 
 143. In every (a, 0) figure there may be four points, but not more than four point!, 
 on the conic 
 
 ax 2 + %hxy + by 2 + i (a'x 2 + 2h'xy + b'y 2 ) = e. 
 
 If l+sdi + iil '+#2) and m + x 1 tana+i(wi'+#2tan#) be substituted for x and y 
 and the real and imaginary parts of the resulting equation be equated to zero, there 
 are no values of I, I', m, m', a, and /3 which render the two equations thus obtained 
 identical or which make one of the equations identically zero in the general case. 
 
 If 1=1' =m=m'=0, the two equations are 
 
 x-fT - Zx^zC -x£T x =c, (1) 
 
 x^T + 2x 1 x i C -x 2 2 T{ = 0, (2) 
 
 where =a+h (tana + tan/3) + 6 tanatan/3, T =a +2A tana+5 tan 2 a, 
 C' = a'+A' (tan a + tan /3) + V tan a tan /3, T' = a' + 2h' tan a + V tan 2 a, 
 T t =a +2h tan /3 + b tan 2 /3, 
 T{ = a' + 2 h' tan + V tan 2 0. 
 Looking upon (1) and (2) as the equations of two conies, the coordinates of a 
 point on which are x 1} x%, it is seen that for given values of a and /3 there are not 
 more than four pairs of values of x x and x 2 . Hence the result follows. 
 
 Equation (1) is that of a conic and (2) that of a pair of straight lines. Only real 
 values of x x and x 2 are required. Hence for such to exist (2) must represent a pair " 
 of real lines and be equivalent to 
 
 T' (x^X^ (x 1 -\ 2 x 2 )=0, (3) 
 
 where Xi and X 2 are real. 
 
190 The Imaginary in Geometry 
 
 From (1) and (2) as? (CT+C'T')-xJ (T 1 C+T 1 'C') = cC. (4) 
 
 Writing this equation as x^A—x^B^cC, (5) 
 
 the values of x 2 are given by 
 
 ^ = A^TB ° r AX^B ( 6) 
 
 The condition for the existence of points in a given (a, /3) plane is that these 
 expressions should.be positive. 
 
 It may be noticed that if x t +ix 2 , a^tana+ia^tan/S be substituted •for x and y in 
 an equation, 
 
 (1) if ix l and ix 2 be values of Xi and x 2 , these values give'a point ( — x 2 , x t ) in 
 the (/3, a) plane ; 
 
 (2) if a pair of values ±x 2 correspond to a given value of x x , the points so 
 obtained are a pair of conjugate imaginary points ; 
 
 ,„. .„ , y Xtana+itanS 
 
 (3) if Xi-Xvs, * = T — -. ?-. 
 
 X A + l 
 
 Hence the point x, y lies on this imaginary line through the origin. 
 
 144. In the case of the conic 
 
 ax 2 + 2hxy + by*+i (a'x^+^h'xy + by) = c, 
 if there are any points in the (a, /3) or (/3, a) planes, the sum of the number of points 
 in these planes is four. 
 
 Since there are points in one or other of the planes, equation (2) of the last 
 article can be expressed as equation (3). Hence the values of x 2 are given by (6). 
 These values are either real or purely imaginary. The real values give points in the 
 (a, /3) plane and the purely imaginary values give points in the (ft a) plane. The 
 sum of the numbers of these points is four. 
 
 145. Special case of an imaginary conic. 
 
 If in the case of the conic considered in the last article the lines 
 a'x 2 + 2h'xy + b'y*=0 are a pair of conjugate diameters of the conic 
 am? + 2hxy + by 2 = c, the relation (2), Art. 143, is satisfied, when a and /3 
 are such as to determine this pair of conjugate diameters. In this case 
 T' = 2\' = G = 0. Hence there are in this case an infinite number of 
 points on the conic in these (a, /3) and (/3, a) planes. 
 
 The corresponding values of x y and x 2 are given by 
 x 1 "T-2x l x 2 C'-x 2 !l T 1 = c. 
 
 The conic in this case is not generally imaginary by displacement. 
 
 If a pair of real sides of the semi-real or real quadrangle of an 
 imaginary conic are taken as the axes of coordinates and a new figure is 
 obtained by projecting the side of the self-conjugate triangle opposite 
 to their point of intersection into the line at infinity, it may happen 
 
Special Imagiixary Conic 191 
 
 that the vertices of the semi-real or real quadrangle of the imaginary 
 conic so obtained are determined by a conic of which the axes of 
 coordinates are a pair of conjugate diameters. In this case the equation 
 of the imaginary conic is of the form 
 
 ax 2 + by 2 + lihxy = c. 
 
 Substituting x± + iy 2 tan 7 and ,Xx tan « + iy 2 for x and y, it is found 
 that 
 
 a (x? - y? tan 2 7) + b (x^ tan 2 a - y 2 2 ) - 2hx 1 y 2 (l + tan a tan 7) = c, . . .(1) 
 
 and x 1 y i (ata.ny+ b tan a) + h (xf tan a — y£ tan 7) = (2) 
 
 If a«= 7 = 0, (2) is satisfied and (1) becomes 
 ax^ — by 2 — 2hx 1 y i = c. 
 Hence there are an infinite number of points in the (1, i) and {i, 1) 
 planes of the conic. 
 
 Conic through five points in the same (a, 0) plane. 
 
 If a conic be described through five points in the same (a, |3) plane its equation 
 by expanding the determinant, obtained by eliminating the constants, is found to be 
 
 * + 2hxK+b(&\ + 2gx + 2fK+c = 0, (1) 
 
 or ax 2 — 2ihxy — by 2 + 2gx — 2ify+e = 0, (2) 
 
 where a, b, c, f, g, h are real. 
 
 Let A< be the discriminant of this equation, and A the discriminant of (1) when 
 the " i'h " are omitted. Let C; and C be the corresponding minors of c. 
 
 Then, it will be found that 
 
 A,= — A and C i —-C. 
 
 Hence the equation referred to parallel axes through the centre is 
 
 ax' 1 - by 2 +-^, — 2ikxy=0. 
 
 The axes x=0,y = are conjugate diameters of the curve ax 2 — by 2 + -~=0, and 
 therefore the conic considered comes under the special case of the imaginary conic. 
 
 Conic imaginary by displacement. 
 
 Consider the conic 
 
 ax 2 + 2i hxy - by 2 + 2gx - 2 ify + c = 0, 
 
 where the axes are rectangular. 
 
 Referred to parallel axes of coordinates through the centre, the equation with the 
 
 notation of the preceding is 
 
 - ax 2 +2ihxy-by 2 + 7 ,=0. 
 
 2ih 
 Turn the axes of coordinates through the imaginary angle | tan ~ ' -—r . 
 
192 The Imaginary in Geometry 
 
 A 
 
 The equation is then a'%; 2 + &' 2 « 2 + 7v = 0, where a! and V are given by 
 
 ' v * 
 
 a + b = a' + b' and a&+A 2 =a'&'. 
 
 Therefore a' - 6' = ± J(a - bf - 4A 2 . 
 
 If (a - 6) 2 > 4h\ the values of a' and b' are real and the conic is imaginary by 
 displacement. If this condition is not satisfied the axes of the conic are conjugate 
 imaginary quantities. 
 
 In form (A), Art. 139, in cases (1) and (2), if h is purely imaginary the conic has 
 a pair of conjugate imaginary axes. The same holds in case (3) except that when h 
 in i.h is <1, the axes of the conic are real and the conic is imaginary by displace- 
 ment. 
 
 ' 146. Modulus of reduction in an (a, 0) figure. 
 
 In an (a, /3) figure let P represent a point and let PN be y, ON be x, and the 
 angle PCW bed. 
 
 Let /3 — a be <a. Then 
 
 PN=x — — ; -^r=Kx (suppose). 
 
 sin (<o - 8) x rr 
 
 Then 
 
 OF 1 graphically=« :! + J ff'V+3^ 2 ^ cos a, 
 
 OP 2 actually =x 2 -kV + 2i^K cos <a. 
 
 actual value of OP 2 _ 1 - Z 2 + 2iK cos o> 
 
 graphical value of OP* ~ 1 + R 2 + 2K cos a> 
 
 = B? (suppose). 
 
 2 _ sin 2 (<» — 8) — sin 2 6 + 2i sin (co — 5) sin cos o> 
 
 sin 2 (<o — 8) — sin 2 6 + 2 sin 5 sin (<o - 8) cos a> 
 
 _ sin a sin («o - 28) + 2i sin (to — 6) sin cos <o 
 
 sin 2 (u 
 
 = cos 28 — c. sin 25 + 2i sin 8 . c (cos 6 — c sin #X where c = cot a>, 
 
 = cos 20 {1 +w 2 } + c sin 2(9 {i- l}-ic 2 . 
 
 If 
 
 If 
 
 If 
 If 
 
 Hence by means of R, the modulus of reduction, the actual distance from the 
 origin of a point in an (a, /3) figure can be deduced from its graphic representation. 
 
 The imaginary angle represented by a real graph. 
 
 Graphically a triangle ONP, in which ON is a, NP is ib, and ONP is a real angle 
 o>, is represented by a real triangle GiT./* in which NP is 6. Let the angle PON of 
 the imaginary triangle be 8 t and that of the real triangle 8. 
 
 " = 2« 
 
 71^ = 008 20. 
 
 ft- a 
 
 jja I COS to 
 1+COSO) 
 
 0=0, 
 
 #»=1. 
 
 8 = a, 
 
 iJ 2 =-l. 
 
Critical Lines 193 
 
 Then - = ^-s and *-*£- v 
 
 a shi(0 + <b) a sin (^ + a) 
 
 Therefore . *f°* -, .*■*"* -fg (suppose) (1) . 
 
 sin^j+a) sin(0 + ») s ^ rr ' 
 Hence 
 
 , . lA'sinu , , „. 2iiTsin<»(l-j'A'cos(») ,„> 
 
 tan(9 i= = — ^ and tan20 i = — — = i t^ <p- (2) 
 
 1 - iK cos a * 1 - 2iA cos w — A^ cos 2o> 
 
 From (1), if ^ = a + s<p, it is found by equating real and imaginary parts of the 
 equation that 
 
 , „ -A 2 sin2w 2iKsma> .„. 
 
 tan 2a = — and tan2sd> = , (J) 
 
 1+A 2 C08 2<B 1+A 2 
 
 The value of tan2(a + s$) derived from (3) is in agreement with the value of 
 tan %&i derived from (2). 
 
 Given any imaginary length A+iB measured along a line, it is possible to deter- 
 mine this length by means of (a, 0) axes inclined at any given real angle a. 
 
 Let a and ib be the coordinates of the end of the length A+iB. 
 
 Then (A + iBf = a 2 -b 2 + 2i'a& cos a>. 
 
 Assume that " A 2 - B 2 = a 1 — b 2 and 2* A B—%iab cos a. 
 
 Then A 2 -B 2 = a 2 -b 2 and AB=ab cos u. 
 
 .-. a 2 + b 2 =+./(A 2 -B 2 ) 2 + 4^-^ = +K, where A" is >A i -B*. 
 ~ V COS 2 a> ~ 
 
 .-. 2a 2 = (4 2 -5 2 )±/f and 26 2 = -{A 2 -B 2 )±K. 
 
 If the positive sign of the square root is taken, a and b are both real. If the 
 
 negative sign is taken for the square root both a and 6 are purely imaginary, that , 
 
 is, a is imaginary and ib is real. 
 
 147. (a, B) figures for the critical lines of a point. 
 
 Probably the two most remarkable of all the properties of the critical lines of a 
 point are the following : 
 
 (1) If any pair of lines at right angles through the point be taken as axes of 
 coordinates the equation of these lines is x 2 +y 2 =0. . If real lengths be measured 
 along the axis of x and imaginary lengths along the axis of y a graphic representa- 
 tion of the lines can be obtained. But if any other pair of lines at right angles 
 through the point be taken as axes of coordinates the equation of the pair of 
 critical lines is still x 2 +y 2 =0 and they can be graphed in the same way. 
 
 (2) Each of the critical lines may analytically be regarded as making the same 
 angle with every real line in its plane (Art. 78). 
 
 The first of these properties can however be explained at once and the second is 
 an immediate consequence of the first. 
 
 In Art. 76 it was shown that if the equation of a pair of imaginary lines is given 
 in the form y 2 + m 2 x 2 = 0, i.e. if the axes of coordinates are the pair of real bisectors 
 of the angles between the lines — which are of course at right angles — a graph exists 
 in all the (a, /3) planes for which a and /3 satisfy the relation tana tan/3= - m 2 . If 
 for the lines y 2 + m 2 x 2 =0 be substituted the lines y 2 +x 2 = 0, this condition becomes 
 
 h. I. G. 13 
 
194 The Imaginary in Geometry 
 
 tanatan£f= - 1. Hence the axes for the graphs of these lines must be at right 
 angles and for all pairs of axes at right angles a graph exists. Hence the property 
 of the critical lines set forth in (1) is simply that for them as for other imaginary 
 lines (a, £) figures can be constructed. For all pairs of axes at right angles the form 
 of the equation of the critical lines is the same (Art. 96). 
 
 In the general case a pair of conjugate imaginary straight lines have only one 
 pair of harmonic conjugates (real) which are at right angles.' These are the bisectors 
 of the angles between the lines. In Art. 77 it was shown that the graph of an 
 imaginary line derived from its equation referred to two (a, /3) axes makes with 
 either of the bisectors the same angle as that deduced from its equation referred 
 to the bisectors as axes of coordinates. In the case of the critical lines there are 
 an infinite number of these bisectors. This result shows that the angle made 
 by a critical line with any of these bisectors must be the same as that made by it 
 with any 1 one of another pair of bisectors, i.e. with any real line in the plane. 
 Hence these two remarkable properties of the critical lines are only particular cases 
 of well established theorems. 
 
 It is instructive to work out directly, by substituting x t + ix 2 and x t tana +t> 2 tan^ 
 for x and y, the (a, /3) figures of a pair of critical lines (see Art. 76). 
 
 EXAMPLES 
 
 (1) Show that the line joining the points Xi + ix 2 , y\ + iy% and x± — ix%, —yi + iy^ 
 lies in the (1, i) plane. 
 
 (2) Prove that the vertices of the real or semi-real quadrangle of the conic 
 
 ax 2 + 2hxy + by 2 + i (a'x 2 + %Kxy + Vy 2 ) = c 
 lie in one or other of the planes (1, 1), (i, i) or (a, 0), where a, give the directions 
 of the common harmonic conjugates of ax 2 + 2hxy + by 2 =0 and a'x 2 +2h'xy +b'y 2 =0. 
 
 (3) Prove with the notation of Art. 143 that the points of the conic, which lie 
 in the (a, /3) plane, are situated on the pair of straight lines whose equation is 
 
 T,'(y-#tan a) 2 +2Ci(y-xtim a) (y-x tanj3) + T'(y-#tan'j3) 2 =0. 
 
 (4) Prove that the graph of the curve , 
 
 ax 2 +by 2 — l+i(by — Ix) (ly+ax)=0 
 in' the (a, /3) plane in which it has a continuous graph is the real part of the conic 
 bx 2 ay 2 xy(l 2 + ab) 1 
 
 p+F ~ W+a~ 2 + «/W+a~ 2 V J*+F ~ lT +ak ' 
 referred to the a, /3 axes as axes of coordinates. 
 
 (5) The locus of points, from which pairs of tangents can be drawn to a hyper- 
 bola such that the sum of the angles which they make with the axis of x is — , is a 
 portion of a rectangular hyperbola. 
 
 (6) The locus of a point, the tangents from which to the ellipse ^ + ^-1=0 
 make angles with the axis of x the sum of which is ^ , is the rectangular hyperbola 
 
 x 2 -y 2 -a 2 +b 2 =0. 
 
Examples 195 
 
 (7) Prove that the point on the line ax + iby=\ which is in the (a, /3) plane is 
 
 a — ib tan a ( a — ib tan ff) tang 
 
 a 2 +6 2 tanatan j3' a 2 + b 2 tan a tan $ 
 
 (8) Show that if a, a', h, k, h', and k' are finite quantities and \ in the limit 
 approaches the value zero, the connectors of the two points Q x and Q 2 whose co- 
 ordinates are respectively r-H-A, -r-+k and rr-+h', ^-r — \-k', to the, origin are the 
 
 XX XX 
 
 critical lines of the origin. 
 
 (9) Hence show that the points Q : and Q 2 ma y be regarded as representing the 
 circular points at infinity in the most general case and that these points in a 
 particular case are a pair of conjugate imaginary points. 
 
 (10) Hence prove that in the general case the measures of the distances of the 
 circular points at infinity from the origin are indeterminate quantities. 
 
 13—2 
 
CHAPTER VII 
 
 THE IMAGINARY IN SPACE 
 
 i 
 
 148. In Chapter I an imaginary point was defined as a double point 
 of a real overlapping involution situated on a real straight line. The 
 real lines, on which such involutions were situated, were assumed to lie 
 in a plane, so that the bases of the involutions were intersecting straight 
 lines. In this chapter this restriction is removed so that the points 
 considered, both real and imaginary, are situated in space. 
 
 A straight line may be regarded as a fundamental conception or may 
 be defined as a locus such that one and only one straight line can be 
 drawn to pass through two given points, and a plane as the surface generated 
 by straight lines all of which intersect in pairs at points at a finite or 
 infinite distance. It follows that two straight lines cannot intersect in 
 more than one point. If they so intersected, two straight lines could 
 be drawn to join their two points of intersection. It is assumed that two 
 intersecting straight lines uniquely determine a plane. 
 
 A plane is determined by any three points A, B, G, which are not 
 collinear. Join B and G to A by straight lines. Then, if P and Q be 
 any two points on these lines, the straight line PQ intersects AB and 
 A G and therefore the three lines lie in the same plane. 
 
 It follows that a point and a straight line determine a plane, and that 
 if two points on a straight line lie in a plane, every point in the straight 
 line lies in the plane. 
 
 The locus of points common to two planes is a straight line. 
 
 Let A and B be two points common to two planes a and a'. Only 
 one straight line can be drawn to join A and B. Every point on the 
 line AB lies in the plane <r and also in the plane a. Hence the theorem 
 follows, for if there were a common point of the planes a and a, which 
 was not on the line AB, the planes a and <r' would coincide in the 
 plane determined by this point and the line AB. 
 
 A plane and a straight line — which does not lie in the plane — 
 determine a point. 
 
Correlative theorems 197 
 
 Let a- be the plane and I the line. Take A any point in the plane 
 a- and construct the plane a through A and the line I. This plane 
 intersects the plane a- in a line <rcr'. I being in this plane meets <t<t' in 
 one point. This point is the point of intersection of I and o\ 
 
 Planes which intersect in the same straight line are termed coaxal 
 planes. 
 
 Three planes which are not coaxal determine a point. 
 
 Let the planes be <t, a', a". Then a, </ intersect in a straight line I, 
 and I meets a" in a point. This is the one and only point common to 
 the three planes. 
 
 Hence the following correlative relations are obtained: 
 
 Three non-collinear points de- Three non-coaxal planes deter- 
 
 termine a plane. mine a point. 
 
 Two points determine a straight Two planes determine astraight 
 
 line. line. 
 
 A point and a line determine a A plane and a line determine a 
 
 plane. point. 
 
 Two intersecting straight lines Two intersecting straight lines 
 
 determine a point. determine a plane. 
 
 If in the above the determining elements are real, the line, point or 
 plane so determined is real. , 
 
 Parallel planes are such as contain more than one system of lines of 
 the one parallel to a system of parallel lines of the other. Parallel planes 
 form a coaxal system of planes the axis of which is entirely at infinity. 
 
 Analytical. 
 
 Let P be any imaginary point in space. Let P' be its conjugate 
 imaginary point. Then PP' is a real line, for P and P' are by definition 
 the double points of a real involution on a real straight line. Let M be 
 the mean point of P and P', which is real. Through M draw a real plane 
 a perpendicular to the real line PP'. Then PM is a purely imaginary 
 quantity. The position of M in the plane <r is determined by two real 
 coordinates. Hence the coordinates of P are of the form a, b, + ic, and 
 those of P' of the form a, b, — ic. If the planes of the coordinate axes 
 are changed the coordinates of P and P 7 are of the form 
 
 a + ia', b + ib', "c + ic, 
 
 and a — ia', b — ib', c — ic. 
 
198 
 
 The Imaginary in Geometry 
 
 It follows that if the equation of the plane, which contains three given 
 imaginary points, is of the form 
 
 (I + il') x + (m-+ im') y + (n + in') z = k + ik', 
 that of the plane which contains the thr.ee conjugate, imaginary points 
 is of the form 
 
 (I — il') x + (m — im') y + (n — in') z = k — iW. 
 
 These planes in the equations of which the sign ofi the imaginary 
 is changed are termed conjugate imaginary planes. 
 
 A line is determined by the equations of two planes which p x ass 
 through the line. These may be transformed into the shape 
 x-(a + ia') y-(b + ib') z — (c + ic') 
 
 (k + ik') - (a + ia) 
 
 1. Every imaginary point has 
 a conjugate imaginary point. 
 
 If the coordinates of the point 
 
 are 
 
 a + ia', 
 
 i + ib , c+ic, 
 the coordinates of the conjugate 
 imaginary point are 
 
 a — ia', b — i¥, c — ic. 
 
 2. Through every imaginary 
 point there is one and only one real 
 line, which is the connector of the 
 point to its conjugate imaginary 
 point. Through this line pass all 
 real planes which contain the point. 
 
 The equation of the line join- 
 ing the point a + ia', b + ib', c + ic 
 to its conjugate imaginary point is 
 
 l '-_ar— a — ia' _- Vj^— ' b — ib' 
 
 ~T~ 
 
 a — ia —a — f,a 
 
 or 
 
 _y-< 
 
 a'- b' c' ' 
 
 This line is obviously real 
 
 (I + il') -(b+ ib') I (m 4- im) - (c + ic) ' 
 
 Every imaginary plane has a 
 conjugate imaginary plane. 
 
 If the equation of the plane is 
 ax +by + cz+d 
 
 + i (a'x + b'y + c'z + a") = 0, 
 that of the conjugate imaginary 
 plane is 
 ax + by + cz + d 
 
 - i (a'x + b'y + c'z + d') = 0. 
 
 In every imaginary plane there 
 is one and only one real line, which 
 is the line of intersection of the 
 plane and its conjugate imaginary 
 plane. This line is the locus of all 
 real points in the plane? ' 
 
 The real points in the plane 
 ax + by+cz + d 
 
 + i (a'x + b'y + c'z + d') = 
 are given by 
 
 aa) + by + cz + d = 
 and a'x + b'y + c'z + a" = 0. 
 
 These points lie on a real 
 straight line, which is also the real 
 line in the conjugate imaginary 
 plane. 
 
 ■ib'-b-ib' 
 
 :—C — ic' 
 ■ic' — c — ic' 
 z — c 
 
Correlative Theorems 
 
 199 
 
 3. The line joining twoimagin- 
 ary points isihe conjugateimaginary 
 line of the line joining their con- 
 jugate imaginary points. 
 
 4. Two conjugate imaginary 
 lines, determined as the connectors 
 of two pairs of conjugate imaginary 
 points, meet at a real point, if the 
 real lines through the two pairs of 
 conjugate imaginary points are con- 
 current, i.e. if the two pairs of con- 
 jugate imaginary points lie in the 
 same real plane ; otherwise they do 
 not meet at all. 
 
 The condition that the line 
 
 x — a- 
 
 la 
 
 y—b — ib' 
 k + ik' — a — id I + il' — b —ib' 
 
 z—c- 
 
 ■%c 
 
 m-ji-im —c — ic 
 
 should intersect the line obtained 
 by changing the sign of i, is 
 
 a V c' 
 k'' V m' = 0. 
 k — a l — b m — c ! 
 
 This is the condition that the 
 real lines through the imaginary 
 points determining the lines should 
 meet at a point (see 2 above). 
 
 If this condition is not satisfied 
 there is no real point on the con- 
 jugate imaginary lines and they do 
 not intersect. 
 
 5. If two imaginary points are 
 such that the real lines through the 
 
 The line of intersection of two 
 imaginary planes is the conjugate 
 imaginary line of the line of inter- 
 section of their conjugate imaginary 
 planes. 
 
 Two conjugate imaginary lines, 
 determined as the lines of inter- 
 section of two pairs of conjugate 
 imaginary planes, lie in a real plane, 
 if the real lines in the two pairs of 
 planes are concurrent, i.e. if the two 
 pairs of conjugate imaginary planes 
 meet in a real point ; otherwise they 
 do not lie in a plane at all. 
 
 The condition that the line 
 ax + by + cz + d 
 
 + i (ax + b'y + cz + d') = 
 kx + ly+mz + n 
 
 + i (k'x + Vy + m'z + li) = 
 should intersect the line obtained 
 by changing the sign of i, is 
 abed 
 k I m n 
 a' b' c d' 
 k' V ml n' 
 This is the condition that the 
 real lines in the determining planes 
 should meet at a point, i.e. that 
 they should lie in a plane (see 2- 
 above). 
 
 If this condition is not satisfied 
 there is no real plane through the 
 conjugate imaginary lines and they 
 do not lie in a plane. 
 
 If two imaginary planes are 
 such that the real lines in the planes 
 
 = 0. 
 
 \ 
 
200 
 
 The Imaginary in Geometry 
 
 points are not concurrent, no real are not coplahar, no real point 
 
 plane can be drawn through their exists on their line of intersection, 
 connector. 
 
 This follows from 4. This follows from 4. 
 
 149. Systems of lines and planes 
 through a real point. 
 
 A real point P contains 
 
 (1) An infinite number, of real planes, 
 which intersect in an infinite'number of 
 real lines through the point P. These 
 planes are determined by P and any 
 other twp real points. 
 
 (2) An infinite number of planes 
 determined by P, a real point A and an 
 imaginary point Q. These planes may 
 be real, in which case the plane con- 
 tains Q>, the conjugate imaginary 
 point of Q. 
 
 All those planes which pass through 
 a real point .A as well as through P 
 intersect in the real line AP. 
 
 (3) An infinite number of planes 
 determined by P, Q and R, where Q -and 
 R are imaginary points. If a particular 
 plane contains the conjugate imaginary 
 points of Q and R, it is real. Otherwise 
 it is imaginary. 
 
 Planes of this system intersect in 
 imaginary lines through P. 
 
 Systems of lines and planes through 
 an imaginary point. 
 
 Through an imaginary point P there 
 passes one real line, which is the con- 
 nector of P to its conjugate imaginary 
 point P'. 
 
 Through the line PP' pass an infinite 
 number of real planes, all of which con- 
 tain P and P'. These are the only real 
 planes which contain P. Hence if a real 
 plane passes through an imaginary. point, 
 it also passes through the conjugate 
 imaginary point. 
 
 Systems of lines and points in a real 
 planp. 
 
 A real plane o- contains 
 
 (1) An infinite number of real points, 
 the connectors of which are an infinite 
 number of real lines in the plane o\ 
 These points are determined by a- and 
 any other two real planes. 
 
 (2) An infinite number of points 
 determined by <r, a real plane a-' and an 
 imaginary plane cr x . These points may 
 be real, in which case the point con- 
 tains <7i', the conjugate imaginary 
 plane of o-j. 
 
 All those points which are contained 
 by a' as well as by <r lie in the real 
 straight line tr<r'. 
 
 (3) An infinite number - of points 
 determined by <r, <r x and o- 2 , where o- t and 
 <72 are imaginary planes. If a particular 
 point contains the conjugate imaginary 
 planes of u x and o- 2 , it is real. Otherwise 
 it is imaginary. 
 
 Points of this system lie on imaginary 
 lines in a. 
 
 Systems of lines and points in an 
 imaginary plane. 
 
 In an imaginary plane o- there is one 
 real line, which is the line of intersection 
 of the plane with its conjugate imaginary 
 plane o-'. 
 
 On the line io-u' there are an infinite 
 number of real points, which are in the 
 planes <r and a. These are the only real 
 points in the plane a-. Hence if a real 
 point lies in an imaginary plane it is also 
 in the conjugate imaginary plane. 
 
Imaginary Projection 201 
 
 Through the line PP' also pass an On the line tra-' are also an infinite 
 
 infinite number of imaginary planes, nuinber of imaginary points, which are 
 
 which are conjugate in pairs. All of conjugate in pairs. All of these points 
 
 these planes have the line PP' for their have the line <ra for their real line and 
 
 real line and therefore contain no other therefore contain no other real line, 
 real line. 
 
 In all these planes an infinite number Through these points an infinite 
 
 of imaginary lines can be drawn through number of imaginary lines can be drawn 
 
 the point P. in the plane <r. 
 
 Through the point P and any other By means of the plane a- and any 
 
 two points, real or imaginary, an infinite other two planes, real or imaginary, an 
 
 number of imaginary planes can be infinite number of imaginary points can 
 
 drawn. These imaginary planes intersect be constructed. These imaginary points 
 
 in an infinite number of imaginary lines determine an infinite number of imagin- 
 
 through P. ary lines in the plane it. 
 
 150. Projection from an imaginary centre. 
 
 Projection from .an imaginary centre of points in one real plane into 
 points in another real plane. 
 
 Let S .be the centre of projection, S' its conjugate imaginary point, 
 o- the first plane and a the second plane. 
 
 (1) All real points in one plane are projected into imaginary points 
 in the other plane with the exception of the points where the line SS' 
 meets the planes and the real points on the line of intersection of the 
 planes a and a. The points which correspond to these points are real. 
 
 Consider any real point P in the plane o- other than the special 
 points mentioned above. SP is an imaginary line since it does not pass 
 through S', the conjugate imaginary point of S. It contains one real 
 point P and therefore cannot contain any other real point. Hence P', 
 the point where SP meets the plane & ', is an imaginary point. 
 
 (2) Every conic is projected into a conic. ' 
 
 Take A, B, C, D and P any five points on a conic L in the plane a. 
 Let these points be projected into the points A', B', C, B' and P'. 
 Then (P . ABGD) = (P . A'B'G'B'). Hence as P moves along the conic 
 L, the point P' will "move along the corresponding conic L'. 
 
 (3) A real conic can always be projected into a real conic. 
 Consider the conic L in the plane o\ Take any real point Q in the 
 
 plane a outside the conic L, i.e. so that real lines can be drawn through 
 the point which meet the conic in imaginary points. ' Draw any real 
 
202 
 
 The Imaginary in Geometry 
 
 line s through outside the plane a and take two conjugate imaginary 
 points $ and S' on this line. Take *S for centre of projection. Through 
 draw any real straight line, a in the plane a- to meet the conic L in 
 conjugate imaginary points A x and A 2 . Then the plane SS'AiA^ is 
 real and &4j is an imaginary line in this plane. It therefore contains 
 a real point A 3 . 
 
 Similarly draw through in the plane a two other straight lines 
 b and c. From these derive as above points B^ and d and on their 
 connectors to, S determine the real points B 3 and C 3 . 
 
 Take the plane A 3 B 3 G 3 as the plane o-'. 
 
 By construction the three imaginary points A u B lt C^ on the conic L 
 project into the real points A s , B s , C s in the plane a, which must be on 
 the conic L' into which the conic L is projected by (2). Also the conic 
 L meets the line a<r' in a pair of points real, coincident or conjugate 
 imaginary. Through these points the conic L' passes. Hence the 
 conic L' contains three real points and a pair of points either real or 
 conjugate imaginary. Hence the conic L' is real. 
 
 Instead of seeking the projection, in which A u B lt C x are projected 
 
 l'(S J 
 
 into A 3 , B 3 , G 3 , it is possible to project A lt A 2 , B 1 into real points A a , 
 A t , B 3 , where A it B t , C t are the points corresponding to A iy B it G 2 . 
 
 It should be noticed that A 3 is the point of intersection of SAi and 
 S'A* and that tjhe points B s and C 3 may be constructed in a similar 
 manner. 
 
The Real Conicoid 203 
 
 Let SS' meet A 3 A 4 in N and let A 3 A t meet a in M. Since Jf (LA^ 
 is harmonic, M is a point in the polar of with respect to the conic. 
 Also NOSS' is harmonic and therefore' N is fixed when S, 0, 8' are 
 fixed. 
 
 Consider the plane A 3 A t B s . It meets a in M a point on the polar 
 of 0. Also -4 S 4 4 and 5 3 JS 4 intersect at N and therefore B t lies in this 
 plane and is the projection of i? 2 on the plane. Hence the plane 
 A 3 AiB s B t meets a in the point If, where B s B t meets b. Hence it 
 intersects o- in the polar of 0. 
 
 Similarly the plane B^iC^i intersects a in the polar of 0. Hence 
 since B 3 , B 4 are common to the planes A 3 A 4 B s B t and _B 3 .B 4 C 3 C4 these 
 planes coincide. Hence (1) all pairs of conjugate imaginary points 
 obtained by drawing real chords through to meet the conic are projected 
 into real points, and (2) for all positions of S the line aer' is the polar 
 of 0. 
 
 • (4) All projective theorems, which are true for real points on a conic, 
 are also true for imaginary points on a conic. 
 
 Construct the projection as in (3) so that the real points for which 
 the theorem holds are not the point SS' . cr or the points on era'. Then 
 the theorem is obvious. 
 
 Similarly it follows that projective theorems, which are true for real 
 points on- real straight lines, are true for imaginary points on real straight 
 lines. 
 
 This affords a general confirmation of the results obtained in the 
 earlier chapters. 
 
 151. Graphic representation of the imaginary in Solid Geo- 
 metry. 
 
 Let the axes of coordinates be rectangular and let the coordinates of 
 any point P he oci + iao 2 , y x + iy«, z^ + iz^ 
 
 Let P x be the point x x , y lt z x , and let P 2 be the point x 2 , y 2 , z 2 . Let 
 be the origin and let OPi and 0P 2 be r^ and r 2 . Let the direction 
 cosines of OP^. and 0P 2 be respectively cos a, cos /3, cos 7 and cos a', cos ft', 
 cos 7'. Then the coordinates of P are r x cos a+ir 2 cos a', r x cos ft+ir 2 cos ft' 
 and 7 1 ! cos 7 + ir 2 cos 7'. These values may be substituted in the equation 
 of a Surface in order to find the principal coordinates of the point P, 
 which are r x and r 2 , measured in the directions given by cos a, cos/3, 
 cos 7 and cos a, cos ft', cos 7'. 
 
204 The Imaginary in Geometry 
 
 (A) To trace, a real plane^ 
 
 Substituting in the equation in its usual form it is found that 
 
 r 1 ! (I cos a + mcos8 +ncosy)—p = (1) 
 
 and r 2 (I cos a' + m cos 8' + n cos 7') =0 (2) 
 
 From (2) it follows that the direction of the imaginary coordinate 
 may be any direction parallel to the plane and that the coordinate r 2 
 may have any value.' From (1) it follows that the point P± may be 
 anywhere in the plane. Hence from the origin a real vector r^ may be 
 drawn to any real point in the plane and through that point an imaginary 
 vector may be drawn in any direction in the plane and of any length 
 to determine a point in the plane. 
 
 (B) To trace a real conicoid. 
 
 . Let the equation of the surface be 
 
 a? y" z* , A 
 
 Substituting in this equation it is found that 
 
 (cos 2 a cos- 8 cos 2 7) 2 (cos 2 a' cos s /3' cosi Vl_ n 
 
 •(1) 
 
 , (cos a cos a' cos 8 cos ff cos 7 cos 7') 
 
 and ^{— ^+^V— + ^V-^r°--( 2 > ' 
 
 Consider the line OP^ It is a diameter of the conicoid whose 
 direction cosines are cos a, cos 8, cos 7. (2) is the condition that the 
 line OP e should lie in the diametral plane of OP t . If this condition is 
 fulfilled (2) is satisfied. 
 
 Let the point P s be given. Then P^P^ is parallel to the diametral 
 plane of 0P 1 and 
 
 r, U' 
 
 cos 2 /3 cos 2 7\ 
 -I — rs — h — ^— — 1 
 
 cos 2 a cos 2 8' cos 2 7' 
 
 1 -— -1 — 
 
 a 2 6 2 c 2 
 
 .(3) 
 
 , T cos 2 a' cos 2 .i8' cos 2 7' 1 
 
 where r is the length of the, semi-diameter in which the line through 
 the centre, whose direction cosines are cos a', cos/3', cos 7', meets the 
 conic in which the diametral plane intersects the conicoid. 
 
 ™ .. . „ ( „ /cos 2 a ' cos 2 8 cos 2 y\ , ) , . . 
 
 Therefore r 2 2 = r 2 | n 2 (^-^ +-^ +-^J - lj (4) 
 
The Real Conicoid 205 
 
 Hence the imaginary points on the conicoid which have Pi,, as repre- 
 senting the real parts of their coordinates, lie in a plane through P x 
 parallel to the diametral plane of 0P X and at such distances from P, 
 that they are situated on a conic similar and similarly placed to the 
 section of the ellipsoid by the diametral plane of 0P X . 
 
 As the point P, moves along OP^ the linear dimensions of the conic 
 on which the points lie vary as 
 
 / „/cos 2 a cos 2 /8 cos 3 7\ _ /%,* w, 2 z? , 
 
 If OP x be taken as axis of z and the axes of x and y are any pair of 
 conjugate diameters in the diametral plane of 0P lt and cos a, cos a', ... 
 are the direction ratios of r^ and r 2 , then 
 
 cos a = cos B = cos 7' = 0. 
 . , „ /cos 2 a cos 2 B cos 2 7\ z? 
 
 Also n^ + T + VW 
 
 , 1 cos 2 a' cos 2 6' 
 
 and - = — — + ■ 
 
 r 2 a 2 ¥ 
 
 Hence (4) becomes 
 
 , /cos 2 a' 
 
 cos 2 )8' \ _ 2. _ , 
 
 Therefore - % - %' + % = 1 . 
 
 a 2 o 2 c 2 
 
 This is the surface which is the graph of the branch in question of 
 the conicoid. 
 
 This result may be obtained, as follows. Take QP± as axis of z and 
 
 two conjugate diameters in the diametral plane of 0P X as axes of 
 
 coordinates. Then, if x, y, z are the coordinates of points on the branch 
 
 in question, of the surface, z is real, and x and y are imaginary. Hence 
 
 , x, y, z are respectively ix 2 , iy 2 and z. Substituting in the equation of 
 
 • x 2 v 2 £ 2 
 
 the surface it is found that the graph in question is — ^ — ^ + — = 1. 
 
 This is an hyperboloid of two sheets which touches the real ellipsoid 
 where it is met by the diameter' OP^. The same surface is obtained 
 whatever pair of conjugate diameters in the diametral plane are taken 
 for axes of x and y. 
 
 If OPi is taken as an. .imaginary axis, the coordinates of a point on 
 the corresponding branch are x u y± and iz 2 and the equation of the 
 
206 The Imaginary in Geometry 
 
 graph is -\ + tj — \ = !• This is an hyperhploid of one sheet, which 
 
 touches the real ellipsoid where it is met by the plane 2= 0. It is the 
 
 same whatever pair of conjugate diameters in the plane z = are taken 
 
 as axes of coordinates. Hence for each diameter and its diametral 
 
 plane, there are in addition to the (1, 1, 1) alnd (i, i, i) branches, two 
 
 hyperboloidal branches of the form (i, i, 1) and (1, 1, i). These two 
 
 x 2 y 2 S 2 
 hyperboloidal branches have a common asymptotic cone — + y^ — - = 0. 
 
 Hence it is seen that the parent branch of the conicoid 
 
 » 2 f z* -, r> 
 
 consists of eight parts, viz. the (1, 1, 1) (i, 1, 1) (1, i, 1) (1, 1, i) (1, i, i) 
 (i, l,i) (i, i, 1) (i, i, i) branches. The branches 2 to 4 and 5 to 7 are of 
 the same type. 
 
 The typical branches are 
 
 (a) (1, 1, 1) branch which is the ellipsoid 
 
 -.+£+'-1-0. 
 
 a? o 2 c 2 
 
 (b) (1,1, i) branch which is an hyperboloid of one sheet 
 
 a? + b* c 2 ' 
 
 (c) (i, i, 1) branch which is an hyperboloid of two sheets 
 
 _^_2/!+i 2 _i = o 
 
 a? 6 2 c 2 
 
 (d) (i,i,i) branch which does 1 not exist in this case. Its equation is 
 
 a? y 2 z* ., „ 
 
 a 2 ¥ & 
 
 The hyperboloids (b) and (c) have a common asymptotic cone 
 
 « 2 y 2 z\ n 
 
 — \-- = 
 
 a? b* c 2 
 
 The conjugate loci may be obtained by taking any point. A' on the 
 ellipsoid and associating with QA' as axes of coordinates any pair of 
 conjugate diameters OB', '00' in the diametral plane of Oil'. ~ 
 
 The equation of the, ellipsoid is then 
 
 ■£ + £+■--1-0 
 
The Real Gonicoid 207 
 
 As in the case of the parent branch there are eight branches of the 
 curve, of which four are typical. 
 
 (a) (1, 1, 1) branch is the same ellipsoid as before. 
 
 (b) (1, 1, i) branch is an hyperboloid of one sheet which touches 
 the ellipsoid where it is met by the plane z = 0. Its equation is 
 
 a' 2+ &' 2 c' 2 
 
 (c) (i, i, 1) branch is an hyperboloid of two sheets, which touches 
 the ellipsoid at the ends of the diameter, which is the axis of z. Its 
 equation is 
 
 a' 2 b 2 c 2 
 
 (d) (i, 1 i, i) branch does not exist in this case. Its equation is 
 
 When it exists it is the same as the corresponding part of the parent branch. 
 The two hyperboloids (b) and (c) have the common asymptotic cone 
 
 Consider the eonicoid 
 
 a' 2 + b'* c' 2 
 
 ^2 fl»2 ^2 
 
 a 2 fi z c 2 
 
 and any real plane parallel to the yz plane. Let the plane be x — h=0 and let h< a 
 This plane meets the eonicoid in the conic 
 
 V 2 2 2 , h 2 
 
 — H — = 1 . 
 
 6 2 c 8 a 2 ' , 
 
 This is an ellipse. It has a real branch which corresponds to a real branch of the 
 eonicoid. It also has imaginary branches. Consider the branch of the ellipse for 
 which y is real and z purely imaginary. This branch gives points on the eonicoid of 
 the form (1, 1, i). ' To obtain the principal coordinates of suph points the x and y 
 coordinates must be combined into a single real coordinate. This gives a real 
 coordinate — say OP\ — in the xy plane and an imaginary coordinate perpendicular to 
 this plane that is parallel, to the axis of z. For this particular branph the coordinates 
 of all points on the eonicoid will be of this form. As however OP\ will generally be, 
 different for different points, these points will not generally lie on the same branch 
 of the surface. 
 
 The hyperboloids may be graphed in the same way as the ellipsoid. 
 The branches will be found to be of the same nature. The paraboloids 
 may also be graphed. 
 
208 
 
 The Imaginary in Geometry 
 
 (C) To trace an imaginary plane. 
 Let the equation of the plane be 
 
 Ix + my + nz + iK (l'x + m'y + n'z) = 0, 
 where P+m 2 + n 2 = l and I' 2 + m' 2 + n' 2 = 1. 
 
 The equation of the conjugate imaginary plane is 
 Ix + my + nz — iK (l'x + m'y + n'z) = 
 and the combined equation of the two planes 
 
 (Ix + my + nz) 2 + K 2 (l'x + m'y + n'z) 2 = 0. 
 
 ^ 
 
 Let Ix + my + nz = be the plane X, l'x + m'y + nz = the plane \', 
 and let s be the real line of intersection of the planes. 
 
 Let Pj and P 2 be as previously defined. Then on substituting in 
 the equation of the plane it is found that 
 
 rj. XI cos a- r 2 K . 2£' cos a =0 (1) 
 
 and r 2 . Oleosa +^K .%l' cosa = 0, (2) 
 
 where "21 cos a = I cos a. + m cos /3 + n cos 7, 
 
 %l ' cos a = l' cos a + m' cos /8 + n' cos 7, etc. 
 
 From (1) and (2) 
 
 XI cos a . XI cos a + K 2 . XI' cos a! . XV cos a = (3) 
 
 This is the condition that the line 0P 2 should. lie in the diametral 
 plane of OPx with respect to the pair of conjugate imaginary planes, 
 that is, if the plane determined by P^ and s be <r u arid <r s be the plane 
 harmonic conjugate of a\ with respect to the pair of conjugate imaginary 
 planes, OP^ must lie in the plane a 2 . 
 
The Imaginary Plane 209 
 
 Wherever the point P, is situated in the plane a x , P 2 must lie in the 
 plane a 2 ; and conversely wherever P 2 lies in the plane a 2 , the point P, 
 must lie in the plane ov Hence the relation between the points P, and 
 P 2 and the planes a, and o- 2 ' is reciprocal. 
 
 Give Pj a definite' position in the plane a,. Then r lt cos a, cos/3, 
 cos 7 are given. In the plane a 2 draw a line 0P 2 in any direction 
 determined by cos a', cos/3', cos 7', so that relation (3) is satisfied. 
 
 Then from (1) 1 = j^¥^-. = ^- , 
 Kr 2 . 2,1 cos a K .p 2 
 
 where p x is the perpendicular from P, on \ and p 2 is the perpendicular 
 from P 2 on \'. But p, is given and therefore p 2 is constant. Hence the 
 locus of P 2 is a line (in the plane cr 2 ) parallel to s. If P, moves along 
 a line parallel to s this line is unaltered. Hence as P\ or P 2 moves in 
 the planes cr, and <r 2 along a line parallel to s, the other point describes 
 another line parallel to s. The distance P,P equals 0P 2 , where P is in 
 the plane a 2 , which can be drawn through P 1 parallel to <r 2 . Hence 
 the locus of P is a straight line obtained as the line of intersection of 
 planes through P, and P 2 parallel to o- 2 and a,. 
 
 If Pi moves along OP,, i.e. if cos a, cos/3, cos 7 are constant but r, 
 varies, then the locus of P 2 , for each value of r lt is a straight line parallel 
 to s. The same is true in regard to the locus of P. 
 
 Consider a section of the figure by a plane OP±PP 2 , where OPi 
 varies but the directions of OPi and 0P 2 are fixed. 
 
 Then from (1) 
 
 OP 1 OP, K$l' cos a' 
 
 7TS = p-o = -^Ti = a constant. 
 
 OP 2 P^P it cos a 
 
 Hence, as Pi moves along OP,, the locus of P is a straight line OP 
 .through 0. This is true for all planes OPzPP,. Hence the locus of P, 
 when Pj has any position in the plane a,, is a plane determined by one 
 position of P and s the real line in the pair of conjugate imaginary 
 planes. When both these conjugate imaginary planes are considered 
 P may Be in either of two such planes. In the preceding a, and cr 2 may 
 be any pair of planes harmonic conjugate of the given pair of conjugate 
 imaginary planes. Hence the graph of a pair of conjugate imaginary 
 planes consists of a system of pairs of planes which pass through their 
 real line. 
 
 The coordinates of a point on such a graph may be constructed as 
 follows. Let ffj' and a 2 be the {a, /8) planes which are real harmonic 
 
 h. 1. g. 14 
 
210 The Imaginary in Geometry 
 
 conjugates of the plane and its conjugate imaginary plane. Take P x any 
 point (real) in ov Draw through P 1 a plane parallel to <r 2 . It meets 
 the graph in a line parallel to the real line in the imaginary planes. 
 The coordinates of any point on this line are 0P 1 real (where is the 
 origin) and P X P imaginary. 
 
 Thus the coordinates of a point of the graph are any real vector in 
 the plane a^ and an imaginary vector parallel to the plane a 2 . 
 
 The preceding may be illustrated as follows. 
 
 Consider two conjugate imaginary planes o- and a', which iiitersect in a real 
 line «. Draw a real plane X perpendicular to s to meet s in 0' and er and tr in a pair 
 of conjugate imaginary lines I and I'. Consider the figure in the plane X. In this 
 plane there are an infinite number of real pairs of harmonic conjugates of I and I'. 
 Let a and /3 be a pair of these lines. The graph of either of the lines I and I', when 
 a and /3 are taken as axes, is a straight line in the plane X. Let P be any point of 
 this graph and let OP x and P X P (x x and iy 2 ) be its coordinates. Then the planes 
 si, si', sa, s/3 form a coaxal harmonic pencil. Referred to any point on s as origin, 
 the coordinates of P are real lengths 00' and x x — which combine with a real length 
 OP x — and the imaginary coordinate iy. These are the principal coordinates of P. 
 The locus of P is obviously one dr other of two planes through s. 
 
 152. Solid perspective. 
 
 Consider any point S and any plane s (the centre and plane of perspective). 
 Take any point in space A. Join S to A to meet s in A T . Take a point A' on SA 
 such that (SNAA')=\ (a constant). Then A' is termed the corresponding point of 
 A in the perspective. 
 
 In this way if A be any point of a solid figure a, then the point A' of another 
 solid figure o-' corresponds to A. 
 
 If A be a point on a straight line a, the point A' will be on another straight line 
 a' which is said to correspond to a, and as the point A moves along the straight line 
 a the point A' will move along the straight line a'. This follows from considering 
 the plane perspective in the plane Sa. The corresponding lines a and a' intersect on 
 the plane s. * 
 
 If two lines a and b in the figure <r intersect, they determine a plane. Take any 
 point P in this plane. Then any line I in this plane through P meets a and b. To 
 P corresponds a point P'- To I corresponds a line V through P' which intersects 
 the lines a' and V which correspond to a and b. Hence P' lies in the plane a'b', which 
 corresponds to the plane ab. Therefore to a plane corresponds a plane and corre- 
 sponding planes intersect on s. 
 
 If any line Sff be drawn through S to meet s in N and a point V be taken on 
 
 SN such that • 
 
 (SNVn)=\, 
 
 SV 
 then the locus of Fis a plane parallel to s, since ^™=X (a constant). Also to every 
 
 point on this plane there corresponds a point at an infinite distance in the second 
 
Solid Perspective 211 
 
 figure. But it has been shown that generally to a plane corresponds a plane. Hence 
 it is assumed that all points at infinity in the second figure, and consequently in 
 every solid figure, lie in a plane, which in this perspective corresponds to the plane 
 in which V is situated, i.e. to the vanishing plane v. 
 
 The points at infinity in the first figure o- correspond to points in the second 
 figure determined by 
 
 ifw sv sw 
 
 ° r ^ ^W = ^' "'" ~NV ' ~WW~^" Hence the ratios of V and W are the same with , 
 respect to S and If, when these points are interchanged. Hence the distance of V 
 from S is the same as the distance of W from If. These lengths may be measured 
 , on the perpendicular from S to, s. Hence the locus of W is a plane w parallel to s 
 and v. 
 
 Therefore if S be the centre, s the plane, and v and w the vanishing planes of 
 a solid -perspective, 
 
 (1) All points in the planes v and w correspond to points at an infinite distance 
 in the other figure. ' 
 
 (2) If a plane a meets the vanishing plane v in the line av, then S . av is 
 parallel to a', and a' is the plane parallel to the plane (viz., S . av) drawn through 
 the line a . s. 
 
 From this it follows that, from a certain point of view, all points at infinity may 
 be regarded as lying in a plane. The method here sketched may be employed to 
 find the properties of solid figures. 
 
 To prove that by a rotation round its real line, through an imaginary angle, an 
 imaginary plane may be superposed on a real plane. 
 Every imaginary plane has an equation of the form 
 
 (ax+by+cz+d)+i(a'x + b'y + c'z + d')=0, (1) 
 
 and the equation of its conjugate imaginary plane is 
 
 (ax+by+cz+d)-i(a'x+b'y+c'z+d')=0 , (2) 
 
 The real line s in these planes is the line of intersection of the real planes ' 
 
 ax + by + cz+d=0, (3)- 
 
 and a'x + b'y + c'z + d'=0 (4) 
 
 Through any real point A on this line draw a real plane o- perpendicular to this 
 line. It will meet the planes (3) and (4) in real lines s x and s 2 an d the conjugate 
 imaginary planes (1) and (2) in a pair of conjugate imaginary lines *{ and s{. In the 
 real plane n- through the point of intersection of the conjugate imaginary lines s{ 
 and s{ pass the pair of real lines Sj and s 2 , and as in Art. 51 the line s{, by a rotation 
 round A through an imaginary angle, may be brought into coincidence with the real 
 line « x . By this process ^he imaginary plane (1) is brought into coincidence with 
 the real plane ax + by+cz + d=0. 
 
 Hence it follows that there is nothing essentially different between a real' and an 
 imaginary plane. As in the case of real and imaginary straight lines, the difference 
 lies in their relation to points outside themselves. 
 
212 The Imaginary in Geometry 
 
 EXAMPLES 
 
 (1) Show that in an imaginary projection all the real points on a real straight 
 line except two can be projected into imaginary points on another real line. 
 
 (2) Prove that the real lines in the imaginary planes, which pass through an 
 imaginary line, form a system of surfaces of the second degree. 
 
 (3) Prove that the real lines through the imaginary points, which lie on an 
 imaginary straight line, form a system of surfaces of the second degree. 
 
 (4) Prove that an imaginary point a+ia', b+ib', c + ic', contains a purely 
 imaginary line the equations of which are 
 
 ix-\-o! _ iy+b' _ iz + c' 
 a b c 
 
 (5) Prove that a purely imaginary plane contains a purely imaginary line which 
 is parallel' to the real line in the plane. 
 
 (6) In Art. 25 prove that if S=Sx=S 2 , the values of all the anharmonic ratios 
 of the four points are either -a x or —a> 2 , where <»i and a 2 are the two imaginary 
 cube roots of unity. 
 
 (7) Prove that if (ABCB) = - »j, then (ABCD) = (ACI)B) = (ADBC). 
 
INDEX OF THEOREMS 
 
 The reference numbers refer to articles 
 
 Angle between real and imaginary lines, 
 sine, cosine and tangent of, 56 
 between two imaginary lines, sine, cosine 
 
 and tangent of, 61 
 between real and imaginary planes, 153 
 purely imaginary, construction of, 66 
 measurement of, 67 
 representation of ' trigonometrical 
 functions of, 68 
 imaginary eccentric, 74 
 real bisectors of, 66 
 Angles, imaginary, relations connecting 
 trigonometrical functions of, 69 
 subsidiary angles of, 69 and 70 
 
 use of meridional tables to obtain, 70 
 trigonometrical function of, used for 
 integration, 75 
 Anharmonic ratio, definition of, 5 
 of real pencil, 10 
 of imaginary pencil, 82 
 of real and imaginary points, 23 
 of four collinear points, condition that 
 
 it may be real, 24 
 of pencil formed by connectors to four 
 points, 23 
 Anharmonic ratios of four collinear points, 
 
 relations connecting, 25 
 Axiom I, 1 
 Axiom II, 51 
 
 Bisectors, real, of an imaginary angle, 66 
 Brianchon's theorem, 98 
 
 Carnot's theorem for general conic, 101 
 Ceva's theorem for real triangle and imagi- 
 nary transversal, 16 
 for imaginary triangle and , imaginary 
 transversal, 65 
 Circle, square of imaginary tangent to, 32 
 met by chord through fixed point in 
 points product of whose distances 
 from fixed point is constant, 32 
 real, through a real and an imaginary 
 point, 32 
 Circles, determine, same involution on 
 radical axis, 33 > 
 determine same involution on line at 
 
 infinity, 33 
 orthogonal, determine inverse points on 
 
 any diameter, 31 
 points of intersection of, 36 
 Circular points at infinity — see Critical points 
 
 Coaxal circles, ratio of tangents to, from 
 variable point on coaxal circle, 
 constant, 32 
 Common conjugates with respect to two 
 conies (general) of points on fixed 
 line, a conic, 103 
 
 .self-conjugate triangle of two conies, 104 
 Conic, general, definition of, 87 
 
 determined by five points, 88 
 
 met by a line in two points, 89 
 
 one tangent at a point to, 89 
 
 correlative of anharmonic property of, 
 90 
 
 correlative of, a conic, 91 
 
 projective ranges and involutions on, 92 
 
 pole and polar with respect to, 94 
 
 determines involution on every line in 
 its plane, 95 
 
 Pascal's theorem for, 97 
 
 Brianchon's, theorem for, 98 
 
 Desargues' theorem for, 100 
 
 Carnot's theorem for, 101 
 
 projection of real branch into imaginary, 
 151 
 
 imaginary and real branches in perspec- 
 tive, 96 
 
 contains real or semi-real quadrangle, 107 
 Conic, real, definition of, 106 
 
 when real, 106, 119, 125 
 
 construction of from real data, 120 
 
 involution determined by on a straight 
 line, 38 
 
 variable chord through fixed point meets 
 in points product of whose dis- 
 tances from fixed point is in con- 
 stant ratio to squares of parallel 
 diameters, 42, 43 
 
 connectors of points of intersection of 
 two straight lines with, 45 
 
 intersection of imaginary straight line 
 with, 49 
 Conic, imaginary, contains a real or semi- 
 real quadrangle, 107 
 
 cannot have more than one real or semi- 
 real self-conjugate triangle, 108 
 
 every real point has one real conjugate 
 with respect to, 111 
 
 locus of real conjugates of points on a 
 straight line with respect to, is 
 eleven points locu^s of line and real 
 or semi-real inscribed quadrangle, 
 111 
 
214 
 
 Index of Theorems 
 
 Conic, imaginary (cant.) 
 harmonic locus of, 112 
 foci of, 122 
 
 focal properties of, 122, 124 
 equation of, 138 
 equation of projection of, 139 
 by nature or displacement, 141 
 tracing of, 142 
 four points on, in (a, (3) and (/3, a) 
 
 planes, 143 
 special case of, 145 
 real conjugate points with respect to, 
 
 111 
 real conjugates of points on line with 
 respect to, locus of, 111 
 Conies, real points of intersection of, 48 
 Conies, conjugate imaginary, intersect in 
 real or semi-real quadrangle, 107 
 have real or semi-real circumscribed 
 
 quadrilateral, 107 
 harmonic locus of, » real conic, 112, 
 
 113, 114 
 anharmonic locus of, a real curve, 114, 
 115 
 Conies, general, with two points of inter- 
 section, intersect in four points, 99 
 with one point of intersection, intersect 
 
 in four points, 102 
 twOj locus of common conjugates of 
 points on fixed line with respect 
 to, a conic, 103 
 have common self-conjugate triangle, 104 
 intersect in four points, 105 
 eleven points, loous of, 105 
 Conies having double contact at imaginary 
 
 points, 125 
 Conjugate imaginary points, 2 
 
 determined by circle on a straight line, 31 
 Correlative properties, of imaginary points 
 and lines, 12 
 of imaginary points and planes, 148, 
 149 
 Correspondence, real and imaginary, 83 
 Critical lines, 22, 78, 147 
 Critical points, all circles meet line at 
 infinity in", 34 
 all conies through, are circles, 34 
 nature of, 22, 78 
 
 Desargues' theorem, 100 
 
 Distances, measurement of, 3, 56, 61 
 
 Duality, principle of, in space, 147, 148, 149 
 
 Equation of coDic, point on which sub- 
 tends pencil of constant anhar- 
 monic ratio, imaginary, at four 
 fixed points, 117 
 imaginary, 138, 139 
 of lines and planes in space, 149 
 
 Focal properties of an imaginary conic, 
 122, 124 
 
 Foci of a conic, construction of, 46 
 of imaginary conic, 132 
 
 Geometry of the sheaf, 150 
 
 Harmonic property of semi-real quadrangle 
 and quadrilateral, 18 
 imaginary quadrangle and quadri n 
 lateral, 84 
 conjugates, common, of two pairs of 
 points determined, 8, 31 
 of given pair of points with given 
 mean point determined, 31 
 perspective, general conic in, 96 
 
 Imaginary coordinates, direction of, 126 
 Infinity, plane at, 153 
 Intersections of a real and an imaginary 
 straight line, 10, 76 
 of STeal conic and an imaginary line, 49 
 of two general conies, 99, 102, 105 
 of real and imaginary straight lines with 
 real conic, 136 
 Involution, extended conception of, 7 
 imaginary, 7 
 when real, 8 
 property of semi-real quadrangle and 
 
 quadrilateral, 19 
 pencil, 22 
 
 determined by circle on line, construc- 
 tion of, 29 
 determined by conic on line, construc- 
 tion of, 38 
 on conic, 92 
 on conic determined by chords through 
 
 fixed point, 93 
 determined by conic on every line in its 
 plane, 95 
 Involutions, any two, real, in plane per- 
 spective, 20 
 
 Lines, system of, through a point, 80 
 imaginary, in space, 147, 148, 149' 
 
 Locus of real points at which a pair of 
 conjugate imaginary points and 
 two imaginary points subtend a 
 pencil of real anharmonic ratio, 116 
 
 Measure of an imaginary length on a real 
 line, 56 
 an imaginary line, 61 
 Measurement of an imaginary length on 
 • a real line, 3 
 on an imaginary line, 56, 61 
 of imaginary angles, 67 
 Measures of sides of closed figure on real 
 line, sum of, zero, 58 
 imaginary line, sum of, zero, 62 
 Menelaus' theorem for real triangle and 
 imaginary transversal, 15 
 for imaginary triangle and imaginary 
 transversal, 65 
 
Index of Theorems 
 
 215 
 
 Parallel lines, 53 
 
 Pascal's theorem, 97 
 
 Pencils with imaginary vertices, 82 
 
 Perpendicular lines, 54 
 
 Perspective, solid, 152 
 
 Plane at infinity, 152 
 
 Plane, imaginary, 148, 149, 151, 152 
 
 Pole and polar with respect to a conic, 
 
 94, 95 
 Poncelet figure of a circle, 30 
 
 of two circles, 37 
 
 of a real conic, 39 
 
 of foci of a conic, 46 
 
 of critioal lines, 157 
 
 of imaginary lines, 76 
 Principal coordinates of a point, 8 
 Projection of an imaginary length on a 
 real line, 55 ' , 
 
 of points on an imaginary line, 81 
 
 of two real and a pair of conjugate 
 imaginary points, 26 
 
 of two pairs ^of conjugate imaginary 
 points, 26 
 
 of points into the critical points, 85 
 
 from a real centre, 85 
 
 from an imaginary centre, 85, 150 
 
 of real branch of conic into the imagi- 
 nary branch, 150 
 Projective ranges and pencils, 21, 81 
 
 on » conic, 92 
 
 Quadrangle and quadrilateral, semi-real, 17 
 harmonic property of, 18 
 involution property of, 19 
 
 Eatio of imaginary points, 4 
 
 Eight angles, imaginary lines at, defined, 22 
 
 Self-corresponding elements of projective 
 ' ranges and pencils, 99 
 
 Solid perspective, 153 - 
 
 Square, semi-real, 27 
 
 projection of quadrangle into, 86 
 Straight line, imaginary, defined, 9 
 
 tracing of, 76 
 Sum and difference formulae for angles be- 
 tween imaginary and real lines, 60 
 imaginary lines, 64 
 Superposed projective ranges and pencils, 
 21, 81 
 pencils with critical lines for self-cor- 
 responding rays are equal pencils, 
 35 
 
 Tangent, only one at point on real conic, 
 
 129 
 Tracing of real line, 136 
 
 of imaginary straight line, 76 
 
 of a real conic, 127 
 
 of circle and ellipse, 127 
 
 of lines and conies, 134 
 
 of real conic, any origin, 136 
 
 of imaginary conic, 142 
 
 of special case of imaginary conic, 145 
 
 modulus of reduction in, 146 
 
 change of origin in, 133 
 
 general consideration concerning, 131,132 
 Triangle, semi-real and imaginary, 14 
 
 construction of, when one side greater 
 than sum of other two, 50 
 
 with two real lines for side's, 59 
 
 with three imaginary lines for sides, 63 
 
INDEX OF TEEMS AND DEFINITIONS 
 
 The reference numbers refer to articles 
 
 (a,,p) figures, 125 
 
 Anharmonic ratio of real and imaginary 
 points, 5 
 
 Conie, real, 106 
 nest of, 130 
 parent branch of, 127 
 general, 87 
 with real branch, 28 
 imaginary, 106 
 
 by displacement, 141 
 by nature, 141 
 special case of, 145 
 Conjugate imaginary points, 2 
 imaginary lines, 9 
 loci, 30 
 Critical lines and points, 22, 78 
 
 Diameters of a conic, 41 
 
 Eccentric angle complex, 74 
 
 Foci of a conic, 45 
 
 . Imaginary point, 1 
 length, 1, 51 
 straight line, 9 
 angle, 51, 52 
 
 Imaginary (cont.) 
 correspondence, 83 
 conic, 138 
 
 Measure of an imaginary length,. 56, 61 
 
 of an imaginary angle, 70 
 Modulus of reduction, 146 
 
 Nest of a conic, 130 
 
 a pair of imaginary straight lines, 1341 
 
 Perpendicular lines, 22 
 
 Pole and polar, 94 
 
 Poncelet figures, 30 ', 
 
 Principal coordinates of an imaginary 
 
 point, 8 
 Projection of an imaginary length, 55 
 Projective ranges, 6 
 
 Semi-real triangle, 14 
 quadrangle, 17 
 quadrilateral, 17 
 , square, 27, 86 
 ■Subsidiary angle, 68 
 
 Trigonometrical functions, 56, 6l 
 of purely imaginary angles, 67 
 
 CAMBRIDGE : PRINTED BY J. B. PEACE, M.A., AT THE UNIVERSITY PREfS