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 University of California. 
 
 
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 Zbc mniversiti? of Cbicago 
 
 FOUNDED BY JOHN O. ROCKEFELLER 
 
 THE PROBLEM OF THE ANGLE 
 
 BISECTORS 
 
 A DISSERTATION 
 
 SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF 
 
 SCIENCE OF THE UNIVERSITY OF CHICAGO IN CANDIDACY 
 
 FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 
 
 (department of MATHEMATICS) 
 
 BY 
 
 RICHARD PHILIP BAKER 
 
 THE UNIVERSITY OF CHICAGO PRESS 
 CHICAGO, ILLINOIS 
 
 «ik 
 
THE UNIVERSITY OF OHIOAGO PRB88 
 OHIOAGO. ILLINOIS 
 
 Bgents 
 THE BAKER & TAYLOR COMPANY 
 
 HEW TOBK 
 
 OAMBRIDGB UNIVERSITY PRESS 
 
 LONDON AND EDINBUBOB 
 
Zbc mnipersiti? of Cbicago 
 
 POUNDED BY JOHN O. ROCKEFELLER 
 
 THE PROBLEM OF THE ANGLE- 
 BISECTORS 
 
 A DISSERTATION 
 
 SUBMITTED TO THE FACULTY OF THE OGDEN GRADUATE SCHOOL OF 
 
 SCIENCE OF THE UNIVERSITY OF CHICAGO IN CANDIDACY 
 
 FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 
 
 (department of MATHEMATICS) 
 
 BY 
 
 RICHARD PHILIP BAKER 
 
 THE UNIVERSITY OF CHICAGO PRESS 
 CHICAGO, ILLINOIS 
 
Copyright ign By 
 The University of Chicago 
 
 All rights reserved 
 
 Published March igii 
 
 Composed and Printed By 
 
 The University of Chicago Press 
 
 Chicago, Ilh'nois. U.S A. 
 
TABLE OF CONTENTS 
 
 The History of the Problem 
 Method and Results 
 
 V 
 
 vi 
 
 I 
 
 1. The Internal Problem 
 
 2. The Order of the Problem 
 
 3. The Elimination 
 
 4. The Parametric Fields 
 
 5. Two Special Cases . 
 
 6. Reality of the Solutions .. 
 
 7. The Elementary Theory of Equations Applied 
 
 8. Reality of the Roots for Real Angle-Bisectors 
 
 9. A Practical Method for Approximate Solutions 
 
 10. Multiple Points 
 
 1 1 . The Transformations 
 
 12. Finite Multiple Points 
 
 13. The Discriminant 
 
 14. The Monodromie Group of the Equation F (cr, a, j3) = o 
 
 15. The Equation fofthe-Sides 
 
 16. The Monodromie Group of the Equation for 'the Sides 
 
 17. The Reduction of the Equation for <r in the Case of Equal Bisectors 
 
 18. The Reduction of the Equation for the Side in the Case of Equal Bisectors 
 
 19. The Surf ace F (»■, o, 0) = o 
 
 II 
 
 1. The External Problem 
 
 2. The First Elimination 
 
 3. The Group of the Equation 
 
 4. The Second Elimination . 
 
 5. Multiple Roots 
 
 6. The Nodal Curve . 
 
 7. Finite Multiple Points 
 
 8. The Discriminant 
 
 9. Interrelations of the Two Equations 
 
 10. The Transformations 
 
 11. The Determination of the Sides 
 
 12. The Case of Equal Bisectors (External) 
 
 I 
 I 
 2 
 S 
 7 
 8 
 8 
 
 13 
 16 
 16 
 20 
 43 
 45 
 46 
 
 49 
 52 
 53 
 55 
 59 
 
 60 
 60 
 61 
 
 63 
 64 
 66 
 67 
 67 
 69 
 70 
 81 
 82 
 
 224663 
 
IV 
 
 TABLE OF CONTENTS 
 
 III 
 
 1. The Mixed Problem 
 
 2. The Monodromie Group 
 
 3. The Equation for the Tangent of a Half-Angle 
 
 4. The Solution of the Real Problem 
 
 5. The Character of the Solutions 
 
 6. The Case of Equal Bisectors .... 
 
 83 
 83 
 84 
 86 
 
 87 
 88 
 
 IV 
 
 1. The General Problem for Real Data 
 
 2. The Problem When a Right Angle and Two Bisectors Are Given 
 
 3. Special Cases of Isosceles Triangles 
 
 4. The Identical Relations among the Six Bisectors 
 
 5. The Indirect Proof 
 
 6. Generalizations of the Problem 
 
 89 
 89 
 93 
 95 
 96 
 
 98 
 
 Vita 
 
 99 
 
THE HISTORY OF THE PROBLEM 
 
 The problem of constructing a triangle when the lengths of the bisectors of the angles are 
 given has been an outstanding problem among geometers probably from the time of Pascal' 
 and certainly from the time of Euler.'^ 
 
 Brocard^ has summed up the literature, dealing almost entirely with special cases, of 
 which the most extensive treatment appears to be the solution of the problem when one angle 
 is a right angle due to Marcus Baker. ^ This problem is of the sixth order. Among many 
 special treatments that appear in the smaller journals the fundamental problem of determining 
 the character of the algebraic irrationality involved is not mentioned. As a result apparently 
 conflicting statements occur as to the order of the equation concerned, this depending on acci- 
 dental choice of the parameter field. 
 
 The only paper dealing with the general case where the internal bisector formulas are 
 used is P. Barbarin's.' The case where the external formulas are to be used and the case 
 where two of the assigned bisectors refer to the same vertex is not treated in general in any 
 paper known to the writer. 
 
 Barbarin proved that the general internal problem could be solved by the solution of an 
 algebraic equation of order not greater than twelve. The irreducibility and group of the 
 equation are not discussed, and as the equation itself is not set out explicitly further reduction 
 of the order of the problem is not precluded. 
 
 The method of attack used by Barbarin is to solve first the problem when an angle and 
 two bisectors are given, and to use the result as a basis for attacking the general problem. 
 The necessary sacrifice of" symmetry prevents any explicit comparison with the solution given 
 in this paper except at the cost of labor disproportionate to the result. 
 
 The problem must have an extensive domestic history in the schools: Barbarin charges that 
 E. Catalan was among those who have proposed it as an elementary exercise, and from a Russian 
 scholar the writer learns that it has been there extensively used in the schools as a standard 
 set-back for ambitious young geometers. 
 
   See Barbarin in Mathesis (1896), 143-60; also BnU. de. S.M.F. (1894), 76-80. 
 
 'Pet. Mem., XI (1765). 
 
 ^L'lntermidiaire (1894), 1-149; also in Zeilschriftf. M. u. N. U., 32.444. 
 
METHOD AND RESULTS 
 
 The eliminations necessary in the two most difficult cases are accomplished by a com- 
 bination of the method of symmetric functions and a birational transformation. A complete 
 theory of the role of birational transformations is not at hand. In these cases one of the two 
 parameters enters the resultant in the first order. The birational transformation reduces a 
 rather complicated equation to one of the first order in the quantity to be eliminated. A 
 mere substitution then accomplishes the elimination. In a more general case it might always 
 be possible to reduce by such a transformation to the order in which the parameter entered 
 (choosing the lowest where choice is offered) and materially expedite the process. Whether 
 this is so or not must be left an open question. In this case such a surmise led to a tentative 
 search with successful results. 
 
 The determination of the discriminant at least as to its algebraic factors was conducted 
 in the field of the roots (sides of the triangles) by a geometric process which gave two factors 
 afterward transformed by the transformations of the elimination to the field of the param- 
 eters. It then became possible to find all the factors and check the result. 
 
 The separation of the roots was effected by following graphically and algebraically the 
 plane of the sides as divided by discriminantal lines through the transformations to the multiply 
 covered plane of the parameters. 
 
 In the internal case the algebra combines with the case of three internal bisectors (which 
 has a unique real solution for real data) the cases of one internal and two external bisectors 
 into an equation of order ten with the general group. No further irrationality is necessarily 
 involved, though a cubic proved a convenience. The solution is explicitly given by two methods, 
 one with and one without this cubic. For practical approximation a tentative method not 
 involving an elimination proved superior. 
 
 In the external case the solutions are not always real for real data. The equation is of 
 order six and has the general group. 
 
 In the mixed case there are three permutations each involving an irreducible equation of 
 the tenth order with the general group. The separation of the roots is accomplished by a 
 simple graph and approximate solution by trial with the table of logarithmic sines. 
 
 Special cases of equal bisectors, isosceles and right triangles are discussed. In each case 
 the group is determined. 
 
I 
 
 I. THE INTERNAL PROBLEM 
 
 To construct all triangles whose internal angle-bisectors are equal to three given lines. 
 
 In the absence of a geometrical criterion as to the possibility of constructing an assigned 
 figure with given apparatus (whether ruler, ruler and sect carrier, ruler and compass, three- 
 bar link, etc.) resort must be had to algebra. 
 
 The formulas giving the internal angle-bisectors in terms of the sides are: 
 
 ^'2=(«+*+c) {-a-^b+c)bc 
 
 M^ 
 
 {b+cY 
 
 ) Ka-b- 
 
 •■+ay 
 
 {a-\-b-\-c) {a-\-b—c)ab 
 
 T, {a-\-b-\-c) (a—b+c)ca , . 
 
 "-= ^^^ ^'^ 
 
 (a+6)" 
 
 It is to be noticed at the outset that a change in sign of a leaves K unchanged but changes 
 L and M into the corresponding external bisectors which are given by: 
 
 Tz.^ {a—b-\-c) {a-\-b—c)bc 
 
 K. TZ T 
 
 (h-cY 
 j^ (a+b—c) {—a+b+c)ca , , 
 
 ^=" ic^^ ^'^ 
 
 jrz^_ {—a-\-b-\-c) {a—b+c)ab 
 ~ (a-b)' 
 
 We expect then an algebraic solution of the problem to involve as necessarily introduced 
 extraneities the cases where the assigned quantities appear as one internal and two external 
 bisectors, but not those where one external is accompanied by two internal. 
 
 2. THE ORDER OF THE PROBLEM 
 
 Bezout's rule as to the order of eliminants gives 64 as the number of sets of values of a, b, 
 c, satisfying the three equations. This includes as improper solutions any sets of values which 
 satisfy them independently of the values of K, L, M. That the number of such improper 
 solutions should properly be given as 54 is to be established. 
 
 Geometrically phrased the surfaces denoted by the equations (i) intersect in general in 
 10 distinct points which vary with K, L, M, and have at certain other points contact of various 
 orders. To obtain a proper enumeration we consider slight distortions of the original surfaces 
 whose intersections will be in fact 64 distinct points. Since there are identical relations between 
 the planes represented by the separate factors such a distortion must destroy these, or give 
 independence to these planes. 
 
 Algebraically phrased we write for a, b, c and the other plane factors quantities differing 
 
2 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 arbitrarily but slightly from the critical values and determine the number of intersections in a 
 region near the critical points. 
 
 k I m 
 
 and 
 
 T= 7 — r-;— — ^,-— r- and take a-\-b-\-c=i. 
 {a+b+c)abc 
 
 Then 
 
 , raii-aY , Tb(i-by Tc{i-cy 
 
 I — 2a I — 2b 1 — 2C 
 
 and 
 
 The elimination of c gives three quartics so that the 64 points are still represented. 
 A sufficient independence is attained if we treat the quantities t, a, b, c as independently 
 variable in small limits near the values determined by the equations. 
 
 The point a=i, b = o, c = o, t=oo is critical. Write t=-, a=i+a, b = P, c = y and con- 
 
 sider the region near / = o, a=o, /8=o, y=o. 
 The types of proper approximations are 
 
 kt = a'; U = p; mt = y. 
 
 Hence for assigned k, I, m, there are two intersections in the region and this point is to be 
 counted a double point for every k, I, m since the values of a, b, c are in the limit independent 
 of k, I, m. 
 
 So also 
 
 a = o, b=i, c = o, T= CO 
 and 
 
 a = o, b = o, c=i, T=oo. 
 
 The class gives then six intersections. 
 
 The point a = i, 6=00, c= — <», t=o, approached by b = p+^, c=—p, is critical, and 
 
 writing a = i+a, b=^, c = — , the proper approximations are of the types ka=T, l^=r, 
 R y 
 
 m-f=T with four solutions and six members of the class, in all 24 intersections. 
 
 The point a=oo, b=^, c=— =0, t = o, approached by a=h = p, c=i — 2p, gives in the 
 
 same way proper approximations of the types 
 
 with 8 intersections in the region and three members of the class, in all 24 intersections. 
 
 The total number of points to be counted for the contacts at these critical points is then 54. 
 
 No other combinations of values render k, I, m indeterminate and we conclude that the 
 order of the problem is ten. 
 
 3. THE ELIMINATION 
 
 It is convenient to substitute for the original problem that of constructing a triangle whose 
 internal bisectors are in a given ratio, the further construction being an affair of ruler and compass 
 and unique save as to cyclic order. 
 
THE ELIMINATION 3 
 
 This reduction enables us to take any relation between the sides; in general we take 
 
 a+b+c= I. 
 
 We may then proceed to eliminate c by means of this and b by Bezout's or other methods, 
 arriving at an equation for a. 
 
 The consideration of such an equation (which is obtained in § 15) yields, however, no view 
 of the relation between sets of angle- bisectors and sets of sides; to obtain which the symmetric 
 function transformation of order three is introduced. 
 
 a-\-b-{-c =x=i 
 
 ' ab-\-bc-\-ca = y 3) 
 
 abc = z 
 
 The determination of any quantity of which y z are rational functions gives after solution 
 of a cubic equation a unique triangle; that is as to ratios of the sides. It turns out that this 
 cubic though convenient is not algebraically necessary, the character of the irrationality involved 
 not being essentially altered by its introduction. 
 
 Using the symmetric function n^ethod we write 
 
 K^+L^+M' =p 
 
 K'L'+L'M'+I{'K'=q (4) 
 
 K' D M' =r 
 
 and expressing p q r in terms oi x y z, obtain 
 
 g= (xy-zy ^'^'^^~ syg+^'z-^^y]. (5) 
 
 It is convenient to write 
 
 '"k" '~IJ' "'~M 
 
 * = ^.' ^=fa' '^ = jr.- (6) 
 
 We now write the symmetric functions of the angle-bisectors of order zero : 
 
 q' _ {k-\-l+mY 
 
 « = - = , 
 
 pr kl+lm+mk ' (7) 
 
 g^qi^ {k+l+my 
 '^ r' Mm 
 
 and use the quantities a, j3 as fundamental parameters. 
 Choosing the scale by placing x=i 
 
 (Ay-§yz+z-yy 
 
 {yi- 2y'z+3yz- sz'-z) Uy-8z-i) ' (8) 
 
 (4y'-8yz+z-yy 
 {4y-Sz-iyiy-zyz' 
 
THE PROBLEM OF THE ANGLE-BISECTORS 
 
 The elimination can now be accomplished by means of a birational transformation: 
 
 42 (y- 2Z) 
 
 <t,= 
 
 applying which we have 
 
 iy-z) i4y-Sz~i)' 
 
 z 
 "(y-z) Uy-&z-i)' 
 
 4<^(.^+i)^(«^-t) 
 
 (9) 
 
 (lo) 
 
 i6t4-(-t3(-4o<^+i6)+t"(33</>^-28<^)-|-t(-io<^3+io<^^)-|- </)"(</>+ 1)»' 
 Substituting for t and writing <^-|- 1 = <t 
 
 ibao-""— 4oaj8(r«-)-56a/8o-7-)-33a;SV'— 94a;8^o-s (n) 
 
 -|-(T4(6ia/3='-ioa;8'+4/83) + '^'(40"/8^-4i33) 
 
 -|-a^(-50tt/33+a/8-l-4y8-l)-j-tr(2oa/33- 2a;84-f S/S^) 
 
 The factors a' (cr— i) are removed in the course of the work. 
 This equation will be denoted by 
 
 . F(o-, a, /3) = o 
 
 and referred to as the symmetrical internal equation. 
 
 We note that c is a symmetric function of a, b, c and that if o- be given, 4>, t are uniquely 
 
 determined, v, z being given by 
 
 -<A(</.-r+i) 
 
 y= 
 
 4t(<^— T— i) 
 
 4t(<^-t— i) 
 
 which is unique except for the singular points and lines of the transformation: 
 
 The lines y—z = o( ,. ^ ^, . ^ , 
 
 / ( correspondmg to the pqmt </>= =o t= oo 
 
 4y— 52 — I = o 1 
 
 the line y— 22 = corresponding to the point <^ = o t= — i 
 
 the line z = o corresponding to the point <^ = o t = o 
 
 and inversely 
 
 the line <^ = o corresponding to the point y = o 2 = 
 
 the line t = o corresponding to the point y= 1=0 z= 00 
 
 the line <^— t— 1 = corresponding to the point y= 22= =0. 
 
 If instead of choosing x= i as defining the scale we take 
 
 4xy— 8z— a:J=i and write 
 xy—2Z=K whence x' = 4K— I 
 
 we have by elimination of x and y 
 
 [2(4-c+i) + k]» 
 
 ^= 
 
 2^(8k-i) + 2(s-c-i) + k3' 
 
 [2(4K+i)+k]3 
 (4/C-1) (2+t)'z' 
 
 (9O 
 
 (12) 
 
THE PARAMETRIC FIELDS 
 
 A simpler birational transformation is now available 
 
 with reversions 
 
 4(cz (4*^- i)z ,s 
 
 ^ tf, 
 
 The expressions for a ft and the eliminant are the same as before, the same extraneities 
 occurring. 
 
 This transformation is simpler but has a practical inconvenience in the determination of 
 X from the cubic x^ = 4k—i. 
 
 The triangles for which x — Xi, wXi, oy'x,, are not distinct in ratio of sides. 
 
 We shall use this transformation to obtain certain information but shall not discuss it 
 completely. 
 
 The extraneous factors <^(<^+i)^ need notice. 
 
 <^ = o leads to a' =i 7=02 = 
 a '■ b '■ c=i : o : o 
 K   L '■ M= 1:0:0 
 
 which is not in general a solution. 
 
 <^= — I leads to 
 
 o : 6 : c= I : «> : <»', («>'= i) a complex triangle whose angle-bisectors (internal) all vanish and 
 which though a solution of the ratio problem by way of indeterminateness is not a solution of 
 the original problem. 
 
 4. THE PARAMETRIC FIELDS 
 
 Proceeding to the discussion of the equation 
 
 we note that it is a two-parameter equation of order 10 irreducible in the domain of rationality 
 constituted by a ^ and their rational functions. For u occurs only to the order i and any 
 reduction in the domain must be of the form 
 
 F=MiPa+Q) 
 
 where M, P, Q are rational functions of /3, a- alone. Hence the terms containing a and those 
 free from a have a common factor — which is not the case. 
 
 The problem can be discussed in three fields: 
 
 i) a, /? may be restricted to such values as arise from the assignment of real quantities 
 as the lengths of the angle-bisectors. This may be called the field of real angle-bisectors; 
 
 2) a, /3 may be any real quantities; 
 
 3) 1, y3 may be any algebraic quantities. 
 
 In connection with the first field we have a part of, and in connection with the second 
 field the whole of, a real surface which in virtue of the fact that a is single-valued may be con- 
 sidered as a geographical surface and is properly represented by a set of a contour curves cover- 
 ing the whole y3, <r plane. 
 
6 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 In connection with the third field we have a " variete" in four dimensions to use the phrase 
 of MM. Picard and Simart and in accordance with their theory since the equation is the result 
 of ehmination and of numerical genus zero and geometric genus zero, can be birationally trans- 
 formed into the totality of point pairs of two Neumann spheres. This is accomplished in this 
 case by means of the parameters <f>, t in terms of which o-, a, /3 are rationally expressed (lo) and 
 which are themselves rational in o-, a, j8, viz.: 
 
 <^ = <r-i , ■^=Q   
 
 In the field of real angle-bisectors, the quantities k, I, m are necessarily positive and o /8 are posi- 
 tive and restricted to a certain region of the real a /S plane. 
 
 ds) 
 
 In fact, k, I, m are the roots of the equation 
 
 ru'—qu^+pu—i = o , 
 and p, q, r are positive. 
 
 The discriminant expressed in terms of a, fi is 
 
 /?■(»— 4) + 20^(9— 2a)/3— 27a3 
 
 __ ^ 
 
 which must be positive for real positive roots. 
 
 The quartic curve along which the numerator vanishes will be denoted shortly by 
 
 Z)(a,;8) = 0. 
 
 It has a cusp of the first species at (o, o) where the tangent is /?= o, a cusp of the first species at 
 (3, 27) where the tangent is parallel to 2'ja=/3; asymptotes 0=4 and /8= — ^; a parabolic 
 asymptote /8=4a'; and a real finite inflexion without apparent significance in the problem. 
 
 i 
 
TWO SPECIAL CASES 7 
 
 In making a diagram for psychologic reasons connected with the single valued property 
 of a, we reverse the usual alphabetic order and take a vertical. The region inside the cusp in 
 the first quadrant is the field of real angle-bisectors. Inside the hyperbolic branch in the sec- 
 ond quadrant and between the /? axis and the curve in the fourth the discriminant is also posi- 
 tive, but as either a or ^ is negative the regions represent real squares of angle-bisectors, some 
 of the set being negative. The curve D{a., /3) = o in a proper sense and the axes in an improper 
 sense are loci of equal angle-bisectors (Fig. i). 
 
 S. TWO SPECIAL CASES 
 
 At this stage it is convenient to solve some special cases. 
 First the case of equal bisectors ; K=L = M a = 3 13— ly. 
 On writing <r=$s, F(cr, a, j8) = o becomes 
 
 165'°— i2o5'-|-5657-|-2975'— 28255— 1735^+34853— 1775^+385— 3 = 0, 
 
 which reduces to 
 
 (25-3) (5-1)3 (25^+35-1)3 = 0. 
 
 5= I gives the equilateral triangle. 
 
 5= I gives K=^ s= 00 (using the k, z chain) the sides being a : 6 : c :: i : 00 : 00 ; on 
 the scale a-\-b+c= 1 a   b   c-.^   m : —m-\-\ where m is an infinite quantity. 
 
 This is the limit of a triangle with very small base and very long sides when the vertex is 
 brought into line with the base. 
 
 Since c is negative the B and A bisectors are external. 
 
 This triangle is a constantly occurring triviality. 
 
 — S— 1-^17   T "i+l' 17 5+1-^1^7 
 
 5=—^^ — ' — -' gives a:b.c::—i: ^-^-^ — -   ^-'-^ — - 
 
 4 4 4. 
 
 an isosceles triangle with two external bisectors equal to one internal. The angle A is approxi- 
 mately 25° 20'. 
 
 5= ■^ ' — -' gives a   b   c::—i   - — - — - : ^ — - — ' 
 4 4 4 
 
 which are impossible in the sense that |6|+|c|< la]. 
 
 As the class will need frequent discussion we reserve the word "impossible" for this exact 
 meaning. 
 
 The case a = 4 ^= 54; the point in the (a, /3) plane being the intersection of the asymptote 
 of Z)(a, )8) = o with the curve. 
 
 K : L: M::2 : 2 : i 
 
 (7=0 is a root by inspection. Writing (t = 35 as before the remaining terms reduce to 
 
 (253+25^-75+2)^ (^—2) (5^— 5) = o. 
 5=2gives<|)=5, T=4, y=2z=oo. 
 
 This is the same infinite triangle as in the case a=3 /3=27, a different path of approach to the 
 limit being involved in the different ratio of bisectors. 
 Using the magnitude of o- as an index this root is (2). 
 
THE PROBLEM OF THE ANGLE-BISECTORS 
 
 ^ = <^= — I T=0 y=<X> 2=00 -=o 
 
 2 
 
 This is the limit oi a -.h : c::m : m : — 2m+i when m= oo . 
 The root is (7). 
 
 •5=1/5 gives a -.h : c::2 : \ \—\ : v'S— i the obtuse-angled triangle which occurs in 
 Euclid's construction of a regular decagon. The root is (i). 
 
 ^= — 15 a : b : c:: — 2 : j s-|-i : y s+i the acute-angled triangle of the same context. 
 The root is (8). 
 
 The cubic factor is irreducible and has three real roots which are to be counted twice each. 
 Approximations are 
 
 5=1. 2098 a : 6 : c : : 1 . 947 : . 584 
 5= .3259 1.247 : 045 
 
 ^= -2-5357 665 : .559 
 
 The triangle corresponding to (5) (6) is impossible. 
 
 -I -531 (3) (4) 
 
 - • 202 (S) (6) 
 
 - 233 (9) (10) 
 
 6. REALITY or THE SOLUTIONS 
 
 Returning to the general equation F{<t, a, y8) = o it is first to be remarked that as o- is a 
 symmetric function of a, b, c only real values of o- can lead to real triangles; also that real values 
 of 0-, provided that a (3 are within the field of real angle-bisectors, always lead to real triangles. 
 For real o- and real fi involve real <^ and t and so real y and 2. Hence, a, b, c, are either real or 
 one is real and the other two are complex conjugates. But if a = b, K = L and D(a, fi) vanishes. 
 Now the region of the a, /3 plane where K, L, M are real is finitely separated from the regions 
 where the squares of these quantities are real and in part negative, the intervening regions 
 having some of the squares complex. It must be inferred then that real sides and real bisectors 
 are only found together in the single region inside the cusp of D{a 13) = 0, this curve being dis- 
 criminantal for both sets of quantities ; and that a real o- inside this region means a real triangle. 
 
 It is proper to note however that this argument leaves open the possibilities that (i) other 
 triangles besides the isosceles class occur on the locus D = o; (2) complex or's and complex tri- 
 angles may occur inside the cusp; (3) real sides may occur outside this region, namely with 
 negative squares of bisectors; (4) real triangles inside the cusp may be impossible triangles. 
 
 Of these propositions (i) (3) (4) are affirmed and (2) is denied by the subsequent develop- 
 ments. 
 
 7. THE ELEMENTARY THEORY OF EQUATIONS APPLIED 
 
 The next undertaking is to use as far as possible the elementary theory of equations in 
 separating the roots, determining their reality, and classifying the triangles connected with 
 them. 
 
 On account of the labor involved in some of the operations (such as determining the set 
 of Sturm's functions) being prohibitive, a complete discussion cannot be had by this method, 
 but the results as far as they go are valuable in themselves and as a corroboration of the sub- 
 sequent treatment of the problem. 
 
 We consider first the functions 
 
 ,,, = o-3-j8o-K (8 (16) 
 
 j7j = o-3— y8o--|-2j8 
 
 / 
 
 J 
 
ELEMENTARY THEORY OF EQUATIONS APPLIED 
 
 Using the k, z chain 
 
 x^= 
 
 a 
 
 z = 
 
 4Vt 
 
 (17) 
 
 Assuming a y3 to lie within the real angle-bisector field 
 
 a^i ^>27; D(a,l3)>0 
 
 a change of sign of 17,, rj^ involves a change of sign of x and z, and so by means of 17,, rj^ the 
 roots are classified with respect to their connected triangles. 
 
 17, = and >;2 = o have each three real roots, one negative, for 18—27. 
 
 Considered as curves in the a-, -q plane the families for varying P have common properties 
 and one case may be taken as a type. The graph shows 1?,, 172 for /?= 54 and also F{<r, 4, 54) 
 (Fig. 2). 
 
 F 
 
 By reducing F(a-a ^) to the second order by means of »;, = o and 172 = o respectively it appears 
 that for the roots of the 77's, for <r = o, and <t= i, provided that a^^ /8— 27, F has the following 
 signs : 
 
 A B o cr=i C D E F 
 
 + + ± + + + - + 
 
 The sign at o is + if a >4 and — if a< 4. 
 
 F=o has then at least one real positive root between E and F. 
 
 Since here x,z, and hence y i=[x'+&z+i]4x), are positive the problem has a real solution 
 to be interpreted as referring to three internal angle-bisectors. 
 
 The signs of x,z in the intervals — =0 to yl, .4 to 5, etc., are 
 
 A B o <T=i C D E F 
 
 X 
 
 z 
 
 + 
 
 + 
 
 + 
 
 + 
 + 
 
 + 
 
 + 
 + 
 
 + 
 
 The intervals CD and EF are the only ones in which such an internal solution can occur. 
 No conclusion can be drawn as to CD for Fia-) is positive at both C and D. 
 
 The greater root of V2 is a superior limit for roots of F. A superior limit not so close but 
 more convenient is furnished by ]/ /3. 
 
lO 
 
 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 We next proceed to transform the equation so as to find the number of real roots for inter- 
 vals in which 
 
 (i) iji r?2 are both positive, 
 
 (2) i?i 1/2 are negative and positive respectively, 
 
 (3) Vi '72 are both negative. 
 
 For arithmetic convenience we begin by eliminating o- from F{(t a /3) = o and o-3 = /8(<t— X). 
 F is first reduced to a quadratic; that is a function M rational and integral in o-, a, /8 is deter- 
 mined such that 
 
 F{<J a P)-M[cTi-P{a-X)\ = N 
 
 where N is of the second order in o-. 
 
 This is conveniently done in a tentative fashion and no extraneities are introduced, for 
 the equivalent end term process is 
 
 I F{<r a 13) o 
 
 -M o-3-/3(o— A) I 
 
 and has the determinant of multipliers i. 
 
 Thence the result is obtained by substituting in the general eliminant of a cubic and quad- 
 ratic form. To use the end term process again would introduce an extraneous factor. 
 
 The arithmetic is simplified by using ir = A— i instead of \. 
 
 The result arranged as a polynomial in v is 
 
 
 
 ir~ 
 
 TT* 
 
 ir< 
 
 irT 
 
 n* 
 
 ITS 
 
 jr* 
 
 1,1 
 
 IT" 
 
 IT 
 
 z 
 
 a (33. . 
 
 .... 1 
 
 
 
 
 
 -16 
 
 
 
 
 
 
 
 
 
 P'.. 
 
 
 
 
 
 
 
 
 64 
 
 
 
 
 
 
 
 
 
 a3j3». . 
 
 
 
 
 
 
 81 
 
 -36 
 
 22 
 
 -4 
 
 I 
 
 
 
 
 
 a'p'. . 
 
 
 
 
 
 
 -480 
 
 260 
 
 -76 
 
 12 
 
 -4 
 
 
 
 
 
 affi.. 
 
 
 
 
 
 
 768 
 
 -512 
 
 -256 
 
 -16 
 
 
 
 
 
 
 
 Ifi.. 
 
 
 
 
 
 
 
 
 
 64 
 
 
 
 
 
 
 
 03/S .. 
 
 
 
 
 -1152 . 
 
 800 
 
 968 
 
 -354 
 
 228 
 
 
 
 4 
 
 2 
 
 
 
 a^/S .. 
 
 
 
 
 3072 
 
 -2560 
 
 -2816 
 
 1760 
 
 388 
 
 — 140 
 
 12 
 
 -4 
 
 
 
 oA . . 
 
 i 
 
 1.096 — 2( 
 
 h8 
 
 -5376 
 
 3328 
 
 1648 
 
 — 1224 
 
 -147 
 
 168 
 
 -6 
 
 -8 
 
 I 
 
 (18^ 
 
 For the investigation of region (2) where the values of A range from i to 2 we write 
 
 _i — TT 2 — A 
 ''^"T^^A^ • 
 
 The equation in p will then have positive roots for roots of F(<r) in the region (2) that is 
 between the curves r),=o and 172 = 0. 
 The equation in p is: 
 
 
 p.. 
 
 P' 
 
 pf 
 
 p' 
 
 p' 
 
 p' 
 
 f 
 
 f 
 
 p" 
 
 P 
 
 I 
 
 a/33... 
 
 
 
 
 
 -16 
 
 — 96 
 
 — 240 
 
 -320 
 
 — 240 
 
 -96 
 
 —16 
 
 /33 
 
 
 
 
 
 
 
 
 64 
 
 384 
 
 960 
 
 1280 
 
 960 
 
 384 
 
 64 
 
 a3ft' 
 
 
 
 
 
 
 I 
 
 4 
 
 22 
 
 68 
 
 161 
 
 320 
 
 400 
 
 256 
 
 64 
 
 a'/P 
 
 
 
 
 
 
 -4 
 
 — 20 
 
 — 104 
 
 -168 
 
 -180 
 
 -644 
 
 -1280 
 
 — 1024 
 
 — 288 
 
 alP 
 
 
 
 
 
 
 
 -16 
 
 -368 
 
 -2384 
 
 — 6192 
 
 -7728 
 
 -4688 
 
 -1136 
 
 —16 
 
 ti' 
 
 
 
 
 
 
 
 64 
 
 448 
 
 1344 
 
 2240 
 
 2240 
 
 1344 
 
 448 
 
 64 
 
 a>ti 
 
 
 
 
 
 2 
 
 22 
 
 104 
 
 508 
 
 1490 
 
 3150 
 
 6084 
 
 7120 
 
 3616 
 
 496 
 
 a'fi 
 
 
 
 
 
 -4 
 
 -24 
 
 -188 
 
 -592 
 
 1316 
 
 7240 
 
 6972 
 
 -832 
 
 -2592 
 
 —288 
 
 ai 
 
 
 • 
 
 I 
 
 
 2 
 
 -33 
 
 -48 
 
 399 
 
 330 
 
 — 2015 
 
 -268 
 
 3168 
 
 — 2160 
 
 432 
 
 (19) 
 
ELEMENTARY THEORY OF EQUATIONS APPLIED II 
 
 The transformation is of course effected in the coeflScients of the powers and products of 
 a, /3, separately. 
 
 Proceeding with the determination of signs of the coefficients: 
 
 [p'"] a3 is positive in the region, 
 
 [/o'] 2a»(a+o/3— 2/3) is positive, 
 
 [p'] and [p'l vary and are reserved for later discussion. 
 
 [p'] This coefficient can be arranged as 
 
 -i6(|8+i)I>+a3(-/8»+i2y8-33)+/8^(-4ia3+i84a'-3S2«+384)-304«'^ 
 
 where D is written for D{a /3) which is positive in the region. 
 
 Since a^2> and P^2j this is negative throughout the region. 
 [p5] can be arranged as 
 
 -ga/SD-zS'CsiSaJ- 15600^+23840- i344)-/3(iio2a3-i3i6a^) + a3(-/3»+ 330) 
 
 which is negative. 
 
 Mis 
 
 — 240/8Z)— ^(79903— 41400'+ 6192a— 2240) — y8(333oa3— 72400^) — 201503 
 
 which is negative. 
 
 [p3] is 
 
 — 320i8Z)— 18^(96003- 5ii6a»+7728o— 2240) — 18(255603— 6972a')— 26803 
 
 which is negative. 
 
 [p'] is 
 
 -24o()8+i)D-i28(a-3);8'- (4320)8-34880)0^-32003/3-331203 
 
 which is negative. 
 [p]is 
 
 (-96/3+8o)£»+64(a-3)/3[/3(-o'+Ioa-32) + a'(-j8+2l)] 
 
 which is negative. 
 
 [p»] is 
 
 -i6(/8+i)£> 
 which is negative. 
 
 Returning to the eighth and seventh powers, 
 
 [p7]-4[p8] = 4[-4Z)+4o'/3(-a+3)+4o'y8'(-^+io)-87a3] 
 
 which is negative. 
 
 Hence if [p*] is negative [p'] is also. 
 
 Of the eleven signs either the first two, three, or four are positive and all others negative 
 throughout the region and F{p) = o has one and only one positive root. 
 
 From the last result it follows that F{cr, o, /3) = o has one root and only one root in the 
 intervals A • • • B,C • • • D, E • • • F and since there is one root in £ • • F which leads to a tri- 
 angle with positive sides and therefore internal bisectors in the given ratios and since all such roots 
 fall in these intervals there is one and only one solution of the original geometrical problem for every 
 set of real quantities assigned as the lengths of the internal angle-bisectors. Since this root is greater 
 than 3 and less than | ''/S or less than the greatest root of 0-3- j8<r+2/3=o and no other root 
 occurs in the interval there can be no trouble in separating the root. The sides are to be 
 determined from <t by rational operations and solution of the cubic (P— i'+^y— 2 = 0). 
 
12 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 In a certain narrow sense this is a solution of the problem, but it is necessary to inquire 
 more closely as to the character of the irrationality involved and as to the necessarily intro- 
 duced extraneities and their connection. 
 
 Proceeding with the method in hand we write >l/=k— 2. 
 
 The equation in 1/' will have positive roots corresponding to roots of F=o in the intervals 
 
 — 00 to j4 and D to E. As however the least root of 772 = o is an inferior limit the interval 
 D • - Eis the only one in question. 
 
 In this case the labor required is not justified by the results to be expected, and it may 
 merely be noted that the coefficients of i/*'" and i/'' are positive while the constant term 
 is -i6(,8+i)ZJ. 
 
 From which at least one root results. 
 
 By writing 0= — we have positive ^'s associated with roots of F=o in the intervals 
 
 TT 
 
 B • • C and F • • • -\-<x but as the greatest root of 1;, = o is an upper limit the interval B • • C is 
 alone possible. 
 
 Here taking the coefficients from the polynomial in tt with the proper changes in order 
 and sign • — 
 
 [6^0] is ai and is positive, 
 
 [0*] is 8a3-|-4a^;8— 2a3;8 and is always negative, 
 
 This varies in sign, being negative at (3, 27) and positive at (4, 54). 
 [6'!] is 4a3fi'-^a'p^+4D+{i6a-^y)a^/3+66ai] and is positive. 
 
 [^6] is - l6|8D-f-,5^(-42a3-f2I2a'- 2S6a)-|-a»;8(- 294a-f 388)- I47a3. 
 
 The coefficient of /8" has the value 6 for 0=3 but rapidly diminishes, the derivative being 
 
 — 118, and has negative values for «— 3.05 . . . , while /3 is very near 27 if this coefficient is 
 positive. Hence the whole expression is easily seen to be negative. 
 
 [0^] varies in the region. 
 
 [6^] is (8ia^-48oa-|-768)a|8^-(- (968a- 28i6)a»j8+ 164803 and positive. 
 
 [^3] is 32a»(- 2Sa;8+8o/8- 104a). 
 
 The coefficient of /8 is negative for a >3 . 2 when /8< 40. 
 Hence the whole expression is less than — 1 1 2 (3 2a') . 
 
 [6^] is 384a2(— 3a/3-f 8j8— 14a) and negative. 
 
 The remaining coefficients are positive. 
 The signs then run in the interval 
 
 + -± + -± + + + 
 
 The variations of [0^] and [0^] have then no effect and there may be always six roots in the 
 interval. 
 
 With these results the practical utility of the classical theory seems to end. The direct 
 evaluation of the discriminant in Bezout's form is impracticable, the elements of the ninth order 
 determinant involving a and /3 in polynomials of the fourth order with coefficients of six figures. 
 Still less ie it to be expected that a successful attack could be made on Sturm's functions. 
 
REALITY OF THE ROOTS FOR REAL ANGLE-BISECTORS 
 
 13 
 
 Recourse must then be had to special methods applicable to this problem and to the power- 
 ful general method of following the transformations used in the elimination and interpreting 
 them. 
 
 8. REALITY OF THE ROOTS FOR REAL ANGLE-BISECTORS 
 
 If the bisectors are real k, I, m are positive. Write t={a-\-b-{-c)abc and 3'i = y,, y' — yt 
 
 yj = — , the internal formulas become 
 tnt 
 
 1 — 2* 
 
 y=— 7 r- with a;=a for Ti; x=b ior y^: x=cioTyi 
 
 x{i — x)' 
 
 (20) 
 
 The curve represented by this equation has the line x=i for an asymptote with a cusp at 
 infinity. The x axis is also an asymptote with an inflexion at infinity, the curve approaching 
 with negative y. The y axis is an ordinary asymptote. The curve meets the x axis also at 
 «=HFig-3)- 
 
 
 
 \ ^ 
 
 ^^^c. 
 
 W : CX'^ 
 
 \ 
 
 1b. ; /b. 
 
 A>\ 
 
 Iax ': ^3 
 
 "^ 
 
 '■■ Fig 3 
 
 Consider first t positive and therefore y positive. 
 
 For any I the lines ^i = t". ; y' — Tt' >'3 ~ ~i ^^^^ '-^^ ^^^ curve in one real point. The sum 
 
 rSt It fHt 
 
 of the abscissas is | when t= °o and is o when t = o. Since the curve monotonously approaches 
 the y axis as t decreases the value i is reached once and only once by a-\-b-\-c; hence there 
 is always one and only one solution for a positive /. 
 
 The original problem restricted to internal bisectors has a unique solution, for it is evi- 
 dent that a b c and t must all be positive for this interpretation. As to the possibility of the 
 values a 6 c we note that since the perimeter is i and each side is less than J any two must be 
 together greater than the third. 
 
14 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 Considering negative t's each of the lines yi = T,! ^tc, cuts the curve in three real points. 
 Assuming that k<l<m we call the abscissas 
 
 ai<a2<a^; 6i<62<6,; c,<C2<Cj. 
 
 The product of 27 factors 
 
 P[i-iai+bj+Ck)] i,j,k=i,2,s 
 
 is symmetric in the a's, b's, and c's separately and can be expressed in terms of k, I, m, t. 
 
 This product vanishes for solutions and only for solutions. 
 
 From the other eliminations we know that there are in general ten finite values of / satis- 
 fying P=o. We have seen that one and only one is positive. 
 
 For negative t's we consider the 27 combinations ai-\-bj-\-Ck denoting them shortly by 
 {i,j, k) understanding that the indices in the order written refer to a,b, c. 
 
 Some of the 27 are excluded as being continuously greater or less than i. These are 
 
 (i, I, i)<o 
 
 (2, 2,.2)>f 
 
 (2, 2,3), (2,3, 2), (3, 2, 2)>2 
 
 (2,3,3)> (3, 2)3), {2„2>, 2)>f 
 
 (3, 3,3)>3 
 
 (i, I, 2), (i, 2, i), (2, I, i)<i 
 
 There remain 15 arrangements which may possibly yield solutions. 
 Taking the set (i, 2, 2) aj-t-6j-|-C2=2 for t = o, y= =0 
 
 = — 00 for /= 00, 3^ = 0. 
 
 Hence at least one real root, for the abscissas all decrease monotonously with increasing /, 
 and are finite and continuous in the interval o>/> — 00 . 
 
 Similarly (2, i, 2) and (2, 2, i) each yield at least one real root. 
 Of the six permutations of (123) three can be excluded, namely: 
 
 (i, 2, 3) 
 (i, 3, 2) 
 (2, 1,3) 
 
 (a, -l-62-f c,) > (c, -I-C2-I-C3) > 2 
 {a,^b,+c,)>{]},^-b,+b,) + {c,-b,)>2-\>i 
 (a2+6i+C3)>(c,-|-C2-fc3)>2. 
 
 For (2, 3, i) a2-\-bi-\-Ci = 2 for l = o; for /=—<», y= — o,-the equation 
 
 X^ — 20<^-\-x{l-]r2kt) — kt = O 
 
 has one root J and the others approach infinity with ± 1 — 2kt. 
 
 As k<l<m, a2+bi-\-Ci approaches negative infinity with i+y' —2U — V —2int. 
 
 Hence (231) yields at least one real root. 
 
 In a similar manner it can be shown that {312) and (321) each give at least one real root. 
 
 In the set of permutations of (113) we have in the case (131) ai-\-b^-\-Cx = i for / = o and is 
 negatively infinite when t is. 
 
REALITY OF THE ROOTS FOR REAL ANGLE-BISECTORS 1 5 
 
 To determine whether it ever exceeds i in the range we express the roots of 
 
 3;^ — 2.r'-|-(i-|-2^/).T— ^^ = (21) 
 
 in terms of kt near x = o,t~o and x=i, t = o. 
 
 x,=ki-k^t?-2kH* .... (22) 
 
 ^2 = 1— 1 —kt-\ h . . . 
 
 2 
 
 a;, = 1+1 —kl-\ h . . . 
 
 2 
 
 a,-\-bj-\-Ci=kt-\-i-\-\ —U+mt+ . . . 
 
 Since in the limit as / approaches — o the square root term exceeds in absolute value the 
 sum of the terms of the first order, (131) is greater than one in the region of /= — o, and so gives 
 at least one real root. Similarly (311) gives a real root at least. 
 
 The treatment is not sufficient for (113) as this may not approach minus infinity with /. 
 Reserving this case and taking the i, 3, 3 set 
 
 (133) : a,+b,+Ci>c,+b3+c^>c,+C:,+Ci>2 
 (313) : ai+b, + c,>b,+b,+b3>2 
 
 (331) requires a more detailed examination and is to be considered in connection 
 with (113). 
 
 (113) a,+&,+C3 = I when / = o 
 (331) 03+^3+^1 = 2 when / = o. 
 
 When t approaches minus infinity (113) approaches ^ —2t[ — ]/k — }^ 1+] m] while (331) 
 approaches the same quantity with reversed sign. 
 
 So one and only one of the two combinations gives at least one real root in the interval. 
 
 In terms of the fundamental parameters 1 ^+1 l>ym reads a<4 and in this case (113) 
 has the root, but if a > 4 (33 1 ) has it. 
 
 The boundary case = 4 needs special mention as in this event the first approximation 
 is indeterminate. 
 
 Consider the limit ( — 1/^ — 1 /+]/w)(l — 2/) 
 
 t= — 00 
 
 where k^ is such a number that \/ki-\-\ 'l—\/m = o. 
 We have 
 
 Limit = limit {-y'kV+x-Vl+\/M)y2 
 
 x=o yy 
 
 y = o 
 -J^ limit ^. 
 
 If then the path of approach be along the parabola 2x^ = k,y the limit is i. So we may consider 
 that (113) has a root at the limit. Or by approaching on the other side (331) has a root here. 
 
l6 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 For a = 4 the original equation has a root cr = o which leads by the chain of equations to 
 just such an infinite triangle as is here in question. 
 
 In every case then there are at least ten real roots for finite values of t, and as we know that 
 there are only ten such roots there must be one in each of the classes specified. 
 
 The arrangement of the bisectors in the triangle should be noted. For the internal case 
 the magnitudes of the bisectors and the sides which they meet are in reversed order. 
 
 For the case (231) the internal bisector is M since c^ is negative. 
 
 The least bisector is internal but no further statement can be made as to absolute magni- 
 tudes, for if / is large c, may in absolute value exceed 6,. 
 
 For (312) the medium bisector is internal, for (321) the least. So (122) has the greatest 
 internal, (212) the medium, and (221) the least. 
 
 The set i, i, 3 are all impossible triangles (see § 11) and the words internal, external are 
 without a proper meaning, but we may say, in (131) the medium bisector is associated with the 
 side that is unique in sign {b). 
 
 In (311) the greatest and in (113) the least is so associated. If a>4 (331) gives a possible 
 triangle with the least bisector internal. The want of symmetry of the arrangement by which 
 the least bisector is "internal" in five analytic cases out of nine and in four real cases out of 
 eight (or in three out of seven) is to be noticed. 
 
 9. A PRACTICAL METHOD FOR APPROXIMATE SOLUTIONS 
 
 Following the plan of the last section we have the three equations (21) with k, I, m given 
 and a particular set of three roots, say a^, b^, d whose sum is to be unity. For the sets (3, 2, i) 
 (2, 3, i) (3, 1, 2) (i, 2, 2) (2, 1, 2) (2, 2, i) it is easy to show that the sum decreases monoto- 
 nously as t increases. For the other negative sets this is not so, but the solution occurs on such 
 a slope which is unique. By trial we may find values of / between which the solution lies. 
 
 After the first operation the convergence is satisfactory, the curve ' sum = F{t)' being flat 
 in character. The result carries with it the class of triangle. 
 
 Conceivably this process could be automatically carried out by a machine. 
 
 Parallel bars set at distances proportional to k, I, m and kept parallel by a parallelogram 
 linkage, intersect fixed curves patterned to equation (20). A stretched string with its length 
 greater than the sum in question by a constant would reach a mark as the solution is 
 attained. 
 
 It is to be noted that this method avoids the elimination. 
 
 10. MULTIPLE POINTS 
 
 Considering the fundamental equations in the form: 
 
 Fa=a^ — 2a''-{-{i-{-2kt)a—kt = o , , 
 
 Fb = o, Fc = o, a+b+c=i 
 
 li a,b , c, k, I, m, t are a consistent set of values, for neighboring values we have: 
 
 aF. , aF„, , 9F„, , . 
 
MULTIPLE POINTS 1 7 
 
 and two similar equations. If the point is an ordinary point at which moreover none of the 
 
 dp dp dp 
 partial derivatives 'a" gi g- vanish, we have three equations of the form: 
 
 Sa = ASt+BBk ) 
 
 U = CU+DU I (24) 
 
 hc=EU-\-P^m \ 
 
 First suppose A-\-C-\-E^o and solve 
 
 o = ^a+hb+hc={A+C+E)U+Bhk+DU-\-Fhm. 
 
 This gives one value of 8/ and so one solution in the neighborhood {^k, U, &m) of {k, I, m), that 
 is of (a, ;8). 
 
 Secondly if 
 
 4a^ — 3a+i ^ ^' 
 
 there is a double point in the neighborhood. 
 
 Setting out the complete increment equations after elimination of k, I, m: 
 ^^^_ (, )(, ), ^^_^ ( ).,. 
 
 4a^— 3a+i (a— i)(4a^— 30+1) 
 
 .(26) 
 
 (fl-i)(4a»-3a+i) /(4a^-3a+i) 
 
 (a-i)(4a'-3aH-i) (a-i)(4a^-3a+i) 
 
 6(2g-i) ^ , 
 
 (o-i)(4a^-3a+i). 
 and two similar. 
 
 When 2 ; = o we have by addition an equation of the form : 
 
 40^-30+ 1 •' ^ 
 
 BZk+DU-irP^m = o 
 in which unless t = o or a, b,c have certain special values B, D, F are finite both ways. 
 If o, /8 and -7-^ are assigned the ratios k-\-^k : l-{-U : m+Bm are given. 
 
 Let 
 
 k+Sk l+Sl m+8m 
 
 "1^ = 1^ = -^^ = ^ (^7) 
 
 Then 
 
 ^   Bk+Dl+Fm „, „ , , , 
 
 ^^Bk'+W+Fm'^^^i'^ -^^ ^^^- (^«> 
 
 For an (a P) arbitrarily near a point for which ^+C+£=o the terms of the second order 
 on substitution of these values for 8^ U Sm and the values given by the first approximations 
 for Sa Sb Sc give a quadratic for 8t. 
 
 Such points then are double points and the locus a discriminantal locus. 
 
 Naming it 
 
 4a"— 3a+i ^^' 
 
1 8 THE PROBLEM OE THE ANGLE-BISECTORS 
 
 we recognize that T must be a factor in the discriminant of F{a,k; l,m)=o and also of 
 ^("■j ") ^) =o, F{a, k; l,m) = o being the equation for the side a in terms of k, /, w (§ 15). 
 
 A triple-point locus, that is a locus whose intersections with T = o are triple points, can be 
 obtained by forming the second derivatives of a, b, c with respect to t and expressing the condi- 
 tion that their sum vanishes. 
 In terms of a, b, c this is 
 
 ^ _^ {a-i){2a-i){^a'-ta+2,)ai 
 
 ^-^ (4a-3a-Hi)3 =° ^^o) 
 
 To express this in the y, z plane, however, requires a laborious elimination, and, owing to the 
 practical necessity of multiplying up the denominators the result (which aside from an infinite 
 value of 2 is of the fourteenth order in z) is not free from extraneities, in fact for the point 
 
 both T and 5 vanish, but the point in fact is only a double point. The finite triple points will 
 be determined (§ 12) by another method. 
 
 Multiple points however can occur when T does not vanish, li k = l, a^b 
 
 and in this formula interchange of (8^, 8/) in general changes this value but leaves a-^^, 
 /?-|-8/3, which are symmetric functions of ^+8^, l-\-U, m+8tn unchanged. Hence 8/ has two 
 values and the locus is discriminantal for F{a-, a, /?) =0 but not for F(a, k; l,m)=o. 
 
 The finite number of points where ^ = /, a^b, and £ = Dare given by 4c3-f-4c'-|-c— i =0 and 
 form an apparent exception but as they are discrete points on a continuous locus there are still 
 two values of 8; for points as close to them as we please. 
 
 If ^ = /, a = b, then B = D and U has only one value. 
 
 Naming the locus k = l, a^b zs, D:,=o, we see that for values of a, j3 satisfying the (a, /3) 
 equation of the locus the double root recurs three times. 
 
 For the first two equations of the system (2 1) are identical and if a^jU^, a, are their roots and 
 fli, flj, c are the sides of the triangle o,-(-a2-|-c=i and ai+a2+a3 = 2; whence a^=c-{-i and if 
 we write ^ = i 
 
 I — 2C (c-f-l)c" ^•^ ^ 
 
 Setting aside c = o as not finite, we have a cubic in c 
 
 2c^{m—i)-\-c^{m+T,) —mc-\- 1=0. 
 
 For every m, that is for every a, /3 on the locus, there are three values of c and so three double 
 points. The discriminant of the cubic is 
 
 m{()m^-\-T,?>nC-]-()m-\-2i6) (33) 
 
 If m vanishes the double value of c is i and the corresponding triangle has sides in the ratios 
 
 I _ I 
 
 '   1/3 ' 1/3 ' 
 
MULTIPLE POINTS 1 9 
 
 This is an eight-point but 13 is infinite and so it is not to be listed in the finite multiple points. 
 For the cubic factor, which is irreducible, there are three four-points, one with a value of m 
 near — 5 and two complex values. These are seen later to correspond to intersections of T = o 
 andZ>j = o. 
 
 For the locus k = l, a — b, the locus of isosceles triangles, to be named Z), =0, proceeding as 
 above, 
 
 m = Y nV-t- \77 N (34) 
 
 (2c-i)(c+i)'(c-i) 
 
 If c= I, then m = %, t = o, a, /8 are indeterminate. The triangle has sides as i : o : o. 
 There remain three values of c for every m leading to isosceles triangles given by 
 
 2im-4.)ci+{sm+8)c'-m = o (35) 
 
 The discriminant is 
 
 m(2'jm'+gm+T,2) (36) 
 
 For m = o the double value is c = o, not finite while the complex values of m lead to two complex 
 triangles in which a part of the isosceles solutions coincide. 
 These are 
 
 ^ . h. .   I9£i_r'5 . 19T1 -15 . 28±i - 15 
 
 a : : c : : : : . 
 
 94 94 47 
 
 The points are only double points in the general problem. 
 
 Next are to be considered the points which are singular in the individual equations of 
 
 the system. 
 
 dp 
 If ^ = o the equations take the form 
 
 8^. =C8t+DU (37) 
 
 Be =E8t-\-FSm I 
 
 which give 
 
 {C+Eyk'+[2{C+E){DBl+F8m)-A']8t+[{DBl+F&my-BU] = o (38) 
 
 Since this quadratic has one root which approaches zero with Bk, Bl, Bm as a rule only an 
 ordinary point exists for the problem. If A ' vanishes, which happens for a = i , one of the values 
 in question, the point is a double point at least, but as yS is infinite not among the finite set. 
 
 If / = w, 6=t=c there is a double point as before on the locus i)j = 0. 
 
 If ^Fa/^a and ^Fb/^b both vanish and a = b,Bt is given by a quartic, three of whose roots 
 approach zero with 8^, 8/, Bm. This gives triple points for the triangles 
 
 n   h- r-- 3+V -7 . 3 + 1 -7 . i-V -7 
 a.o.c. 8 • 8 • 4 
 
 and the conjugate values which are on r = o and a : 6 : c : : i : i : — i for which a = oo ,^=00 . 
 If both the i)artial derivatives vanish and a=t=6, 8< is given by a quartic two of whose roots 
 approach zero. 
 
20 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 This applies to the triangles 
 
 a:b:c:^-^±\^:^-^^::^:i 
 
 : : I 
 
 : : I 
 
 8 
 
 ( 3+r -7) . (-5-1 -7) 
 8   8 
 
   i3-v-y) . (-5.+V -7) 
 
 which are double points merely, though they formally satisfy T = o and 5 = 0. 
 
 Three of the partial derivatives cannot vanish together as a-\-b-{-c=i and the only 
 
 ('?±l/ — 7) 
 permissible values are i, 3 — — . 
 
 If a partial derivative is infinite a side has the value f from which k is infinite and /8. 
 
 Finally none of the special values give finite multiple points not on one or other of the dis- 
 criminantal loci T = o, D2 = o. 
 
 The nature of this investigation naturally leaves a doubt as to its sufficiency and may 
 be regarded as a mere reconnaissance which serves the purpose of gathering material to lighten 
 the labor of a conclusive determination subsequently undertaken (§ 12). 
 
 II. THE TRANSFORMATIONS 
 
 Beginning with the configuration in the (a, b, c) plane, take as reference triangle an equi- 
 lateral triangle and as the homogeneity relation a-\-b-{-c= i. Every point in the plane repre- 
 sents an analytic triangle with unit perimeter. Each such triangle has a sixfold representation 
 corresponding to the permutations of a, b, c, the six points having a sixfold central symmetry. 
 
 The lines —a-\-b-{-c = o, a — b-\-c = o, a-\-b—c=o form a proper triangle dividing the plane 
 into compartments. Inside, all the triangles represented have positive sides and are possible 
 triangles. In the regions outside one of the lines the represented triangles are impossible. 
 Outside two of the lines the triangles can be constructed, one side being considered negative, 
 and the bisectors meeting this side being external. The sixth part of the plane shown in Fig. 5 
 has the regions ©, o, 00, 000, 5, 6, 7, as subdivisions of the region of impossible triangles with 
 real sides while the regions 1,2,3, 4, 7', 8, 9, 10, include all the real possible triangles, the region 
 I alone representing the real possible triangles with positive sides, that is triangles in the 
 ordinary sense. 
 
 In the regions 5, 6, 7, the angle-bisectors are real though the triangles are impossible, while 
 in ® , o, 00, 000, the bisectors are pure imaginary quantities. 
 
 Considering in detail the discriminantal curves, we have first 
 
 ^^^ a(i-a)(i-2a) ^^ 
 
 4a^-3a+i ^' 
 
 4y'—2oyz-\-40Z^—y-\-z = o (39) 
 
 In the y, z plane the form is 
 
 As this is an ellipse every real (y, 2) is finite and so every real (a, b, c) is finite and the curve 
 is closed in the {a, b, c) plane. " It touches the sides of the reference triangle at the mid-points 
 and passes through the vertices perpendicular to the medians. 
 
 From the summation form it has no point such that a, b, c are all positive and less than J. 
 
THE TRANSFORMATIONS 
 
 21 
 
22 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 By writing c=i — a—b, a-\-b=2i, a—b=2r] a sextic is obtained which contains i) only 
 as rf, r}* from which any number of points are easily obtained and multiplied by the sixfold 
 symmetry. 
 
 The curve is a trefoil not entering the reference triangle and crossing the compartments 
 o and oo in the fundamental region (Figs. 4, $). 
 
 The set of lines 'w{a—b) = o while not discriminantal are so transformed as to lie on D{a, p) 
 which is discriminantal. 
 
 This locus of isosceles triangles has been named Di. 
 
 The locus of equal bisectors for scalene triangles k = l, a^b has for its equation 
 
 2ab{a-\-b)- {d'+Sab-\-b'')-\-2{a-\-b)- 1 = (40) 
 
 and is to be taken with two other curves obtained by cyclic interchange to constitute the com- 
 plete locus 
 
 A(a, 6, c) = o (41) 
 
 Each constituent has three real asymptotes 
 
 a=h, b=i, c = \ 
 
 with cusps at the infinite point, and no finite singularity. Two branches touch the sides of 
 the reference triangle at the vertices, and externally. 
 
 In the fundamental region (Fig. 5) the various branches separate the regions 5 and 6, 
 10 and 9, 8 and 9, 4 and 3, 3 and 2, 6 and 7. 
 
 Other discriminantal lines are "n"(a-)-6) = o, which leads to j8= co and 'w{a-\-b—c) = o giving 
 )8=o. The loci corresponding to zero and infinite values of a are too compHcated for their 
 utility at this stage and are added later. 
 
 In the fundamental region a-\-b = o separates 00 and 000; 4 and 9; 3 and 8; 2 and 7'. 
 Thelinea-|-6—c=o separates I and ®; 10 and o; 9 and 00; 4 and 000. 
 
 To the (a, b, c) plane and this collection of curves, following the process of the elimination, 
 is now applied the "elementary symmetric function transformation," 
 
 a-\-b-\-c =x, 
 
 ab+bc+ca=y, I (3) 
 
 abc =z. 
 
 Passing to a rectangular system by writing x—i and transforming the various curves the con- 
 figuration in the y, z plane can be set out. The transformation is point for point between the 
 fundamental region in the (a, b, c) plane and the region within the discriminantal curve in the 
 (y, z) plane. This curve is 
 
 D,= (a-b){b-c)ic-a) \ 
 
 in {a, b, c) and / (42) 
 
 A = 4y'—3'^—i8yz-f 272^+42 = ) 
 
 in the (y, s) plane. 
 
 The region of the (y, 2) plane without this curve represents complex points in (a, b, c) the 
 ratios of the "sides" being of the form 
 
 a : b : c : : i : p+q]/—i : p—q\/—i. 
 
THE TRANSFORMATIONS 
 
 23 
 
24 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 Six such points corresponding to the six permutations of a, b, c are represented by each 
 point in (y, z). 
 
 We may say shortly that the transformation brings up from the complex regions those 
 triangles whose sides have ratios which while not real are the roots of a cubic equation with 
 real coefficients (Fig. 6). 
 
 The curve Di{y, s) = o is a semi-cubical parabola and can be written as 
 
 4(33'-0^+(9>'-272-2)^ = o (43) 
 
 The cusp (y=h^ = 'ii) corresponds to the point A in the {a,h,c) plane {a=h = c=^). 
 
 The tangent at the cusp is y' = 32'. The curve cuts the y axis 2^tF {y=\,z = o){a = o,b = c=\) 
 and has ordinary contact with it at the origin corresponding to the vertex of the reference tri- 
 angle in the fundamental region in (a, b, c). We locate also the points 
 
 Z ; a : 6 : c : : 2 : I : I, y= {(,, 2= 3V \ 
 
 H ; a : b : c :: —1 : 1 : 1, y= — i, z= — i f 
 
 v.,.j,.,.. (3--/I7)   (i+F 17) . (i + v 17) .._ (51 17-19) ■_ (5-3l/i 7)( (44) 
 B,a.b.c.. -^ .—- g . ^ , y- , z- ^^ 
 
 C the conjugate of B. 
 
 B and C occur in the solution of the equilateral triangle case and are on both Dt = o and Z)j = o. 
 The locus 1)2=0 becomes in the (y, z) plane 
 
 (y-22)= (y-h22)-42(y-22)-f2 = o • (45) 
 
 This cubic has asymptotes 
 
 y— 22 — i = o with a cusp at infinity, 
 
 y+ 22-|- 1 = with the third intersection at 2 = — J. 
 
 At the origin the inflexion y^+2 = o gives a three-point contact with Z), = o. Z?, and A also 
 touch at B and C. They cut at H and after entering at H the real region (A >o), A remains 
 continuously within Di to the infinite line. The part of D^ without A is of course not repre- 
 sented in the real (a, b, c) plane. 
 
 The locus r=o in the (y, 2) plane is 
 
 4y'—2oyz-\-4oz'—y-\-z = o (39) 
 
 This ellipse meets Di and A at the origin and meets D, at F. It lies partly within and partly 
 without the real region (Z), >o). The cusp of D, the point A is within the ellipse. T=o also 
 touches y— 2 = at the origin. 
 
   The locus 1T(a-|-6) = o becomes y— 2 = 0. (46) 
 
 The locus ■IT(a-|-&—c) = o becomes 4y—8z— 1 = and touches Z?i at F. (47) 
 
 The locus a6c = o becomes 2 = 0. (48) 
 
 Since the transformation of the fundamental region is point for point there is no trouble in 
 transferring the compartment markings from the (a, b, c) plane. For new compartments we 
 have, introducing as dividing curves, 
 
THE TRANSFORMATIONS 
 
 25 
 
26 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 Fi . . 4y'-8yz+z-y = o (a = o, ^=o) (49) 
 
 Ci . . y^— 23/22 -|-3-yz—SS' — Z = (a=co) (50) 
 
 •W, : A<o, y-2<o, Z/',<o 
 Ifj : A<o, 3»— z<o, /7, >o 
 XXi : A<o, y— z>o, r<o, 3;— 23<o 
 XXj : A<o, y— 2>o, r>o, y— 2z<o 
 XXX' : D,<o, y-2z>o T<o, 4y-8z-i<o 
 XXX : A<o, )'-2z>o, T>o, 4>'-8z-i<o 
 X, : Di<o, 4y—8z—i>o, T>o 
 
 X2 : Di<o, 4>'— 8z— i>o, T<o, z>o /(Si) 
 
 Pi : Z>,<o, z<o, D2>o, Ci>o, /7,<o 
 Pj : Di<o, z<o, A>o, Ci>o, Hi<o 
 Pj : D2<o, 2<o, I>2<o, Oo, Hi<o 
 Ml : A<o, z<o, A<o, Oo, Hi<o 
 Pi' : A<o, z<o, A>o, Ci<o, Zr,>o 
 P3' : A<o, 2<o, A<o, Ci<o, £ri.>o 
 W^i' : Z>i<o, 2<o, A<o, Ci>o, ffi>o 
 
 In tracing the intersections of the curves Ci and Hi with themselves and the other curves 
 of the (y, z) plane we notice that Ci passes the origin with an inflexion yi—z = o and so goes 
 from ® to 000. Since the asymptote 4^— 82+1 = lies entirely in 000 in the third quadrant 
 and the curve crosses it at z= — gV ^^^ does not cross the y axis, it must remain in this com- 
 partment to infinity. Leaving the origin in the first quadrant the curve does not cross FZ 
 (431—82—1 = 0) but crossing A enters XXX. As it reaches y=2z at J (2, i) which is outside 
 T it must next cross T, pass through XXX', and enter XXi at / and remain in this region to 
 infinity. This serpentine branch of Ci cuts //, at the origin and in two other real points, one 
 in XXX and one in XXX'. The other branch is parabolic, approximating y'+$z = o, and lies 
 in the third and fourth quadrants. It crosses the z axis at z= — ^, outside A at whose cross- 
 ing z=— 5*7, and meets A at H y= — i, 2= — i where the curves have ordinary contact. 
 There is no further intersection in these quadrants. 
 
 The existence of the compartments named is thus proved. 
 
 In transforming to the (<#>, t) plane we have the birational transformation 
 
 .^(«^-T+i) ^ _ <t>(<t >-r) 
 
 with the reverse 
 
 42(y-22) 
 
 (y-z) (43'-8z-i) ' (y-z) (43;-82-i) 
 
 (9) 
 
 Real (y, z)'s give real (<^, t)'s and conversely. 
 
 Complex (3/, z)'s give real (<^, t)'s only for 3'=2s; <^ = o, t= — i. The whole linear locus 
 31-22 = complex as well as real is thus represented at the point 4> = o, t= — i, but no other 
 complex points in the (y, z) plane give real points in (</>, t). Other singular lines of the trans- 
 formation are 
 
THE TRANSFORMATIONS 27 
 
 z = o which becomes the point ^=0, t=o but as Limit -=y the linear elements at the point 
 represent the various values oi y. y—z = o gives <^ and t infinite with Limit -=—42. 
 
 T 
 
 4y—8z—i = o also gives <l> and t infinite but Limit - = i unless y and z are infinite when ^ — t = i . 
 
 T 
 
 As before noted this line <^ — t— i =0 occupies a unique position inasmuch as to every point on 
 it corresponds the same triangle, namely the limit of 
 
 a -.b : c::i : p : -/>+5 
 when p is infinite. 
 
 The line y—2z = o gives <^ = o, t= — i and has in the limit — ; — = , assigning real 
 
 T+i 42—1 " " 
 
 linear elements to real (y, 2)'s, complex to complex. 
 
 The boundary curves in the {<l>, t) plane are ( Fig. 7) : 
 
 D,= {<t>-ry{<t>-4ry+{<t>-r) (^^^+28<t>T- S27^)+{<I>-t) (3,^- i6t) + <^=o (52) 
 
 The asymptotes are 
 
 ^ <^— T=o, intersecting also at (o, o) 
 
 <I>—T=l, intersecting also at (VV) ■^^)' 
 
 Corresponding to the factors (<^— 47)^ is a paraboHc asymptote 
 
 3(<^-4t)»+i28t=o. 
 
 At <^= — I, T = o is an inflexion <^'3= 54T' which has four-point contact with the curve /8= — 
 
 T 
 
 = 54 at this point. At <^ = o, t= — i is a conjugate point. The point A is represented by a 
 cusp at <^=i, T=-V. At B (<^= -^MillLJl) ^ ^^_ (45+in i7) \ ^^^ ^^^^^ j^^g ^.^^jj^^^,,. 
 
 with D2 and also at C the conjugate of J5 in (abc). The point .ff becomes the infinite point on 
 <^— 4r=o, the axis of the parabolic branch. 
 
 The locus D2 is ' • 
 
 (<t>-T) (<^-4t) (3,^_4t)+<^^ = o (53) 
 
 Its asymptotes are 
 
 <l>— T-j-| = o with intersection at <^=— 5, t=o 
 
 <^—4T+§ = o with intersection at <^ = ^^, t=^ 
 
 3<^—4T— 2 = with intersection at<^=2, t=i 
 
 At the origin is a cusp <^^= i6t3. 
 
 The contacts of the curve with D, have been noted. At <^=2, t=i, which is an infinite 
 point in ia,b,c) and {y,z), D^ touches <^— t— 1 = 0, the line which also falls on D{a, fi) in 
 the (o, /8) plane. 
 
 The locus r= o is a hyperbola 
 
 t(6<^-i)-(6.^'-3<^+i) = o (54) 
 
 the asjanptotes being 
 
THE TRANSFORMATIONS 29 
 
 The complete representatives of £>, and D^ are the irreducible factors above set out with 
 the addition of <^ = o in both cases and also </>— t=o in the case of Di. These factors which 
 vanish with a may be properly set aside and treated with the zero and infinite values of a and 
 P as nonfinite discriminantal factors. 
 
 The {<i>, t) representatives of z = o, y—z = o, 4^—83—1 = have been discussed (p. 27), 
 the curve Hi becomes <^+ 1 = 0. The curve Ci (a = 00 ) becomes 
 
 l6T4+T3(-40<^+l6) + T^(33<^^-28<^) + r(-IO<^+IO<^^) + «^»(<^+l)'= \ , 
 
 The asymptotes parallel to <^— t = o are complex and the infinite point on this line is a 
 conjugate point. The infinite branch corresponding to (<^— 4r)2 is parabolic. As in {y, z) 
 the curve has two separate branches, and therefore at least a square root must be used in express- 
 ing the points. This is suflScient, for writing 
 
 <i>—T=x, <i>—^T = y 
 we have 
 
 y _iia;+2±l'— i44a;5+ii7x'+36a; 
 , 2x gx'— 2X-\- 1 
 
 The quadratic denominator is essentially positive and so y is infinite only for infinite a;'s, 
 that is on the parabolic branch. 
 
 From the radical, limits for the branches are obtained. 
 
 The closed branch has a cusp at the origin, i6r3+^' = o. 
 
 The open branch touches t=o and so A at P(— i, o). 
 
 The closed branch has three-point contact at the origin with Z),, and Z), is closer to the 
 T axis than d for points in the third quadrant near the origin. The remaining intersections 
 with Z>i are the infinite points on <^— t = o, and <^— 4t=o; a two-point contact at /(o,— i) 
 which is a conjugate point on Z?,, and a single intersection approximately at <^=— .27, 
 T= — . 15, which is better determined by the rational value for /3, — \'. There are also two 
 complex intersections. 
 
 The curve Ci (0= 00) meets Di = o at the origin, at infinity on <t>—T=o, at infinity on 
 <A— 4r = o, at apoint where/3= — Y (<^=— .4653 . . . , t= — .0226) and two complex points 
 which with this are the roots of an irreducible cubic. At these three points the curves have an 
 ordinary contact. 
 
 The locus p= = — V passes successively through the cuts of a= 00 and Dj, A, 
 
 T 
 
 <^— T— 1 = 0. The latter point is (-|-2>""5)- 
 
 The cuts of a = 00 Vith <^-|- 1 = o, <^ = o loci for which a = o give as points where a must be 
 determined by the direction of approach, 
 
 / : (0,0), J : (o, — i) for <^ = o 
 and the points 
 
 T=o, —.609. . , —1.245. . , —1.646. . . for<^= — I. 
 
 To complete the essential features of the diagram we notice that Z), has a closer contact 
 with the T axis at the origin than D,. 
 
30 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 The line <^= — i meets D, at t= — 1.014, meets T at t= 
 
 - ■'^2,2,-     
 
 These points enable the ordering of certain compartments to be made clear. 
 
 The (<^, t) plane is covered as by a single sheet of the (y, 2) plane stretched but not folded. 
 Continuity is preserved except for the singular points and lines of the transformation, which 
 become lines and points respectively, and except that as regards the infinite values the usual 
 conventions of the projective plane are to be observed. It is convenient to locate some special 
 points: 
 
 A : <I> = -1, r=V- _ ^ 
 
 ^   0_ (13+31/17) ^_ (45+111/17) 
 
 4 ' 16 
 
 C : conjugate to B 
 D : <f>=2, T= I 
 £:3<^ = 4T=oo 
 
 F:4>=T='X) on <^— T=o i.e., on £>i 
 G:<^ = T=oo on <^=r+| = o i.e., on D, 
 
 ^:<^ = 4r=oo /(S6) 
 
 / : <f>=T=o 
 J : <^ = o, T= — I 
 
 L : <^ = 3i/ 5— I, T= fK 5 and M its conjugate 
 ^'•4'= 5, T=4 : Z, if, iV occur in the case of a=4, ^9=54 
 P:<^=-i, T=o a = 4, /3=S4 
 PF : <^=2.629 . . . r=,886... a = 4, ^=54 
 Z : <I>—T= J, <^= CO on an asymptote of A. 
 
 The region (i) is within the cusp of A at A and reaches to the infinite with F and Z as 
 limits of the branches. 
 
 The region (4) is identified by means of H, D, and W, which are on its boundary, while 
 E is not. 
 
 The region (3) is bounded entirely by D2 and infinite points and reaches D, H, and E 
 and is then inside the branch of A in the first quadrant of (<^, t). 
 
 The region (2) is located by E and N and by its separation from (3) by A- 
 
 The regions (5), (6), (7) all reach C and (5) is bounded entirely by D, and A; (6) reaches 
 / with (5), while (7) does not. (7') which in iy,z) has continuity with (7) through infinity, 
 has in (<^, t) continuity through the point P and joins (4) in the infinite regions in (^, t) just 
 as it joins (4) in {y,z) along y—z = o. 
 
 The regions (8), (9), (10) have contact of their boundaries at B, and (9), being entirely 
 bounded by A, is the inside region in (<^, t). (8) reaches H in (y, z) and so must belong to the 
 parabolic branch of D2 and be the upper one of the three regions. 
 
 Of the regions with real sides and imaginary bisectors which in (y, z) reaches F and Z 
 and is bounded by A and 2 = o in (<^, t) is bounded by Di and <;(> — t = o and reaches I. 
 
 o bounded in (y, 2) by A, 2= o and 43/— 82— i = o in (<^, t) lies in the first quadrant between 
 A and <t>—T = o. 00 is continuous with o through A- 000 in (y, 2) joins 00 along y— 2=0 
 
 which involves <^= 00 , t= 00 with -= — 42 and 2 runs from zero to — j so in (<^, t) these regions 
 
THE TRANSFORMATIONS 3 1 
 
 are continuous through infinity in the second and sixth octants. 000 reaches / by passing 
 through / a singular point. In (y, z) 000 is continuous with XXX' through infinity, in {<t>, t) 
 through the line <^— r— 1 = 0. 
 
 The regions outside Z),, that is corresponding to complex sides, are to be identified as follows: 
 
 The line Hi = o becomes <^+i = o and the origin in (y, 2) has linear elements which cover 
 <^ = o, T >o hence M2 bounded also by A and reaching C is identified. 
 
 For the region W,, the asymptote oi H^ y = i reaches infinity at a point which becomes P 
 and as Wi reaches y = z for all positive values of z in the {<i>, t) plane it must reach all infinite 
 (<l>, r)'s which are the limits of <t>= — 42T and so occupies the remainder of the second quad- 
 rant after M2 is removed. The same argument locates XXi and so XX2 by crossing T but not 
 <^ = o. The regions XXX and XXX' can now be reached through /, the linear boundaries 
 in (y, z) being replaced by the collection of linear elements at /, which is a singular point of 
 the transformation. 
 
 In the (<^, t) plane it is convenient to subdivide these compartments by means of the 
 lines Hi and Ci representing zero and infinite values of a. 
 
 Xi is identified by A , and the cusp on Dt and X2 by crossing T. 
 
 © is reached from XXX by crossing Z>i. 
 
 Ml is reached from 7' by crossing A, thence crossing C, (a= co) we arrive at P^; across 
 Hi (a = o) to Pj', across a= co to W,', across D2 to W2, across a= 00 to P/, across a = o to P„ 
 across a= 00 to P2 (see Fig. 8). 
 
 It is convenient to subdivide also the regions © by <^= — i, 00 by T, 000 at / and again 
 by a = 00 . 
 
 In the (a, /8) plane the discriminantal loci to be traced are: 
 
 £)(a, /3) the representative of £>,, D2 and the line <t> — T— 1 = 
 T(a, /3) = o, the axes and the infinite boundary (Fig. 9). 
 
 On account of the single value of a for given j8, <j and the connection with a model of the 
 surface F{(t, a, /3) = o the a axis is taken vertical. To trace T = o which is a rational curve in 
 (<^, t) the parametric representation, obtained by substituting for t its rational expression in 
 
 <tt, namely — t^^ — r — in the (</>, t) expressions for a /3 (10). 
 (6<^— I) 
 
 This gives 
 
 a= ('^+i)'(6<^-i>(2'»-i) \ 
 
 (57; 
 
 „ («^+iW6<^-i) 
 
 S 
 
 Letting <t> range over all real numbers we have a real branch. It is proper however to 
 inquire whether any other branch exists by which complex (<^, t)'s are represented by real 
 («, /3)'s. 
 
 The general condition that such may be the case is : 
 
 If a=f(<f,)=f{a+bi) = A+Bi 
 and ^=g{<f>) = g(a+bi) = C+Di, 
 
 B and D have a common factor other than b. 
 
 I (58) 
 
32 
 
 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 In this case P is real if 6 = o or if 
 
 3(4a-i)[664-3i,^(i2a^+i7a+5)+(6a-i)(a+i)3] 
 -|-[6J^-(6a^-3a+i)][-J^(24a+i7)+3(8a+i)(a+i)»] = o 
 
 The coefficient of h* is (—72^—120) and a set of terms free from b exists. 
 For a=i the expression becomes 
 
 — 192Z)''— 1066^+72 
 which does not reduce, and hence the general expression does not reduce. 
 
 F.J 8, 
 
 oL ■f\>Mte 
 
 d-* 
 
 c^-^ 
 
 
 ; r 
 
 
 / 
 
 
 11 
 (SJ. 
 
 
 
 /' 
 0-"/ 
 
 +- 
 
 0, + + 
 
 " -4- 
 
 
 »l 
 
 / 
 / 
 / 
 / 
 
 / 
 
 "^ 
 
 ^^-"-^ 
 
 " i^"'"~'7 
 
 \ 
 
 t 
 
 ./"'^^ -4 
 
 J^-- / 
 
 \ 
 
 
 ^.^'^ 
 
 /.y^'r- 
 
 — h 
 
 
 / 
 
 / 
 
 +; 
 
 >' 
 
 
 • 
 • 
 
 
 + - 
 
 
 
 (59) 
 
 For a=o, &' = f , y3 is real but the corresponding expression for a is 
 
 [432&«+2886''-(-ios6"- i][4326i+4286^- 102]- [4326"+ 1646'- i8][3i26'- 15] • 
 
 when a = o, and this is not zero when h'' = \. 
 
 Hence there is no such common factor and r(a, /8) = o is a unicursal curve and <^ is a proper 
 parameter. 
 
 It is moreover in i : i correspondence with the hyperbola in the (<;^, t) plane, and the facts 
 discovered there may be utilized in tracing. 
 
 The curve T meets a = 00 in one point only, for at /, which is apparently an intersection, 
 the linear elements differ and / is singular. 
 
 This gives a single asymptote /8= .0164 . . . approximately. 
 
 j8 is infinite for no real finite ^ but for <^=°o, a=oo, ^=00. 
 
 This gives a parabolic branch with /3=a» as parabolic asymptote. 
 
ON 
 
34 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 For <^= — I, a = o, /8=o the character being given by 
 
 54 iy3a3= 2^.33. 77^2 a cusp. 
 
 At <l>=i, tt = o, ^=0 
 
 2''.3''.;8-'-f 77.0=0 an inflexion. 
 
 Since the curve crosses D2 at the real four-point it must touch D{a, j8). This occurs at 
 a= — .99678. . /3=5.3542 . . approximately. 
 
 For<^ = i; a=o,y8=-V-. 
 
 For 4>=o; a=iV, /3=-i. 
 
 For /8= — I <t>' = o or <^ is complex. This point which corresponds to / is then a turning- 
 point for j8. For j8< — i all the values of <l> are complex. 
 
 There is a maximum value for a at <^= — 33 . . , a= .66 . . , ^= — .33, and a maxi- 
 mum value at "^=.402 . . ; a= — i.oi . . , 18=5.45 . . , and a maximum value at 
 <^= — 1.58 . . ; a=— .8307 . , /8=.3oi2 . . . The curve meets D only at the origin 
 and the four-point. 
 
 These and the negative facts implied by omission are sufficient to establish the general 
 character of the curve. 
 
 In the (<^, t) plane the curves (<^-|-i)^ — /8t = o form a family of cubic parabolas with centers 
 at i'(— I, o) and can be easily visualized in the diagram. By so doing it is seen that the num- 
 bering of the regions i, 2, 3, 4, 5, 6, 7, 8, 9, 10 corresponds to the order of magnitude of the 
 real roots of F(o-, a, j8) = o. 
 
 Since A has three-point contact with the curve /?= 54 of this family at P, and a here inde- 
 terminate has the limit 4 for this approach, the region 7' is seen to contain only values of yS 
 which are greater than 54 and to be continuously joined to (7) along a=4 in the (a, ^) plane. 
 This corresponds to the change of class of the root from (113) to (331) when a passes the 
 value 4 (§8). 
 
 Digressing to complete the comparison of the two classifications of the real roots we have 
 the set (122) [§8] has one negative side and as the sum of the positive sides is not greater than 
 2, the negative side cannot be less than — i. This identifies the set of three with (8), (9), (10). 
 (122) has the greatest bisector internal and opposite a, : hence j a, | is the least magnitude among 
 the sides and as it approaches zero the other sides approach 5, J which is the triangle repre- 
 sented at F. This identifies (122) and (10). (221) can approach H (i, i,— i) and is then (8) : 
 (212) is (9). The set (231), (312), (321) correspond to (2), (3), (4), and (2) reaches the line 
 a+b—c = o or has a side equal to |. This must be c^ for if Oj or bi has this value c, is infinite. 
 Thus (2) is (312). (4) reaches a+b = o or one side has the value i. This must be a^. So 
 (4) is (231) and (3) is (321). 
 
 For the set (5), (6), (7), consider the approach to (i, o, o): only ai+bi+c^ can reach this 
 point with a and b negative and nearly equal : (113) is (5). (131) can approach with a and c 
 negative and very unequal, this is then (6). (311) is then (7), while (7') having two positive 
 sides greater than i and a negative side less than — i is (331). 
 
 The transformation from the (<^,t) plane to the (a, j8) plane effects a 10 : i correspondence 
 and brings as in general complex (<^, t)'s into correspondence with real (a, /3)'s. 
 
THE TRANSFORMATIONS 35 
 
 Beginning with the real regions of the (<^, t) plane it is necessary to determine the limiting 
 
 values of a and ^ at the points where they become indeterminate and also for the infinite values 
 
 of (<^,t) by various paths of approach. 
 
 , , . Am^im—i) 
 
 The hmit of a for <^ = ott= oo is -.-. r, . 
 
 {m—iY{m—/^y 
 
 The limit of a along (<^+i)^— /3t = o for finite )8 as <#> approaches <x is o. 
 
 The limit of a along 4>—mT^ = o is 4. 
 
 The limit of « along m<\>^—r=o is o. 
 
 The limit of a along <^(</>— t) — i = o is 4. 
 
 For finite points : 
 
 At the origin — 
 
 . ±m{m — i) 
 The limit of a along <^— »2t=o is ; . 
 
 The limit of a along A is 4: along A is =» . 
 
 At the point P (— I, o)— 
 
 The limit of a for all rectilinear approaches is o. 
 
 The limit of o for approach on any curve (<^+i)2— ^t=o is 4. 
 
 The limit of a for approach on w(<^+i)^— '■=0 is -, ; — r and the value infinity occurs 
 
 ,. , (20W+1) 
 
 only for negative t s. 
 
 At the point 7(o, — i) — 
 
 The limit of a for approach on t— n<}>+ 1 = o is 7 ; . 
 
 ^^ (3-4«) 
 
 The point J then represents all real points on the line /3+i = o. 
 
 At the point P(— i, o) — 
 
 The limit of P for approach not on a ^ curve is 00 . 
 
 Any value of j8 is reached by approach on the /8 curve (<^+i)^— /3t=o. 
 
 As a preliminary to identifying the compartments the signs of a, j8 may be marked on the 
 (<^, t) diagram (Fig. 8). 
 
 The discriminantal loci in the {<!>, t) plane are A, T and the loci giving zero and infinite 
 values to a and /3. 
 
 In transferring the {<!>, t) compartments to the (a, /3) plane they must be folded at the 
 proper discriminantal lines A and T. (Though D, falls on D[a, /J] it is not discriminantal.) 
 
 For the other loci a special inquiry must be made. For a= 00 , e.g. the points on the same 
 /3 curve, close to and on opposite sides of this locus, correspond toa=+»?,a=— w and in the 
 (a, j8) plane are near only in the projective sense : folding is then not the proper word. 
 
 If such a locus occurred with the vanishing or infinite factor entering with an even exponent, 
 folding would suit, but the only one of this class for finite (<^, t) has a further characteristic 
 which prevents the use of the concept. Namely, the locus <^+i = o contains only points f6r 
 which a vanishes to the second order, while /8 vanishes to the third. The whole locus is a sin- 
 gular line all of whose points find their representation at the origin in the (a, P) plane. It is 
 only possible to say that the sheets are connected at this point. 
 

 36 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 For infinite (<^, t) an example can be found. The region Wi is connected in the (<^, t) plane 
 with XXi by passage through the infinite line along {<t>+iy— I3t=o for any negative /S. In 
 both regions, however, a is positive and as the limit of a along the ft curve is o the order of the 
 vanishing factor must be even and folding occurs. 
 
 In the same way for positive /8, 00, and oooi are folded on tt = o, in the fourth quadrant of 
 (a, /3) since a is negative in both compartments. 
 
 Starting with those regions of the (<^, t) plane which fall on the first quadrant of the 
 (a, /8) plane, the region (i) inside the cusp of Di falls inside the cusp of D. Since £), is not dis- 
 criminantal Xi is continuous with (i) over the line D up to the parabolic branch of T. Along 
 this curve the points are readily identified by the parametric value <l>. Otherwise Xi reaches 
 <f>—T=o, that is a = o, and X2, which is folded on X, at the curve T, reaches •r = o, that is a = 4, 
 ^=00, while X, along <t>{<t>—T)—i = o reaches for infinite <#> the same point (Figs. 9 and 7 
 and p. 3S). 
 
 The line ^ — t— 1 = falls as a whole on D and the part in the first <t>, r quadrant falls on 
 the boundaries of the "cusp" region. The point D (2, i) falls on the cusp and the lower part 
 of the line reaching t = o reaches a = 4, /8= «> . The upper part reaches a= 00 , /3= 00 . 
 
 The line D2 also falls on D and the hyperbolic loop in the first octant of {<t>, r) falls on the 
 boundaries of the "cusp" region. 
 
 The three regions (2), (3), (4) fold alternately and form a sort of pleat. (2) is continuous 
 with X2 along the upper branch in (a, /3). The upper branch of Z?j in (<^,t) falls on the lower 
 branch of D in (a, /3) and vice versa. Next consider the region P2 which falls on the first quad- 
 rant of (a, yS). P2 reaches a = o, /3 = o along (j>= — i and reaches 1 = 4, fi= co along <^(<^— t) — 
 1 = for <^= 00 , and reaches a= 00, ^=00 along the parabolic branch of A. It reaches a=o, ft 
 any positive, along <f> — T—o and ^=0, a any positive at the cut of a= c» , a = o, <^= — i. 
 
 P2 does not contain T in (<^, t) and so in (a, j8) passes continuously over the branches of 
 T without change. The regions (8), (9), (10) join Pi over A in (<^, t) and are pleated in (a, /3) 
 over the cusp region. The branch of D2 between (8) and (9) falls on the lower boundary of 
 the cusp region, for it is continuous through infinity with the boundary of (2), (3). B falls 
 on the cusp. For the region Af, a>4 and moreover a has a greater value than belongs to the 
 branch of Z), which separates Mi from (7')- The region M2 reaches a = 4, /3= 54 at P by approach 
 on the /8 curve and so joins Mi. Further M2 reaches ^ = 0, a any value between o and 4 by 
 approach at P along parabolas (p. 35). Mi reaches /3=o, « any value between 4 and 00 in the 
 same way. M2 reaches a = o, /3 any positive at the points along <^ = o. M2 embraces the 
 regions (5), (6), (7) which are pleated on the cusp region, C falling on the cusp, the fold of 
 (5) and (6) falling on the upper boundary and the fold of (6) and (7) on the lower. (7) for 
 which all points have a not greater than 4 is joined continuously to (7') for which all points 
 are not less than 4. 
 
 Since Mi and M2 do not contain T in the {<i>, t) plane they pass over it without change in 
 the plane (a, yS). With the regions (5), (6), (7), and (7') which are pleated on the cusp region 
 they form a continuous covering of the first quadrant of the (a, /?) plane in the same manner 
 that P2 with the pleated regions (8), (9), (10) does. 
 
 The sheet X2 with the regions (2), (3), (4) pleated behaves in the same fashion, while the 
 sheet Xi containing the region (i) giving the internal solution is without fold at the boundaries 
 of the cusp region. There remains in the first quadrant the pair of regions XXXi and XXXi' 
 between a= 00 and the negative side of <^-f-i = and separated by T. These reach all positive 
 
THE TRANSFORMATIONS 37 
 
 a's for j8=o at the indeterminate points for a (p. 29) and folding on the asymptotic branch of 
 T cover the space between this Hne and the a axis. 
 
 The sheets X, and X, are folded at the paraboHc branch of T where 4>>\ and the complete 
 account of the first quadrant of (a, /8) is: 10 real roots for the "cusp" region, 4 real roots 
 between this and the parabolic branch of T, 2 real roots between the two branches of T, and 
 4 real roots between the asymptotic branch of T and the a axis. 
 
 It is interesting to notice the persistence of the root (i) for a region of the (a, ^) plane 
 much more extensive than the region where it has an interpretation as a solution of the prob- 
 lem for real angle-bisectors. 
 
 Taking up the second quadrant of the (a, /3) plane the region W^ has a not greater than 4, 
 /3 any negative. Wi is continuous with Wi through P and as the /3 curves with P as origin 
 are Pr=ai and the curve a = 00 is 2ot = o-% all the /8 curves for /8< o fall between a= 00 and the 
 <^ axis. For Wi , a is not less than 4. The region is bounded by A which cuts a= 00 at a point 
 for which j8= — ^ , and as this value is asymptotic for Z?(a, /3) = o and A is discriminantal the 
 curve /3= — V touches 0= 00 at the point in question in the (<#>, t) plane. For j8> — V the 
 P curves leave W^' by crossing a= 00 . So PF, and W^' together form a sheet covering the second 
 quadrant of (a, y8) up to the line D. At this line the sheet is folded and returns from the fold 
 as W^- W2 reaches a= 00 for o< j8< — V, ^nd reaches a=o, /3=o along <^+i = o, and reaches 
 j8=oo , o<a<4 at / (o, o). Wi has these latter values in the infinite regions. Wi and W2 
 only join (6) and (7) at 0=4 ^= 00 . 
 
 The (<^, t) regions XXX^, XXX2 are continuous with XXXt and XXXi respectively 
 through the cuts of a = 00 and </)+i = o where ji=o and a depends on the path of approach. 
 These paths are parabolas touching a= 00 since this is of the first order while <^+ 1 = o is of the 
 second, and so in passing these points from one of the regions to the other a does not become 
 infinite. The regions are also continuous with XX2 and XXi respectively through J. XXi 
 is continuous with 00O2' over <^— r— 1 = 0, the non-discriminantal representative of D{a, j8). 
 The whole set forms a double sheet covering the second quadrant of (a, /8) (with the exception 
 of the part inside the loop of T) and the part of the first quadrant up to the asymptotic branch 
 of T, as before mentioned. It is folded on T from the cut of T and a= 00 [asymptotic point 
 of T in (a, 13)] through <^= — i [cusp of T at a = o, /3=o], to J (o, — i) [a= ^\, /3= — i] and up 
 to <^=g, the asymptote of r in (</>, t) [inflexion of r at a=o, /8=o]. 
 
 The sheets reach a = o, /3 any negative as follows: 
 
 o>)8> — I in XX2 (J represents 13— — i as a whole) and in XX i at infinite points on j8 
 curves. They reach /3=o, a any positive at the indeterminate points discussed above. 
 
 The region inside the hyperbolic branch of D{a, /3) is covered by 00O2' in one sheet and the 
 part of XXX2 between the cut of A and a= co and the origin in the other. 
 
 The third quadrant in (a, /8) has as representatives in (<^, t) : 
 
 ©2 and XX X;, continuous over £>,. 
 
 00O2 and XXX/ continuous through J, and since <^— t— 1 = is crossed at J and also 
 between XXX/ and 000, there must be added that part of 000, for which <^+ 1 >o. This part 
 has o > /J > — I but joins 0002 for /8 = — i , o any negative at J. 
 
 As no discriminantal lines occur we have two separate sheets. XXX ^ is continuous over 
 /3=o with XXX3, and XXX/ with XXX/ over the same line. 
 
 P/ and Pi are folded on D and reach /3=o, a any negative at P and the indeterminate 
 point for a. At the fold o >/3> — Y (Fig. 7). These facts locate the regions in (a, /3). 
 
38 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 The fourth quadrant has in (<^, t): 
 
 The second octant of (<^, t) having four regions d, O2, 00,, 0O2 which meet at the four-point 
 and are folded so as to hang together at the four-point in (a, j3). They are joined in pairs along 
 £>(<!, /8) for all values of /8 greater than the four-point value, for lesser values the j8 curves in 
 (^, t) meet D^ in complex points. 
 
 For the lesser p's ooi and 0O2 are folded on T. d and O2 are folded on the other branch of 
 r up to {<i>=r=\) [j8= Y", a=o] after which they pass continuously into X, and X^ which are 
 folded on this parabolic branch of T in the first quadrant of (a, /8). 
 
 The fold of 00, and 0O2 passes through the origin in (a, /8) and is continuous with the fold 
 of the sheets away from the loop of T and reaches the first quadrant as the fold along the asymp- 
 totic branch of T. 
 
 The regions P, and Pj are folded on D2 for positive ;S's and reach a= 00 , /3= 00 with Z)^ 
 and a= CO , y3 any positive on the curve a= 00 . They then cover all the fourth quadrant below 
 the branch of D. They are continuous with P/ and Pj' over the a axis. 
 
 The regions XXX^ and XXX^' are folded on T, reach a= 00 , /3= 00 along <^— 't=o : a= 00 , 
 /3=o at <#>4-i = o but for a= 00 the greatest /3 is .0164 . . the value for the asymptote of T. 
 
 In (a, /3) then these sheets are folded on the lower branch of T and extending upward pass 
 continuously into ® and 000, respectively across Di and <^—t— 1 = 0, both of which fall non- 
 discriminantally on Z)(a, ^). 
 
 Collecting the parts of the plane which are continuously connected in (<^, t) as well as in 
 (a, P) we have, setting aside the region of ten real roots : 
 
 Sheet A containing X,, d • 
 
 Sheet B containing X2, O2 
 
 Sheet C containing Af,, M2, ooj, Wi, W/ 
 
 Sheet D containing P2, 6,, ©2, W^, XXX^, XXX^ 
 
 Sheet E containing 000,, 0002, XX^, XXX,, XXX,, XXX/, XXX/ 
 
 Sheet F containing 0002', XX„ XXX,', XXX,' 
 
 Sheet G containing P,, P,' 
 
 Sheet H containing Pj, Pj' 
 
 Sheet J containing 00, 
 
 Sheet K, the tenth sheet, is not represented outside the region often real roots (Fig. 10). 
 
 The phenomena at infinity are: 
 
 i) a= CO J p any finite value. 
 
 There are ten distinct finite roots except at the asymptotic lines of T and D, any value of 
 o- being reached for at least two finite real values of /8. 
 
 2) /3=oo,a>4. 
 Eight roots are infinite in pairs. For (T'^ = mfi they are given by: 
 
 l6aw'— 40ttm3+33aOT^-|- (4— IOa)m-(- (a — 4) = o. 
 
 Two roots are <^= i, and these belong to the regions (5), (6) if /? approaches infinity from 
 the positive side, and to the regions XXX, and 0002' if /3 approaches on the negative side. 
 
r 
 
 THE TRANSFORMATIONS 
 
 39 
 
 The four pairs of infinite roots are (i) and (lo) : (2) and (9) : (3) and (8) : (4) and (7') for 
 approach with positive /?, while for negative ft all approach in the complex regions. 
 
 The equation in m has no negative roots for a>4. 
 
 3) fi— 00 , a = 4. One root is zero for any ;3. This is (7) or (7'). 
 
 Three roots are finite. These are easily determined from the 4>, t equation for = 4 when 
 /8 is infinite and <i> finite, for then t=o. 
 
 The equation is 
 
 ^[(<^-0(3<^-4'-)'+4(<^--^)(3<^-4'-)-<^] = o. 
 
 Neglecting the indeterminate solution t = o, there are three values for </> : o, '-^ , ^—^ . 
 
 '% 
 
 AO- 
 
 <^=o belongs to (5), the negative value to (6) and the positive to (4) for positive /J. For 
 negative /8 they fall in XXX2, W^, 000/ respectively. 
 
 The values obtained for a (or <^), however, depend on the path of approach. If /3= 00 
 a = 4 the roots are eight infinite and two indeterminate, while for approach along the lower 
 branch of D whose asymptote is a— 4 = the root (6) and the root (7) are continually equal 
 and attain the limit 4>=—\. 
 
 For 3<a<4 as /3 approaches positive infinity there are four real roots, in the sheets AT,, 
 Xi, Pj, Mj. Of these X2 and Af, have the limit <t=i, the others being infinite. The three 
 pairs of complex roots have an infinite limit. 
 
 For tt< o as y8 approaches positive infinity six roots are real and two of these, 02 and ©2,. 
 are finite. 
 
 For P approaching negative infinity the number and ordering of real and complex roots is 
 essentially different, /3= 00 being a discriminantal line. For this approach and a>4 two roots 
 are real instead of ten. For a<4 but positive four roots are real and the finite pair are con- 
 
40 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 tinuously connected X2 to W2 and M^ to XXX2. For negative a only two roots instead of six 
 are real and these are the finite pair 00O2 (continuous with oo^) and ©^ (continuous with o^). 
 The discontinuity at a = 4, ^8= 00 is essential, the original equation having its last three coeffi- 
 cients indeterminate for these values. 
 
 The region of ten real roots terminates at the cusp where (2)= (3)= (4), (s)=(6)=(7), 
 (8)= (9)= (10). A positive circuit of the cusp permutes the roots by the substitution (243) 
 (567) (8, 10, 9). 
 
 On account of the essentially incomplete and non-analytic character of the real field a thor- 
 oughgoing application of the devices of a Riemann surface is of course impossible, nevertheless, 
 as a means of presenting the complicated state of facts in a condensed form, it seems best to 
 make a tentative use of them. 
 
 Drawing a barrier along <i = s from /i=2j to ;8= co and marking the above substitution 
 on it, we name the ten sheets in correspondence with the ten real roots in the cusp region : 
 
 ABCDEFGHJK 
 
 127843s 10 96 
 
 This is consistent with the barrier and gives the plan for the real roots shown in Fig. 10. 
 
 The complex roots need further inquiry. We conceive as many sheets as needed laid 
 over the regions in question and marked with the proper complex values. Z)(a, j8) = o is a 
 locus of points such that three pairs of roots are equal. For the real region and also up to the 
 four-point on the lower parabolic branch these have been determined. For the rest of the curve 
 there are two equal pairs of complex roots. [For k = l=i D is rational in m and the roots 
 corresponding to A are those of a squared cubic factor, whose discriminant vanishes at the 
 four-points (one real) and for m = o which corresponds to 1 = 4, /8= 00 .] (See §17.) 
 
 It is proper so to name the complex roots that after a real pair become complex and of 
 course conjugate the pairing should remairt unchanged except by discriminantal points. 
 
 The part of D2 where two complex pairs are equal will effef t then an interchange of part- 
 ners among the pairs, or rather the monodromie may be so ordered as to effect this. 
 
 In crossing j8=o ten roots become zero. In all cases, however, four are real and merely 
 have their order of magnitude reversed. The three complex conjugate pairs have the order 
 of the pairing inverted, the root with positive imaginary part and that with negative becoming 
 interchanged in each pair. 
 
 In crossing a = o six roots become infinite, and form a cycle. If we choose a pair to be 
 real they must not be conjugate but opposite in the cycle. This effects a re-pairing discussed 
 in the monodromie for ^=27 (§14). Revising the numbers in respect of the barrier (3, 4) 
 (5, 6) (9, 10) leave the real axis as conjugates and for a path with fi constant crossing was 
 proved impossible. Hence if (9, 4) be the real pair the hexagon is 
 
 / \, 
 
 \/ 
 
 6 
 
 and (3, 6) (5, 10) are paired after crossing. 
 
THE TRANSFORMATIONS 4 1 
 
 This is for positive /3. A similar re-pairing occurs for negative ^. 
 
 The phenomenon is unlike anything occurring on a Riemann surface for an analytic function 
 of a complex variable. There crossing a branch cut implies completion of a circuit round a 
 discriminantal point: this passage is only a half-circuit in the a complex plane. 
 
 The cycles given for a= oo (i, 2, 5, 8, 4, 3, g) (6, 10, 7) to be interpreted as cyclic substi- 
 tutions invert the order of pairs but keep the pairing of complex roots. 
 
 For fi= 00 to connect the aproaches with the two signs the following system of interchanges 
 is needed: 
 
 0<a<3 (l) = (2), (3) = (4), 
 
 (5, 6) {f, 10) change the order in the pairs, 
 (7) (8) change the relative order of magnitude. 
 
 That is a half-circuit affecting pairs as on y8=o. 
 
 3<o<4 in addition to the above the barrier is to be extended with (2, 3, 4) (7, 6, 5) 
 (8,9,10). 
 
 o>4 and to end of the upper branch ol D, (i)= (2), (6)= (5), (10) =(9), (7) = (8).«' 
 
 Fora=oo,a'>- up to the end of J's parabolic branch (3)= (4), (7) = (8), (i)=(2), and 
 4 
 
 also the half-circuit. 
 
 For the last two fegions and also on the rest of the line /8= 00 , a >o a barrier is needed with 
 the substitution (2,6) [or (i, 5)]. For negative a the order of magnitude is reversed and a 
 barrier (i, 7) [or (2, 9)] is to be applied. This barrier and the (2, 6) barrier for a>o is needed 
 in view of the occurrence of that part of D where two complex roots become equal. 
 
 With this set of conventions a consistent plan of the sheets can be drawn. In Fig. 11 
 the discriminantal lines and the barriers are shown, with the equalities and conventional changes 
 in brackets [ ] and the pairing of complex roots in parentheses (), the first of the pair being 
 the root with positive imaginary part. The six-cycles at tt = o are symbolically indicated by 
 hexagons. 
 
 Taken in connection with Fig. 10 for the real roots and the identification in p. 40, it 
 is to be considered as a condensed expression of the various connections between the roots of 
 the equation for the field of real a, y8. 
 
 Fig. 10 for the real roots may serve the purposes of a model of the surface /^(ct, a, /3) = o 
 as far as the order of magnitude and number of the real roots for the various values of (a, /3) 
 is concerned. 
 
 There are, however, many questions which are proper to ask concerning the connections 
 of the real roots which it does not answer or answers only with difficulty. For this reason a 
 model is in order to complete the concise expression of the facts (§19). 
 
 The representation of the facts discovered in the field of complex a, yS is of course out of 
 the question. For instance there are two complex four-points, at which the loci T=o andZ)=o 
 intersect. These loci are, however, continua of two dimensions existing in a space of four dimen- 
 sions and their intersection is a point merely. 
 
 To say that this field consists of the totality of point pairs of two Neumann spheres though 
 a useful device for the presentation of certain general arguments is not of course a representa- 
 tion which enables special facts like the one mentioned to be concisely recorded. 
 
42 
 
 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 ot=«o Cvjdei LIA, 5,6.'^.'5,9]Lb,10,7J 
 
FINITE MULTIPLE POINTS . 43 
 
 Infinite points of multiplicity greater than two occur at : 
 
 Thepoint£ ; a : b : c :: i : — : , 3' = z= — 4, 3<^— 4t = o<^=oo 0=4 j8=oo . Here 
 
 13 13 
 the sheets (2) (3) (8) (9) have a common root. 
 
 Thepointff ; a : b : c :: i : i: — i, y = z= — i, <^ = 4, t=oo, a= 00, /?= 00 where the sheets 
 (3)? (4)) (7'). (8) unite. The line a=o has six infinite roots. For the approach a>o, /3>o all 
 six are complex, and so for a<o, ^<o. If a and /3 have opposite signs two roots have a 
 real approach. The triangles are complex in all cases. 
 
 The origin is an indeterminate in (a, /?). 
 
 For a = o, ^=0, - = o there are ten zero roots. The triangles are indeterminate as t can 
 
 be assigned at will. 
 
 For T = o)'=|, 2=00 and the triangle can be approached by way of 
 
 a=o)oH j — c=—b{o)+i)- 
 
 20)+ I 20)+ I 
 
 where «>'= i and b increases without limit. 
 
 ForT= — i,y=^,2; = o. This is the point F which in (<^, t) is represented as the line <^—t=o. 
 
 No other triangle in the infinite set is real except F (a = b = ^ c = o). The sheets involved 
 are (3), (4), (5), (7), (8), (9), (10), and (9) is only reached with t= co . 
 
 The other approaches to the origin give in some cases finite values for o- but all the triangles 
 are complex, as none of the sheets involved fall inside Z>, in the y, z plane. 
 
 /3=o a^ro has ten zero roots independent of a, but these can only be approached in the 
 k, I, m plane with complex values and give of course complex triangles. 
 
 12. FINITE MULTIPLE POINTS 
 
 To determine all the finite multiple points (other than double points) a start is made from 
 the intersections of D^ and T. 
 
 The corresponding values of <t> are given by 
 
 2<^»— <^+i =0 for the triple points, and 
 S4<^— S7<)l)'4-24<^— 4 = for the four-points. 
 
 The approximate values for the real foiu--point are: 
 
 <i>= .4144425 • • T= 529566 . . . 
 
 a = -.99678 . . ^=5.3542 . . 
 
 To find the other multiple points we write F(<r, o, j8) = in the form 
 
 d. , aF ' e, , ^F e, .^ . 
 
 a = -— and 7i- = o as a = — and 7r~ = o as a = — i (60) 
 
 l/'l OCT xf>2 "<'■ Vi 
 
 Eliminating a in turn between each pair of equations we obtain expressions (01), (02), (12) 
 which must vanish for points of multiplicity three or higher. 
 
 If we then write P=— and revert to the (</>, r) plane by writing o- = <^+i, (01), (02), (12) 
 
 are of order 5 in <^ and 4 in t. 
 
 (01) represents the discriminant and so T and D, in their (<^,t) form must be factors. 
 These forms are given in equations (53) and (54). 
 
 As a fact (01) has no other factors. 
 
44 
 
 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 The form (12) is: 
 
 
 ,ps 
 
 ^4 
 
 03 
 
 0' 
 
 01 
 
 00 
 
 t4 
 
 
 
 
 
 — 1920 
 
 — 240 
 
 t3 
 
 
 
 
 3840 
 
 1488 
 
 
 r' 
 
 
 
 — 2712 
 
 -2105 
 
 164 
 
 -77 
 
 T' 
 
 
 792 
 
 948 
 
 60 
 
 -18 
 
 18 
 
 T" 
 
 -72 
 
 -13s 
 
 -54 
 
 8 
 
 — 2 
 
 — I 
 
 (61) 
 
 and the form (02) is: 
 
 
 05 
 
 04 
 
 03 
 
 0" 
 
 01 
 
 00 
 
 t4 
 
 
 
 
 
 624 
 
 -48 
 
 t3 
 
 
 
 
 -1584 
 
 112 
 
 -32 
 
 T» 
 
 
 
 1377 
 
 — I 
 
 - 3 
 
 16 
 
 T' 
 
 
 — 462 
 
 — 102 
 
 36 
 
 - 18 
 
 
 T" 
 
 45 
 
 39 
 
 - 6 
 
 I 
 
 I 
 
 
 (62) 
 
 Di can be written 
 
 i6t'' = 32T^</i— i9T<^^+3<^3-|-<^2. 
 
 Reducing (12) and (02) to quadratics in t by means of D^ we have from (12) the form I: 
 
 0S 04 03 0J 01 00 
 
 T' 
 
 
 
 -432 
 
 196 
 
 164 
 
 -77 
 
 T' 
 
 
 432 
 
 -414 
 
 45 
 
 - 18 
 
 18 
 
 TO 
 
 -72 
 
 54 
 
 9 
 
 8 
 
 — 2 
 
 — I 
 
 and from (02) the form II: 
 
 
 05 
 
 04 
 
 03 
 
 0» 
 
 0. 
 
 00 
 
 T' 
 
 
 
 -36 
 
 88 
 
 -67 
 
 16 
 
 T' 
 
 
 54 
 
 -91 
 
 71 
 
 -18 
 
 ' 
 
 TO 
 
 -18 
 
 21 
 
 — II 
 
 — I 
 
 I 
 
 
 (63) 
 
 (64) 
 
 EHminating T we have a polynomial in <^ of order 16. 
 
 Of this the factors 
 
 (S4<^-S7<^'+24<^-4), (2<^^-<^+i), (2.^^+13^+2), (<^- 2) 
 are known, the second quadratic and the last factor entering as triple-point factors in the "equi- 
 lateral" case. 24>— I appears as a thrice repeated factor and the cubic factor is repeated. The 
 residue is g<^^+<^— i. 
 
 These must include all triple, quadruple, etc., points on D,. 
 
 In the elimination between (12) and (02) which have the form 
 
 common solutions of ^2 = and i/'3 = o enter which need not satisfy (01): 
 
 " -Ac ^Z 
 
 There are five such common solutions finite both ways and these give rise to the factors 
 (2<^— 1)3 {q<l>^-\-<\> — i) after the reduction by ZJ^. The points they denote on Z)j are not in fact 
 more than ordinary points on D^, that is threefold double points. 
 
 a = ~=— and a = 
 
 "A. "As 
 
 .(6^^-3'/'+i) • 
 
 (6<^-i) 
 
 in (12) and obtain a polynomial 
 
 For multiple points on r = o we write t = 
 
 of order 9 in <^, which has the factors 
 
 (54<^^-57<^'+24<A-4) (2.^"-<^+i) (12^^- 16<^+7) (<^-i) (6^+1) 
 
THE DISCRIMINANT 
 
 45 
 
 Similar operations on (02) give as factors the same cubic and quadratic and a zero and 
 infinite factor. These and the two linear factors of the first set not being common are extraneities 
 and the only new points are given by i2<^'— 16<^+7— o and T = o. These are triple points and 
 not on D3. 
 
 These last points are singular points on r = o; being complex the usual classes are without 
 significance. 
 
 13. THE DISCRIMINANT 
 
 Since T = o and D = o are discriminantal loci T and D are factors of the discriminant of 
 F{<^,a,^)=o. The order of r in u is 4, in ;8, 6. The order of D in a is 3, in y3, 2. This is seen 
 directly in the case of D (15) and from the parametric form and the elimination rule for T (57). 
 To obtain a closer view the equation may be transformed by writing, 
 
 *ZS, /3 = 
 
 = 4^^^^=<f. 
 
 The result is 
 
 5'°— iofo*+i46j'+336V— 94J^5s+(6i6=— 3663 — J'<f)i'"+ 
 (i56^»3+6^(/)j3-f-(_2oo63+463(i)j='+(8oJ3-8W)5+4W = o 
 
 (65) 
 
 The order of the coeflScients in the first derivative with respect to 5 and a homogeneity 
 factor, is in respect of d given by the rows: 
 
 (o) (I) (i) (3) (4) (S) (6) (7) C8) (g) 
 
 From this the order of the elements in Bezout's form can be readily obtained and inserted 
 jn the determinant form, 
 
 The maximum order is 13 occurring in the secondary diagonal of the first s-square and the 
 complete last 4-square. A short calculation shows that this order actually occurs. The factor 
 d cannot divide the result as <i = o is not discriminantal for 64=0. 
 
 The order in d is the order of a as far as it depends on the factors T and D; a = o correspond- 
 ing to d=<x> being represented by the defect of this order from 18 the order of the general case. 
 As a = 00 is not a discriminantal locus as must occur as a factor of the discriminant of F(a; a, p) 
 = 0. The orders of am T, D being 4, 3 we have 4tM+3«= 13 where m, n are the exponents of 
 T, D in the discriminant. The only solution is w= i, « = 3. 
 
 As to the powers of j8 a count of order in the Bezout form for F{<j, a, /3) gives 44 as the 
 maximum and of these 30 can be divided from the rows and columns. Since ^ enters T.D^ to 
 the 1 2th order the exponent of /8 is either 30, 3 1 , or 3 2 . The fact that f or ;8 = o no complex roots 
 become real or vice versa bars 31. 
 
46 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 To decide between 30 and 32 it is necessary to examine the monodromie cycles for /3=o. 
 In the neighborhood of /3 = o the equation to a numerical factor is properly approximated by 
 
 [a3+J/3] [<Ti+t,P] l<J'+hl3] [<T+w/3] = 
 20ft 
 
 where m= , r and t^, t^ are the roots of 4t' — gt+4 = o. 
 
 (a -4) 
 
 Of the 45 squares of differences of the roots 9 are differences of roots belonging to the same 
 3-cycle. These differences each vanish with /8 and to the order 5. The product of their squares 
 vanishes to order 6. There are 27 differences of roots from different 3-cycles and these together 
 contribute Z?"'. The 9 differences of roots of a 3-cycle and the odd root are also of order § and 
 contribute )8*. 
 
 The totar order to which the discriminant vanishes with j8 is then 30. It is clear that this 
 method of determining the exponent of a discriminantal factor is general, and depends for its 
 effectiveness only on the determination of the cycles. The only converse to the theorem is 
 that odd exponents and odd substitutions of the roots are correlated. 
 
 The complete discriminant has the form 
 
 The determination of the numerical factor N has proved impracticable. 
 
 14. THE MONODROMIE GROUP OF THE EQUATION F(<T, a, ^) = 
 
 To simplify the numerical work write (r = 2s, o. = 2,a, li = ib. 
 The equation becomes 
 
 1605'°— i2oa65'-|-56a65'+297a6^5^— 282c6^55-j-6ifl&V— 2']oahis^-\-j,()his^-\-T,()cah^s^ \ , , 
 — 1 263^^—1 SoaZ)''i"+8iaZ)V— io86V+2oa635 — 54a6''5 -1-726''^ -fgai)''— 126'' = j 
 
 We begin by placing b=i 
 
 F(5,a, i)=a(i65'°—i205*+56.J^+2975*— 28255 — 2095"-}- | 
 
 36053 — 695^ — 34.y+9)-fi2(3s''— 53 — 9s^-|-65—i) = o ( 
 
 For a = I , i.e., the cusp on D{a, /?) = o the values of s are given by 
 
 25-3=0, (5-l)3=D, (253-1-35-1)3 = (§S). 
 
 For a= — I also on D they are given by 
 
 (453-45^ — 55-1-2)^ = 0, 53—25-1-3 = 0, S-\-2=0. 
 
 Reference should be made to the graph. Fig. 12. 
 
 To discuss the connections of the roots at 5 = 1, a—i, b = i we write 5=i-t-^', a = i+a', 
 b = i+b' and retain terms of the first three orders as this is a triple point. 
 We have as a proper approximation to the curve, dropping the accents, 
 
 6a—2b+2Sab+i4as+bs—iib^+64abs+4S^s'—2ob's — i6si+43'^b'—i6bi = o. 
 
 If the origin is approached along the plane 
 
 6a — 26 = the (a, s) projection is 
 1705- 99a^-|-i2a^5-|-ii7a5'— 1653 = 0. 
 
MONODROMIE GROUP OF F {o; a, ^)=o 47 
 
 The approximations at the origin are given by Newton's parallelogram as 
 1705— 99fl" = o and —1653+17^5 = 0. 
 
 The first branch gives a stationary root while the second has a parabolic factor correspond- 
 ing to a possible interchange of two roots. 
 
 In fact i6i^ = i7a and if a the parameter moves round the origin in its own complex plane 
 the complex values of s each move round a half-circuit in the s complex plane and are so inter- 
 changed. 
 
 We have then for this point and this approach an interchange of a pair of roots entering as 
 part of (a cycle of) a substitution of the roots which is an element of the monodromie group. 
 
 At the same point but by an approach on a = kb, k+l gives 8^^= (3^— 1)6 and a circuit in 
 the b complex plane interchanges three roots cyclically and these three roots must include the 
 former two, by the principle of the continuity of the roots, and the fact that the singular points 
 are necessarily distinct. 
 
 The pair of roots is real for real a's and of the set of three one is real. 
 
 The same values of the parameters a and b give two other triple roots, one set being at s 
 
 — - — ' — — . A similar treatment gives in this case also a cycle of three for the general approach 
 4 
 
 and a transposition for approach in the tangent plane. So also for the conjugate set. 
 
 Since the tangent planes are distinct in the three cases we have as elements of the group for 
 a general approach a substitution of the form (234) (567) (89, 10), while for the tangent 
 plane approaches we have the elements (23) (567) (89, 10); (234) (56) (89, 10); (234) 
 (567) (89) respectively. The roots entering the transpositions may be any pair of the corre- 
 sponding set of three. 
 
 After the cycles at a= i, b=i the monodromie path is taken along b=i. To determine 
 whether any multiple points are encountered we eliminate a between F(s, a, i) and its derivative 
 
(l) 
 
 5=1 
 
 •57 
 
 (2) 
 
 S = 
 
 •347 
 
 (s) 
 
 S = 
 
 ■333 
 
 (8) 
 
 s= - 
 
 -1.9 
 
 48 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 with respect to 5. The resultant which is of order 13 in s must include all finite multiple points 
 on 6=1. Of these 5= i and the roots of 25^+35— 1 = are known, and in fact they are repeated. 
 The remaining septimic has as a factor 45^—45^— 55-I- 2, which gives the values at a= — J on Z). 
 The residue is 
 
 185"— 753— 5452-1-455—12 = 0. 
 
 This quartic has two complex roots and two real whose approximate values are 5=1. 4598 . . 
 and 5 = — 1.924 . . which correspond to a =1.47 . and a = — 1.27 . respectively. (These 
 points are on the upper and lower branches of r= o.) 
 
 There being no multiple points between a= i and a = o we note that at a=o the four real 
 roots are 
 
 which at a= I had the value i . 5 
 which at a = I had the value i 
 which at a= I had the value . 28 
 which at a= I had the value — i . 78 
 
 Since the equation is of the first order in a roots can only cross at double points. This 
 applies to complex roots as well as real for no different a's can have the same value of 5. 
 At a = o six roots which were complex at a= i become infinite. 
 The finite roots for a= o being given by 
 
 (35-i)(53-[35-i]6) = o 
 
 the root marked (5) has a fixed value independent of b. 
 
 The cubic factor has equal roots at b=\ when (i) and (2) become equal and as the origin 
 is approached with a constantly zero the three roots (i) (2) (8) can be made to take part in a 
 cycle by means of a circuit of b round its own origin in the b complex plane. 
 
 To determine the configuration of the roots at a = o, s= 00 we notice that at a= i+c', a' 
 small the roots (3) (4) are conjugate complex roots of 8x^—30' = o and as in the monodromie path 
 from a=i to a = o we leave the point i by decreasing a, a' is negative. The real root is then 
 less than i, and the complex roots have a real part greater than i. We may choose (3) as the 
 root with positive imaginary part. 
 
 In the same way it may be shown that the roots (5) (6) (7) have real parts less than i and 
 (6) may be taken as the complex root with positive imaginary part and (5) as the real root. 
 
 For (8) (9) (10) the real part is negative and we take (8) as the real root (9) as the complex 
 root with positive imaginary part. 
 
 As a leaves the point i and decreases the paths of the six complex roots in the s complex 
 plane start in the order indicated in the diagram (Fig. 13). 
 
 
 "1 
 
 Now F{s, a, i) contains a in the first order only, so no different a's can have the same value 
 of 5, and if the paths cross it must be at a multiple point. There are however no multiple points 
 
EQUATION FOR THE SIDES 49 
 
 for 6 = I and a between i and o. The six complex roots then reach the infinite point of the s 
 plane in the reversed cyclic order and since by writing 5 = - the proper approximation is 
 
 40+9/''= o 
 
 they there enter into a cycle (3, 4, 7, 10, g, 6). 
 
 If we pass along 6= I for a >i we reach a double point at a= 1 .47 . . where s=i .4508 . . 
 is the double value. 
 
 Since at a= 1 the root (i) has the value i . 5 and the root (2) the value i and no double 
 points occur in the interval we add the transposition (12) to the elements of the group. 
 
 At a= — 1 . 27 . . is another double point. To identify the roots here we suppose that 
 the six-cycle at a = o is so passed as to leave (3) and (10), which are opposite in the cycle, in the 
 real positions — 00 and + 00 respectively, when the added element will be (3, 8). 
 
 At a=—i there are three pairs of equal roots of which (25) and (i, 10) are two. The 
 third pair must be opposite in the six-cycle at a = o and is then either (94) or (67). It is in 
 fact immaterial which is taken. 
 
 As elements of the monodromie group we have: 
 
 A 
 
 (12) 
 
 ata= 1.47 
 
 5, 
 
 (23) (567) (89> 10) 
 
 at a= I 
 
 B, 
 
 (234) (56) (89, 10) 
 
 
 B^ 
 
 (234) (567) (89) 
 
 
 C 
 
 (234) (567) (89, 10) 
 
 at a= I 
 
 D 
 
 (347. io> 96) 
 
 at a= 
 
 E 
 
 (i, 10) (25) (49) 
 
 ata=-| 
 
 F 
 
 (38) 
 
 at a= — 1.2; 
 
 The transposition (12) exists. By transforming it by Bi we add (13). 
 Transforming this by C (14) is reached. 
 Transforming (13) by F we reach (18). 
 Transforming (18) by B^ we reach (19) and (i, 10). 
 Multiply £ by (i, 10) and obtain (25) (49)-. 
 Transform (12) by this and obtain (15). 
 Transform (15) by 5, and obtain (16) and (17). 
 
 We have now every (i, n) from which every single transposition and the symmetric group 
 can be obtained. 
 
 By Jordan's theorem the group of the equation is the symmetric group. 
 
 15. THE EQUATION FOR THE SIDES 
 
 To obtain an equation for the value of the side of a triangle with given internal angle- 
 bisectors we write a-\-b-\-c= 1 and use ratios. We have 
 
 „_l ^ i-2a b{b-iY 
 ~k~a{a-iY I- 2b , 
 
 m I — 2a c{c—iy 
 
 5=v= 
 
 k a{a—iy I — 2C 
 
50 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 Using N=—, r- as an abbreviation we obtain 
 
 a{a—iy 
 
 {l-2b) 
 
 Write now bc = p and note that b+c=i — a so that b, c are the roots of 
 
 u'—{i — a)u+p = o (69) 
 
 Expressing the symmetric function of b, c in terms of the coefficients of this equation 
 
 R+S = N . [-a-+a'+P(-2«-+3a-i)+4P-] ^ ^ 
 
 4P+2fl— I ' 
 
 Similarly 
 
 4P+2a— I ^' 
 
 We now vfTite RS=p, R+S=q, and p = m—a 
 
 N'ni'im—a) _ 7V[4m'— w(2a'+5a+i)+a+i)'a] 
 
 4W— (2a+i) 4OT— (20+1) 
 
 If now -= r we have a cubic and quadratic for m. 
 
 -m 
 
 Nm^-\-m^{— aN — i^r)-\-mr{2a^-\- ^a-\-i) — r{a-\-iya = o 
 4Nm'—in{N[2a'+sa-\-^]-\-Aq)+N[a{a+iy->rq{2a-\-i)] = o 
 
 in) 
 
 Using the indicated end-term multipHers whose determinant is 4 no extraneities are intro- 
 duced and a second quadratic results. 
 If this is 
 
 A'm^-\-B'm-\-C' = o (74) 
 
 and the previous quadratic is 
 
 Am^-\-Bm-\rC = o 
 
 the second order determinants which enter the eliminant are 
 
 AC'-A'C=Aq'{2a+i)+i()qr{2a+i)-Nq{&a^-\-i2a'-\-']a+i) 
 
 -m{2a^+Sa'+Sa'+^a'+a) (75) 
 
 ' AB'-A'B=i(yq'-(i/^qr-\-j^NqUa'+Aa+-i)-\-N'{Aa'-\-?'ai+a'+2a+i) (76) 
 
 BC'-B'C=q'(,2a-\-iy+2Nql2a+i){a+iya-\-^qr{-4a^-T,a-i)+N^{a+xYa^ (77) 
 
 Collecting the coefficients of the powers and products of N, q, r the eliminant takes the form 
 
 Q= — ibq^ria) 
 
 -q^'N{2a-\-\y{2a-\) 
 
 ^-bAqVia) 
 
 -f49^'-iV(2a+i)^(a-i) . 
 
 -9W^(2a+i)(6a3-j-5a2-a-2)a ' ^' ' 
 
 -qm{a-\-iy{6a>+a'—2a-i)a' 
 
 +griV^(20-t-i)(6a3-f-7a^+4a— i)(a— i) 
 
 -N^{a+iy{a-i)a\ 
 
 and writing for N its value in terms of a, multiplying throughout by (a— i)' a^ and putting qr = p 
 we have the eliminant as an equation of the tenth order in a, containing two parameters p, q 
 among its coefficients. Arranged in powers and products of ^, 9 it is : 
 
EQUATION FOR THE SIDES SI 
 
 F(a : p,q) = q'{2a+iy(2a—iy(a—iya 
 — i6pq'(a— i^a^ 
 + 64/>^(a-i)'a3 
 
 -4pq(2a+iyi2a-i){a-i)''a , , 
 
 -q'i2a+i){6ai+Sa'-a-2)i2a-iy{a-iya ' ^'^' 
 
 +9(6a3+a'+2a- i)(2a- i)^(a+ i)3(a- i)a 
 +/>(2a+ i)(2a- i)^(6a<+a3- 30^- sa+ i)(a- i)^ 
 -{a+\y{2a-\ya^ . 
 
 The result may be checked by substituting />= i, 9= 2, i.e., k = l=m when it reduces to give 
 
 1 (3=^V 17) (i*Jl£7) (l^Jiil) 
 4 ' 8 ' 8 
 
 which agrees with the previous result for this case. 
 
 To obtain 6 as a rational function of a the order of eltmination must be changed. With 
 
 this order m and hence be is obtained as a rational function of a. 
 
 R 9 
 
 If we write ^=/; T;=i?> we have two cubics in b: 
 
 N N 
 
 b}-2b'+bii+2f)-f=o, 
 
 b>+b'{:ia- i) + 6(3a^— 2a+ 2g)c+ {ai—a'+ 2ag— g) = o, 
 
 by carrying the elimination to the penultimate step by the end-term method we have 
 
 6[-2/'-|-2g^(2a-i)-4/g(a-i)+g(8a3+3a^-2a-i)-2/(a+i)^a+(a'-i)(3tf+i)a"] 
 + [g"(2a-i)'+2/g(-6a'-i)+/'+^(2a-i)(2a3— 2a'-3a-i)+/(-6c''-3a3+3a'+2a) 
 (a^— i)(a'— 2a— i)a"] = o. 
 
 In this expression 
 
 IV (a-i)'g ~| mV {a-iya "1 
 
 ■^~^L(-2a+i)J   ^~'^ L(-2fl + i)J 
 
 making these substitutions and dividing by , ,^ we have unless a= i, 5, or o 
 
 b[— 2t'{a— lya— 4lm{a— i)''a+2W^(a— i)J(2a— i)a-|-2/^(a-|- i)^(a— i)(2a— i)a— 
 
 mk{a—i) (2a— i) {&a^+sa'—2a—i)+k^{a+i) (30+1) (2a— i)^a] \ ,„ •. 
 
 + [l'(a-iya+2lm(a-iy(-6a'-i)a+lk{a-i){2a-i){6ai+Sa'-Sa-2)a+k^{a+i) ( 
 (a'— 2a— i)(2a— iya-\-m'{a— i)-'(2a— iya—mk{2a— i)^(a— i){2a^— 2a^— 3a— i)] = o ' 
 
 In obtaining approximate solutions substitution in this expression is more laborious 
 than the solution of a cubic. Hence on this ground the solution by the o- chain (11) and (3) 
 is preferable. In practice however the method of trial on the equations (22) is more expedi- 
 tious than either. 
 
 Theoretically we note that the sides involve merely an irrationality of the tenth degree 
 and the cubic which occurs in the o- chain is not necessary but convenient. The cubic irra- 
 tionality is then not accessory in the technical sense. 
 
 p{a : p, q) is of course unsymmetric m {k, I, m) and as /> = , ^ , q= — 7 — , it can be denoted 
 by F{a, k : l,m) and is associated with two other equations by cyclic change. 
 
52 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 That these equations which have the same form are all irreducible can be established thus: 
 
 Suppose they reduce in {k, I, m), choose a corresponding set of factors, and from them form 
 the equation whose roots are o- expressed as a symmetric function of a, b, c. 
 
 The coefficients of this equation will be symmetric functions of k, I, m of order zero and 
 hence rational in a, /8. 
 
 For 
 
 kl-\-lm-\-mk klm 
 
 and writing k+l+m = P, kl+lm+mk = Q, klm = R every coefficient when the denominators 
 are cleared will be the sum of multiples of such terms as R^Q^P^ where on account of the homo- 
 geneity 3^+2);+^=w. 
 
 On dividing such terms by P" we obtain terms of the form 
 
 R^Q^ _ R^Q^ 
 
 Pl-i Pii+2r\ 
 
 = l3-(a- 
 
 The equation will be satisfied by the corresponding values of o- if the equations in a, b, c 
 are satisfied and will be of an order less than lo. But F((t, a, yS) which will be the result of 
 carrying out the above process on all the factors oi F {a,k : l,m), etc., does not reduce, which 
 contradicts the hypothesis of these equations being reducible. 
 
 l6. THE MONODROMIE GROUP OF THE EQUATION FOR THE SIDES 
 
 The form in which the equation emerges from the elimination (79) renders the determina- 
 tion of critical points with the corresponding binomial approximations and cycles of the roots 
 easy. 
 
 We deal first with the values p='xi , q=cc . H p = kq and A is finite the common factors 
 
 of the terms of highest order in q are (a — i )%. For a=i = n,q=- the terms to be considered 
 
 are of orders «', kw^, k^u, k^. 
 
 The first three give 2 two-cycles and the last pair a stationary root at a = i , there is a single 
 root at a=o and four other roots dependent on A. and in general distinct. 
 
 This approach to the point p— 00, 9= <» gives then an element of the group of the type 
 (12) (34). 
 
 If we write ^p = q^ and then 9=- we get the same state of things at fl= i but at a = o there 
 
 is a two-cycle, the other roots are an odd one at a= 5 and a pair not forming a cycle at a= — 5. 
 Since on the Neumann spheres for p, q the points used are as close as we please we may iden- 
 tify the totality of roots at a = i in the two cases. For a real approach the two pairs of roots 
 at a= I are as near as we please two pairs with equal values and opposite sign and as this char- 
 acter is the same for all finite values of A. no discriminantal branch is crossed and the pairing 
 may be identified in the two approaches. By the product of the two substitutions a single 
 transposition is obtained. 
 
 For the approach with A.=o, or by writing q=o and then p=~ we have for a—i = n a 
 
 TT 
 
 binomial approximation of the type «' = 5r" giving a seven-cycle, and for a near o a three-cycle. 
 By the continuity of the roots and the approximate equality of p, q in the cases A.=o and 
 
REDUCTION OF EQUATION FOR O" IN CASE OF EQUAL BISECTORS 53 
 
 X finite but small we may identify the two roots which enter the single transposition with two 
 of those in the three-cycle, and conclude that the seven-cycle contains the four roots which 
 entered the pair of two-cycles in the other approaches. 
 We then have as elements of the group 
 
 A : (afiySe!:r,)(eaK) and B : {Oa) 
 
 At the point q = \p=o there is a four-cycle at a= — i, a second four-cycle at a = ^, and a 
 two-cycle at a = o. This gives the element 
 
 C : {abcd){efgh){ij) 
 
 From combinations of A'' and B the subgroup G(,{6<jk) is generated and one of these roots must 
 be connected by C with some root in the set of seven in A^. By using the powers of A^ as trans- 
 formers every single transposition of the ten roots can be produced and hence the group do! 
 The monodromie group being the symmetric group by Jordan's theorem the algebraic 
 group is also the symmetric group. 
 
 17. THE REDUCTION OF THE EQUATION FOR w IN THE CASE OF EQUAL BISECTORS 
 
 In this case two angle-bisectors are equal. If K = L and Ti^='» 
 
 {m+2Y „ {m+2y 
 
 a= 1 li= . 
 
 Taking m as the single parameter and substituting for a, /S and writing <T = ^[m-\-2) the 
 equation when divided by (w+2)"° becomes 
 
 -|- (40w^-|- iS2w-f 156)^^-1- (w?5— 62W— ii6)^^-j- (—w^-f4W-f-36)f-f (w— 4) = o 
 Since in general y8= — , o-= w-f 2, T=m\s suggested as a solution. This gives i= i which satis- 
 
 T 
 
 fies the equation and dividing out the factor we have: 
 
 i6m^i^-\-\bm^i^-\-{—2/^m^ — 9>om^)iT^{—2^m^—2i\ni')^^-\-{gm^-\-io%ni'-\-i^2m)i^ \ 
 
 + (9OT3-|-i4w"-56m)l'>-f (-w3-38w^-95OT-72)^3+(-»M34-2W^H-57W-f84)^' (82) 
 
 + (2w'— sw— 32)^4- (-w-f4) = o / 
 
 This must be of the form 
 
 where the coefficients are polynomials in m. We see at once that D=i. Recalling the special 
 cases 
 
 a=3, /3=27, w=i 
 
 a = 4, /3=54, w=4 
 where the second factor is 
 
 4^^+o^^-iif-|-3 
 16^34-0^- 2ol-fo 
 
 respectively, and seeing that the polynomials cannot be of higher than the first order in m we try 
 
 4mi'+o^'-(sm+8)i-im-4) (83) 
 
 which divides with quotient 
 
 [2mii+mi'- im+s)i+iY (84) 
 
54 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 The solutions for D{a, li) = o are then 
 
 i) <T=2-\-in, T = m : which gives y=<=^, z=oo, and an infinite triangle whose sides are 
 [5, + 00 , — "» +5] a solution for every m. 
 
 This has been discussed in the case of the equilateral triangle. 
 
 2) The factor 4m^'— (^m+8)i— (m— 4) gives three isosceles triangles. Its discriminant 
 
 being 
 
 28 
 
 -m{2'jm'+gm+52) (85) 
 
 there are three real roots for positive m's, one real root for negative m's. 
 The distribution of m over D(a,l3) is as follows: 
 
 o<m<i from a = 4, /3= 00 to the cusp 
 
 i<m from the cusp to a= 00 , /3= 00 
 
 o>w> — 5 the hyperbolic branch in the second quadrant 
 
 — §>w> — 2 the branch in the third quadrant 
 
 — 2>m the parabolic branch in the fourth quadrant 
 
 The origin is w= — 2, the asymptotic points m = o, m=—\. 
 
 For o<m< i the three real roots of this factor are those called 1, 5, 10. 
 
 For m>i they are i, 7, 8. 
 
 The root (i) gives an isosceles triangle with internal bisectors equal. 
 
 The root (5) gives an impossible triangle with real bisectors. 
 
 The root (10) gives a real possible isosceles triangle with equal external bisectors at the 
 base which is the smallest side. The larger bisector is internal. 
 
 The root (7) gives an impossible triangle, while (8) has a real isosceles triangle, the equal 
 bisectors being external, the smaller one internal. . 
 
 For negative m's if w> — | the point representing the real solution (which gives a complex 
 triangle) falls in XXX 2. In the other cases the triangle is real and impossible, f or — 2 < w < — 5 
 the point falls on the boundary of ©2 and XXX ^, for w< — 2 on the boundary of ©, and XXX y 
 
 3) In the case of the squared cubic factor the discriminant is 
 
 — [gw^+aSW-t-QW-f 216] (86) 
 
 27 
 
 For m>o there are always three real roots. The cubic factor of the discriminant has one real 
 root only, 
 
 w.= -4.987 . . , a=-.997 . . , /3=5.3S4 . . . (See §12.) 
 
 For m<m, there are three real solutions, for o>w>Wi only one. Along the latter 
 part of D{a,P) two pairs of complex roots become equal. The point w, is the crossing point 
 of T, D2 in the <i>, r plane and a point of tangency of T and D in the a, y8 plane: the real four- 
 point. For o<w<i the solutions fall in (2, 3), (6, 7), (8, 9). 
 
 The (2, 3) solution is a real possible triangle with one of the equal smaller bisectors internal. 
 The (6, 7) solution is impossible. The (8, 9) solution like the (2, 3) has one of the equal smaller 
 bisectors internal but the smallest side is much smaller, the opposite angle being always < 20°. 
 
 For I < w < 00 the solutions fall in (3, 4), (5, 6), (9, 10). 
 
 The (3, 4) case has the less bisector internal. The (5, 6) triangle is impossible. 
 
 The (9, 10) case has one of the larger bisectors internal. 
 
REDUCTION OF EQUATION FOR THE SIDE IN THE CASE OF EQUAL BISECTORS 55 
 
 In the case of isosceles triangles it is obviously unnecessary to solve two cubics. The 
 cubic for the side a is 
 
 4(w — 4)a^— (9OT — i6)a^+2(3w— 2)a— »j = o 
 whose discriminant 
 
 — (27OT'+9W+32) (87) 
 
 27 
 
 is algebraically identical with the i discriminant involved. 
 In this case b = a,c=i — 2a completes the solution. 
 
 [For K = L,a-=b,mM' = K^] 
 
 18. REDUCTION OF THE EQUATION FOR THE SIDE IN THE CASE OF EQUAL BISECTORS 
 
 2W m' 
 
 If l=m, q = -j-, p = -^. 
 
 Writing 9 =2/?, p = R' in the equation the coefficient of R'' vanishes identically and (2a— i) 
 is a factor of the remaining terms. 
 The residue is 
 
 8i?3(2a+i)^(a-i)sa^+i?^(2a+i)(2a-i)(a-i)3(-i8ai-i9a3+a^+3a+i) ( ,gg, 
 
 + 2/2(2a-i)Ka+i)2(a-i)(6a3+a^-2a-i)-(a+i)H2a-i)3a=' = o ( 
 
 The case of isosceles triangles gives 
 
 SR{a—i)a^—(2a — i){a+i)'' = o as a factor. 
 
 The other factor is RH2a+i)'{a-i)' - 2R{2a+i) (20-1) ia-i)'{a+i)a+ (a+i)' 
 (2a— i)^a^. This is, as it should be, a square. 
 For l = tn v>e have then for a 
 
 i) a = i giving the infinite triangle discussed in (§ 5) 
 
 2) ;t(a+i)^(2a-i)-8w(a-i)a^ = o (89) 
 
 for the three isosceles triangles, b = c = . 
 
 3) Three triangles each a double solution from 
 
 k{a+i) (2a— i)tt— w(2a+i) (a— i)" = o (90) 
 
 In the last case after finding a we have a quadratic for b and c. 
 
 /For 6+c=i— a and since h — nic; Diib, c) = o 
 {b + cY-2bcib + c) + 3bc-2ib + c) + I=0 (40) 
 
 from which 
 
 a' 
 bc=— , - and with 6+c=i— a 
 2a+i 
 
 a quadratic for b, c. 
 
 Hence the solution of a cubic and quadratic is sufficient. 
 
 It will now be shown that no simpler solution exists for a general value of -: . 
 
 If we discuss the corresponding factor for F(a, k : l,m) for k = l the method of elimination 
 (§ 19) obviously fails to distinguish a and b in any way and we get a sextic factor for the Dj 
 case k=l, a^b. 
 
REDUCTION OF EQUATION FOR THE SIDE IN THE CASE OF EQUAL BISECTORS 57 
 
 This factor gives all the a's and b's needed to make up the three triangles involved, the 
 connection of the pairs being determined by 
 
 D,{a,b) = ia+by-2ab{a+b)+sab-2{a+b) + i=o (40') 
 
 This rational relation holding for three pairs of the /oots the group of the sextic reduces. 
 In fact we have shown how by solving a cubic for c to find a and b by solving a quadratic. If 
 the sextic is irreducible, the group is transitive and cannot further reduce than is indicated by 
 this solution. 
 
 The sextic, obtained by a process similar to that which afforded the corresponding cubic, is 
 OT'[(a-i)3(8a»-4a-i)a]+OT/[(a-i)(-8ai+4a'-4a'+Sa-i)o]+/"(3a^-i)^ (91) 
 
 The other factor is 
 
 w(4a—i)(a—i)'— 4/(20— i)'a (92) 
 
 the tenth root being infinite. 
 
 The irreducibility of the sextic is easily established. 
 
 If the roots are paired as (i, 2) (3, 4) (5, 6) we may write the rational relation Diii, 2) = 
 shortly as (12) =0. 
 
 (The function (i 2) + (34) + (56) = is in G^i and distinct from its conjugates.) 
 
 The group is as in general in the case where a general cubic with a parameter and a quadratic 
 give rise to a sextic on eHmination G^g generated by the substitutions (12) : (i35)(246) : (i3)(24). 
 
 As the Galois resolvent may be taken the equation of degree 48 rational in m which has for 
 roots 
 
 a, — ^,+(0(02 — 62)+«^(a3 — 63) and its conjugates. 
 
 Each root may be rationally expressed in terms of any one of these. 
 
 It is interesting to note what happens to the rational expression of b in terms of a in general 
 valid for the tenth-degree equation for a. 
 
 In this case k = l if we solve F{c, m : k,l) = o for c, we have a cubic for c if a=t=6. 
 
 The rational expression (80) for a becomes 
 
 [4ck-mic- 1) {c^-2c-i)]^^{2c=i=i)k+m(c+i)c'^i2c-iy 
 
 m(c-i)(3c+i) /■-, 1 -M 1 ~^,-L,^,2 
 
 = 5, a is indeterminate, the limiting valu« 
 (§5)- 
 
 m(c-i)(3c+i) (2c+i)*+ot(c+i)c» (2c-i)» 
 
 for c = 5, a is indeterminate, the limiting value leading to the infinite triangle previously discussed 
 
 For the isosceles case 
 
 whence a = b = . 
 
 2 
 
 For the case a^b however 
 
 k^ (c-i)(c+i)' 
 m 8 
 
 m (2C-I-1) 
 and the expression for a becomes indeterminate. The limiting value gives 
 
 (2C3-C»+l) 
 
 a= — 
 
 (2C+I)(C-I)' 
 
 V 
 
DO- 
 
 c 
 
 >o 
 
 
 C" 
 
 
 I 
 II 
 
SURFACE r (o-, a, 0)=o   59 
 
 whence 
 
 , 2C' , , ' 2C'i2C' — c'+l) 
 
 (2C+l)(c-l) (2C+l)'(c-l)' 
 
 This value for ab is however inconsistent with Diia, b)=o which gives 
 
 ab = 
 
 (2C+I)- 
 
 Hence the rational expression fails as was to be foreseen from the group theory. 
 
 19. THE SURFACE Fi<T,a, /3) = o 
 
 Since a is single valued we take the « axis vertical. /8 and o- to the right and up respectively 
 in the plane of the diagram. (Fig. 14.) 
 
 This shows the ridge lines T = o, Z>3 = o and the lines Z),=o and <^— t— 1=0 marked <f> 
 where the discriminantal cylinder has an ordinary intersection with the surface. 
 
 The projection of the asymptotic cylinder a = 00 on the /3, o- plane is marked. The cross- 
 sections give a descriptive idea of the surface. (Fig. 15.) 
 
 Drawing to scale is unfortunately impossible as the small loop of the asymptotic cylinder, 
 only extends to ^= .016 . . and the real region commences at 18=27 where the ridge lines 
 have a triple tangent. 
 
II 
 
 I. THE EXTERNAL PROBLEM 
 
 The formulas for the external bisectors being 
 
 ^^^(^b+c)ia+b-c)bc 
 
 {b-cY 
 
 {c—a 
 {—a-\-b-\-c){a—b-^c)ab 
 
 {c—a) 
 
 ( — nJ-h 
 
 {a-by 
 
 as in the internal case we use ratios and write 
 
 K':L':M'::l:^:- 
 k I m 
 
 a+b+c=i 
 
 }il-\-lm-\-mk klm 
 
 Expressing a, ft in terms of x, y, z elementary symmetric functions of the sides and writ- 
 ing it;=i we have 
 
 Uy'-y-3zy 
 
 o. ^ 
 
 4y*—y^—6y'z+gz'—syz+z 
 
 Q^ -(4/ - y-32> '^^^ 
 
 z{^y—%z—i){—^y^+y-\-iS,yz—2']z^—^), 
 
 The cubic expression in the denominator of /3 is 
 
 P*=[{a-b){b-c){c-a)]' 
 
 the discriminant of the cubic whose roots are the sides. 
 We notice also that 
 
 Da 
 
 J^ / \ 
 
 4y* —y3— 6y'z + 92' — 3)'2 + z 
 
 Hence all isosceles triangles have a =4, /3= 00 . 
 
 Points on 0=4 for which /34=oo are reached only in the y 2 plane along 43)'— 32 = as limits 
 for y= 00 , which gives infinite sides with complex approach. 
 
 As in the case of the internal problem it is convenient to eliminate in two ways. 
 
 Writing 
 
 2. THE FIRST ELIMINATION 
 
 60 
 
THE GROUP OF THE EQUATION 6l 
 
 we obtain 
 
 4-°^ 3(4P<^+3'^+/>+i) \ 
 
 y3(4-«) p(<T-p)(p-a+l)' i^^' 
 
 8io tr(— 9p»+9po— 9p+8(T)(3or— 3p+i) / 
 
 From the first of these 
 
 , (3tr+l)(l-a) 
 3"= (I + -) ' 
 
 and by substitution in the second 
 
 ^{a-4)^{a+iy(i+ao-)[~a'(a+iy+a{ga'+io<T+s)-4] . » 
 
 This equation of the first order in /8 with no common factor of the coefiicients of /S" and /S" 
 is necessarily irreducible in R{a, /8). 
 
 Since all subsequent operations must depend either for their necessity or their effectiveness 
 on the nature of the group of the equation it is proper to determine this in advance if possible. 
 
 3. THE GROUP OF THE EQUATION 
 
 For a=o the equation (6) reduces to /8o-((r -f-i)'=i which has a binomial approximation 
 5*=j8 for 0-=- at 5 = 0, /3=o, and hence a cyclic substitution of order three among the roots. 
 
 27   
 
 At y3= , (T= —\ 3. double root occurs giving a two-cycle and establishing the symmetric 
 
 4 
 group on these three roots as a subgroup of the monodromie group. 
 
 For /8=o the equation reduces having a pair of squared factors. Further equalities occur 
 at a =0, I, 3, 4. These facts may be represented in a graph where the upper curve is double 
 (Fig. 16). 
 
 For a4=o, 1,3,4 a proper approximation is of the form «/?= (o-— a,)" where <r, is one of the 
 doubled roots. The two values of o-, lead to a substitution of the form (12) (45). At a=i, 
 o-= — I and at 0=3 0-= — |, the crossing points of the curves, no three-cycles occur, but at 0=4 
 the approximate forms are 3(125'+ a)'=— 4 Pas' forcT=o+5and" = 4+fl, and also 485'+ a = o at 
 0-= —^+5. These give a substitution of the form (14) (25) (36). The two substitutions may 
 be denoted by U and V respectively. 
 
 For a=3, /3=27, the equal bisector point, the equation reduces to a perfect sixth power. 
 The approximation is 
 
 29165'— loa where <r=—\-\-s, a = 3+a, P=2j. 
 
 At this point we have a six-cycle. 
 
 Without further specifying the identity of the roots we may now prove that the group is 
 the symmetric group on six letters. 
 
 Assuming that the six-cycle is (123456) the three-cycle falls in this either with an adjacent 
 pair or in alternate positions. That is either i, 2,occurin (a,6, c) or (a, 6,c) is (135). 
 
 In the first case transforming (12) by the six-cycle we get (23), (34), (45), (56), (61) and 
 
62 
 
 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 compounding with (12) successively (13), (14), (15) and having now all single transpositions, all 
 the substitutions of the symmetric group follow. 
 In the second case we have 
 
 5= (i 23456) :r= (13s) and also (13), (35), (15). 
 
 By using 5 as a transformer we add (246) : (24) : (46) : (62) and the roots fall so far into two 
 disconnected sets. The cube of S is (14) (25) (36) and V is of the same form. If the two are not 
 identical V either keeps the same division or connects the odd and even. In the latter case the 
 transforms of the transpositions already at hand give all transpositions, in the former cases U 
 can be used as a transformer on V and the same result is reached, namely the symmetric group. 
 As in the previous problem the algebraic group is also the symmetric group. 
 
 The equation in <t, although convenient for determining the irreducibility and the group, is 
 under disadvantages in other respects. The expressions for y, z are rather complicated and fail 
 to give determinate values in rather a large number of special cases. These occur for the 
 following values: 
 
 a=o /34=o when three roots are infinite but p has the factor i-|-a<^ in its denominator 
 and is indeterminate. 
 
 a= I /34:o when o-= — i is a fourfold root and p is indeterminate. 
 
 a=3 when o-= —J is a threefold root, p indeterminate. 
 
 It is not necessary to conclude that a(a— i)(a— 3) is a factor of the discriminant, for the 
 definite values of o- which arise point out a discriminantal point rather than a discriminantal 
 locus. ' 
 
 To avoid these things which cannot all be successfully dealt with by limits a second method 
 of elimination is convenient. 
 
THE SECOND ELIMINATION 63 
 
 4. THE SECOND ELIMINATION 
 
 Writing 
 
 P±Z4)^B:^-^=A • (7) 
 
 a a ' 
 
 we have 
 
 y-4y'+3 z _R xjjx 
 
 zUy-8z-T)-^ ^^> 
 
 (3>'-i)[z(6y-i)-/(43'-i)] _ . /oM 
 
 These are quadratics in z. The elimination is simplified by writing z=(4y'—y)t and 
 eliminating t, after division by (43^ — y) . The values y = o,y = \do not in general give solutions, 
 with 2 = 0. 
 
 Arranged in powers and products oi A, B the result is 
 
 F{y,A,B)= A'B'Uy-inSy-sVy 
 
 +AB' [4y-i]^[3y-i][i28/-96y+i7]3' 
 
 -gAB Uy-iVlsy-'^] 
 
 +8B' [4y-iY[3y-^]'Uy-i]y 
 
 -2yA Isy-i]'' 
 
 —SB [33;-i]3[6y-i]=o 
 
 (9) 
 
 For « = 3, 18=27 • ^ = ~l> B=—g, this reduces to 
 
 36(2^—1)^=0 
 
 giving a unique complex triangle with equal external bisectors. 
 Here y = \,z='h) ^^^ 
 
 a \ b : c=2-\-^\/2 — ^1/4 : 2+w 3y 2 — «)^ •S] 4 : 2-\-oi' ^y. 2 — <o3^/ 4 
 
 where <o= (-^+'^'^) . 
 2 
 
 That this equation is irreducible and has the general group follows from the fact that 
 
 y is a, rational function of <t, a, j8, while A , B are rational functions of a, fi. 
 
 Expressed as a rational function of {y, A, B), z is given by 
 
 (^y-y)[SABUy-i)y+&Bi3y-i)y+gA] , . 
 
 3AB[64f-28y+s\+SB[i8y^-gy+i]+2'jA ^ ' 
 
 This is the form obtained at the penultimate step of the elimination and has the disadvan- 
 tage of being indeterminate for 27^ =B, y = \ in which case the form 
 
 o-.. ., , I 3^(4y-i)(8y-3)+8(3y-i)' ,. 
 
 S2-4y 1+ 8^5(4y^-y)+85(3y-y)+9^ ^"^ 
 
 or 2= J for the special values, may be used. This is identical in virtue of F{y, A, B)=o. 
 
 The rational integral expression for z in y, ^4 , B appears to be too complicated for use. 
 
64 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 5. MULTIPLE ROOTS 
 
 The method is that of the internal case (§ 10). If a-{-b-\-c=i and z = abc, we have, 
 
 z taking the place of < as a parameter in each equation of the set of three but not being a propor- 
 tionality factor as in the internal case, the algebra becomes a little more complicated. 
 For variations near a solution we have 
 
 ia\—\a?-\-iia}—2a{x — 2kz)-\-z{^\ — li)\ 
 -\-dk\2a^ — a-\-\'^z \ (13) 
 
 -\-dz\2a'k-\-a{s,—K)-\-'&]i'^ — Q 
 
 At ordinary points there are three equations of the form 
 
 da=Adz+Bdk, 
 
 db = Cdz+Ddl, ] (14) 
 
 dc=Edz-\-Fdm 
 
 As in the internal case we conclude that 
 
 A+C+E=T 
 
 is a factor of the discriminant and also that k = l, 0^ = gives rise to double roots. Whether 
 or not D2 recurs as in case of the internal problem is more troublesome to determine on account of 
 the complexity of the expressions, and the question is postponed. 
 As before we investigate these discriminantal factors 
 
 ~ -4a5-l-7a''-|-i6a32-4a3-|-(i6z-|-i)a"-8az+i62» ^^' 
 
 Expressing the summand as far as possible in terms of y, z by means of 
 
 a^—a^-\-ay—z = o 
 ^^ 2 [(2 /-8yz-z)a ' -f(43'z-y'-8z''+2)a+(y2-6z'')] ,^. 
 
 ~ i(y-4z)a'-f-(i6y2— 4/+y-S2)a-f43'2-z)] 
 
 (17) 
 
 and the explicit evaluation of the symmetric functions leads to 
 
 z''[— 6i44y^-|-28i6y— 32o]-(-z3[4og6y3— 24ooy^-|-444y— 25] 
 -|-z^[ — 896y''-f 6o8y3— i36y^-f ioy]4-2[64y^— 48y''-|-i2y3— y^] = o 
 
 which reduces to 
 
 r=z(4y-i)[4>'(>'-62)-y-Sz)][4(y-4z)='-(y-52)] = o (18) 
 
 For convenience we call 
 
 4y(y-6z)-(y-sz) : T,, 
 4(y-42)^-(y-52) : T,. 
 
 As in the internal case the vanishing of the denominators must also be discussed. 
 The condition that the numerator and denominator for a should both vanish is 
 
 rj[z»(32y-z)+z(i28y3-64/-y+z)-f(-32'i+2oy3_3y^)] = o (19) 
 
MULTIPLE ROOTS 65 
 
 The second factor is however an extraneity as it does not also consist with o^— a^+flj'— 2 = 0. 
 
 Tj may simply be expressed by ^= — 4 or as a(a— i)+4z = o. 
 
 The latter relation multiplied by the corresponding expressions in b, c and expressed in 
 y, z is z^Ti-o, while if k be replaced by —4 in the equation for a (12) this becomes the square 
 of a(a— i)+4z=o. 
 
 T2 = ois not however a proper discriminantal factor, for the value ^= — 4 has no relation to 
 the homogeneous problem, and if a+b+c^ i the a equation does not become a square. More- 
 over the expression for 8k becomes a perfect square for ^= — 4 and we do not obtain two dis- 
 tinct solutions in the neighborhood. 
 
 Ti in the a, j8 plane is represented by /8(4— a) — 8a = o, in which the factor F' enters so that 
 
 the triangles given along Ti = o are in a sense associated with isosceles triangles. None are 
 
 real or possible. 
 
 (±y^ — y) 
 The factor T, is purely extraneous. If r,=o, ^=(—zf~\ > which is satisfied by a = §, 
 
 ■■\,c=\ : y=i'6,2 = BV 
 For these values 
 
 Namely for T, = o 
 
 /3=co , a = 4 but |8(a — 4) = oo. 
 
 4(i6y-s) 
 
 a-4=- 
 
 T ' 
 
 64/ -5 
 128(3^-0(24^-5)3; 
 
 (i6y-s)^ 
 
 For this set of values the equation for o- has roots o, o, — i, — i, — j, and — | and the 
 last named, a single root, is the value giving the triangle. 
 
 Taking up the factor z we have for 2 = 0, 5= =» , A = — . _ . if y+o, y^j. For B= 00 
 
 the equation Fiy,A,B) = o becomes (4^- i)^3;[i63'3(4.4 +3)^- 8^^(32^^4-5271 + 2i)-f-y(84^"-|- 
 
 147^-1-64)— (9/l-t-8)(yl-|-i)] = oandy= , , is a single root. Hence z = o is not discrimi- 
 
 (4^+3J 
 nantal unless ^'=5 or y = o and these points occur on the other factors. The factor 2 is then 
 extraneous. 
 
 We are left with (4^— i) which is in fact a discriminantal factor and will be referred 
 to as T. In the A, B plane it is represented by 27^1—5 = and in the a, ji plane by 
 
 (a-4)(/?+27) + 8l = 0. 
 
 Using a similar notation to the internal case we write D2 = o as the representative of the 
 equal-bisector non-isosceles locus. 
 
 11 k = l, a^b, there are in the {a, b, c) plane three factors of the form 
 
 D2{a,b) = 2{a-\-by-\-i^ab{a-{-b)-s{a-\-by-2,ab-\-4r(fl+b)-i = o. 
 
 Expressing the (a, b) form in c and z = abc if a-\-b-\-c= i we have 
 
 2C''— c3-(-4C2— 2 = 0. 
 
 Reducing by c' — c'-f-yc— 2 = 0, we have c = o or 
 
 31 — 62 
 
 c——- . 
 
 231—1 
 
66 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 Eliminating c 
 
 Z?2(3',z) = i6)'32— 4}'^+2i62'— i8o3's'+3oy'2+y'+362^— i23'z+2 = o (20) 
 
 This factor of the discriminant has of course the same representation in the a, jS plane as in 
 the internal case. 
 
 For D2 = o the sextic becomes a perfect square. If ^ = /= i, 
 
 £ = w»-2W-8, A = - ^ , , ', . 
 
 (w+2)3 
 
 With these values F{y,A,B) becomes 
 [i6>^(w— 4)^w— 8/(m— 4)(w^— 8w— 2)-)-j'(w3— 2iw'+75W+S3)+(w'— 8w— ii)p (21) 
 
 In the reduction the factor (w+2)'' is removed from both numerator and denominator. 
 For m= — 2, = 0, /8=o and the affair is indeterminate. For this case however A = ^ ,B = o 
 with AB= 00 , AB^ finite, and the limiting values serve. 
 
 We conclude from the set of three pairs of equal roots that as in the internal case D\ is 
 a probable factor of the discriminant. 
 
 An expression for the discriminant in the (^,2) plane may be obtained by equating the 
 
 values of the derivatives — as given by the two equations (8) (8') and eliminating A , B. This 
 
 dz 
 
 process gives 
 
 D{y,z) = {Ay-T)Uy^-y-2>zy{D2{y,z)] = o. 
 
 Hence as ^■f—y—2,z vanishes only for a = o, ;8=o no new factors are obtained by this 
 method. 
 
 6. THE NODAL CURVE 
 
 Among the factors of the discriminant of F{y,A,B) is one which relates to equaUty of the 
 y's only without implying equality of the 2's and hence not discriminantal for the problem. 
 This arises from the nature of the elimination process: the two quadratics from which z was 
 eliminated may become identical. There are two conditions in y,A,B from which y can be 
 eliminated. The result is 
 
 AB{a-A)P=o 
 
 where P expressed in a, ^ is 
 
 P = ^'(a-4)(a-9) + 54^(a-l)(2a-9)-|-729a(a-i) (22) 
 
 When P = o, y is given by 
 
 y=[|8(a-4)(a-9)-27a(a-i)]+[l2/3(a-4)(a-3)] (23) 
 
 Since a= oo does not cause ^ or 5 to vanish the effective factors are (i — a)(a— 4)»P. 
 
 The line a= i is in fact a nodal line on the surface F{y,A,B) = o. P= o is also a nodal line 
 while a— 4=0 gives an infinite cuspidal edge. 
 
 The last locus is discriminantal for the problem while the others are merely so in relation 
 to the choice of y and the elimination method. 
 
THE DISCRIMINANT 67 
 
 7. FINITE MULTIPLE POINTS OF ORDER HIGHER THAN THE SECOND 
 
 If A = o, r=t=o and if no other possible factor of the discriminant vanishes the only possi- 
 bilities are 4-points and 6-points. 
 
 The discriminant of the cubic factor to the square of which F(y,A,B) reduces for D2 = o 
 is to a numerical factor 
 
 (w— 4V(ot— i)*(w'+wj+7). 
 
 There should also be counted m=<x (tt=oo,|8=oo). 
 For m= I : 1=3, ^= 27 occurs the 6-point y=^ already discussed. 
 For »»=4 : 1=4, )8=any finite value, four values of y are infinite. 
 The only real finite y8 on D2 = o and on a = 4 is 54. 
 
 The complex factors are intimately connected with ot = 4 in that if w= , 
 
 3 
 
 (w— 4)(w'+w+7) = 27(»3— i). They give 
 
 a 
 
 _i±VjZl 
 
 y3=i\(-i4±i/-3) 
 
 and are intersections of T and D. 
 
 For r'=o we have y=\ which is a double root for all ^'s when 2'jA=B and a fourfold 
 root when 34'+3^ + i = o. These values give the complex 4-points just mentioned and there 
 is never a 6-point. 
 
 The list of multiple points is 
 
 then 
 
 
 
 ''=3, /3=27 
 
 
 a 6-point 
 
 on D2, P, not on T 
 
 „_(3*l/-3) 
 2 
 
 - 
 
 two 4-points 
 
 on Da and T 
 
 a=4, /3any 
 
 
 a 4-point line 
 
 
 a=o, /8=o 
 
 
 is indeterminate 
 
 
 a 
 
 
 a 3-point 
 
 y=i on T 
 
 a' 
 
 
 a 4-point 
 
 y= 00 
 
 a=oo, j8= 00 
 
 
 a double 3-point 
 
 y = o,y=\ 
 
 a=oo, j8=o 
 
 
 a 4-point 
 
 y= 00 
 
 a=0, /3=oo 
 
 
 is indeterminate 
 
 (a 3-point if JB is finite) 
 
 8. THE DISCRIMINANT 
 
 The following factors have been found: 
 
 AU,fi) = i5'(A + i)»-|-25(9^+7)+27(4^-f3) 
 '   i'U,£) = B^(94+8)-S4^B(9^ + 7)-729^(4^-f3) 
 
 T{A,B) = 2^A-B 
 
 A; B; 4^+3; and J? = 00 is also discriminantal. 
 
68 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 If any other discriminantal loci exist they must have common points with either ^ =o or 
 
 For A=o 54=0 the equation becomes 
 
 -B(43'-i)'(2>'-i)3'-(3>'-i)(6>'-i)=o 
 
 after dividing out the pair of factors (sy—iY which belong toA=o. Further equalities occur 
 ioT B = o, B = co and for values of y which are roots of 
 
 288y — 288y'+ 104^— 163'+ 1 = o, 
 
 the corresponding value of B being given by 
 
 5(32/ — i6y+i) = 9. 
 
 The intersections of ^4 =0 and the locus D2 give for B the quadratic 
 
 5^+145+81=0. 
 
 In the field of the complex roots of this equation the quartic for y reduces and gives the 
 set of four y's, two for each value of B, and as D is the locus of three pairs of equal roots we 
 have no outstanding discriminantal points on ^ = o. 
 
 For B = cc the equation becomes 
 
 ^»(4y-i)(8y-3)^+yl(3>'-i)(i28/-96>'+i7)+8(33)-i)='(2>'-i)=o, 
 
 after the pair of factors belonging to .6 = 00 [(4^—1)^] and the factor y have been set aside. 
 The values of A for which y is a. double factor are — i, and — §. The former belongs to D2 
 and the latter to P, and the system of equalities is in each case what is required of such inter- 
 sections. The discriminant of the cubic in A is 
 
 64^^(4^+3)4U + i), 
 
 and since the term A^ is absent A = <x> must be discussed. 
 
 For A=o one pair of equal roots occurs in keeping with the discriminantal character of A . 
 A= —I belongs to D, and has in all four roots —\ and two y = o. This is in accord with the 
 fact that D2 has normally 3 pairs and B=oo is simply discriminantal. A= —J is itself a 
 factor and the two pairs occurring are expected. A = x has no extra equalities for B = <x . 
 
 We conclude that the factors occurring in the discriminant are A, 4^+3, B, T, P, D,; 
 that ^ = oQ and B = oo are discriminantal and that there are no others. 
 
 The complete discriminant is then of the form 
 
 N • A" • B''   (4^1+3)"* •f'-Pi'- D-',. 
 
 To determine the exponents and the numerical factor N special values of Bezout's determinant 
 are calculated. 
 
 First put A.= —I, a non-discriminantal value, and 5 = 27. The computation is rendered 
 
 comparatively light by transforming y to and multiplying up by 16 to obtain an integral 
 
 4 
 form. This divides the discriminant by 2'°. Writing B = 2']x and dividing by 27 we have the 
 form 
 
INTERRELATIONS OF THE TWO EQUATIONS 69 
 
 The value of the Bezout determinant for this form is if a:= i 
 
 2'5 . 3^4 . 53 and if x = 2, 2^" • 3^' • 13^ 
 
 The algebraic factor of T is x+i, of P, (r'+4.r+i),and of D„ {^x+i). From these values we 
 conclude that d = s, /> = 2, /= i, i = 8. The values of a and m cannot be determined since both 
 A and (4^4+3) are (—1). 
 
 Replacing the factors divided out in the transformation we have 
 
 A= ± 23^ • 35" . yl <■•£»• r • P" •£>» (4^+3)". 
 
 To determine a and m we give A and B simple non-discriminantal values and calculate 
 the residue of the Bezout determinant modulo a suitable number prime to all the factors of A. 
 
 A^i, B = i gives 4 • 7"= — 2 (mod 11) 
 i4 = 2, 5=1 gives 2"+' = — 2 (mod 5) 
 ^ = 2, B = i gives 2"+'" = 1 (mod 7) 
 
 That is provided the positive sign is taken with A. The only solutions permissible on account 
 of the limitations of the order are a = 2, m=2. There is no permissible solution with the 
 negative sign for A. 
 
 On account of the connection in general between the Bezout form and the standard form 
 of the discriminant the result is to be divided by —6" and we have finally 
 
 -2^8 . 2^» . A'   B^ -f • P'   DlUA+sY 
 as the value of the discriminant. 
 
 Q. THE INTERRELATIONS OF THE TWO EQUATIONS 
 
 The variables y and o- are connected by a birational transformation, namely 
 A + i+<T _ 8ABUy'-y)+SBisy-i)+gA i-jy 
 
 '4A+3-3T' B[3AUy-i)iSy-3)+i3y'—iy] y 
 
 (24) 
 
 In general the discriminantal factors will be identical but at certain critical points and lines 
 irregularities may occur. 
 
 For A= — f (a = 3) the equation in a- has three roots equal to — ^ and the expressions for y 
 become indeterminate. The proper values for y are in this case the three roots of the expression 
 for o- considered as a cubic in y. In fact the locus A= — f is not discriminantal for the y equa- 
 tion but a locus of reducibility, the reduced factors being 
 
 5(4>'-i)'(y-i)-9 (3y-i) and J5(8)'3-i2>^-t-3y)-9(3y-i) (25) 
 
 In a similar way for ^ =0 the o- equation has four roots equal to — i and the expression 
 for y is indeterminate, and that for o- fails to give the y values which must be sought from the 
 y equation which has distinct roots for these four places. 
 
 Similar irregularities occur for B = o, jB = 00 , and A=<x> . They are complicated by the 
 fact that the connection between {A, B) and (a, fi) is also birational. These do not call, how- 
 ever, for any special computations. 
 
7° 
 
 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 lO. THE TRANSFORMATIONS . 
 
 To avoid the birational transformation from a,^ to A,B and to keep as close to the tri- 
 angles as possible we consider the transformations leading in a chain from the sides {a,b,c) to 
 the symmetric functions of the sides {x=i,y,z) and to the symmetric functions (a,/3) of the 
 angle-bisectors which may be taken as the data of the problem. The equation F{y,A,B) = o 
 will then be considered as if its coefficients were explicitly written in (a, /i). 
 
 We trace the discriminantal loci in the {a,b,c) plane. Beginning with tt = ao , the repre- 
 sentative is 
 
 ^ab{b-cy{c-a)'ia-b+c) {-a+b+c) = o. 
 
 This is a curve of the eighth order with sixfold symmetry. It is not difficult to establish the 
 following features. The curve has no real infinite branches. The center of the triangle of 
 reference is a conjugate point. The curve has two branches at each vertex touching the sides. 
 It has two branches at the mid-points of the sides touching with inflexion the lines a+b—c = o, 
 etc. The extent to which the branches leave the sides is determined sufficiently by the points, 
 
 a=.205 
 
 b= .045 
 
 c= -75 
 
 a= .46s 
 
 b=-.2is 
 
 c= -75 
 
 a= .06 
 
 b=-.26 
 
 C=1.2 
 
 a= .21 
 
 6= — .41 
 
 C = 1.2 
 
 and the symmetric correspondents (Fig. 17). 
 
 The locus i8=oo has three representatives: (a — 6)(J— c)(c— a); abc, a.nd (a+b—c){a — b+c) 
 i-a+b+c). 
 
 The loci a = o and /8=o are jointly represented by 
 
 ta{b-cy{b+c-a) = o. 
 
THE TRANSFORMATIONS 71 
 
 This curve has sixfold symmetry and is closed. It may be traced by writing a = x+y, b = x—y, 
 c = i — 2x when it takes the form 
 
 2y*+-fii2x'—iix+2) + {i8x*-2ix'+&x'-x)=o. 
 
 As a quadratic in y" the discriminant is 
 
 — 96a;3+iosx" — 36X+4, 
 
 whose one real zero x = . 549 . . . limits the curve to a triangle slightly larger than the refer- 
 ence triangle. 
 
 The curve passes through the vertices parallel to the opposite sides and meets the sides 
 also at the midpoints touching them there. It has no singularities except a conjugate point 
 at the center of the reference triangle. 
 
 The discriminantal factor T is represented by 4{ab-\-bc-i-ca) — 1=0. This is the inscribed 
 circle of the reference triangle. The factor D, is represented by three symmetrical curves of 
 which Diia, b) is 
 
 2ia+by+4ab{a+b)-5ia+by-3ab+4{a+b)- 1=0 (26) 
 
 Writing a-\-b = 2x, a — b = 2y we have 
 
 (8a;'-s^+i)(3a;-i) . 
 ^ 8x-3 
 
 y is real except for ^\<a;<^\. 
 The asymptotes are 
 
 y=|/3a;— Tj\i/3 and the conjugate. 
 
 The intersections with the asymptotes are at a; = iVi!. 
 
 The curve passes through the center of the triangle, the midpoints and the vertices where 
 
 it touches the sides. At the midpoints the direction is »t = — f (Fig. 18). 
 
 By determining these curves and their intersections the diagram of a one-sixth part of 
 the (a, b, c) plane symmetrical and serving as a fundamental region is obtained. It has sixteen 
 compartments (Fig. 19): 
 
 Regions (i), (2) corresponding to real possible triangles with three external bisectors, 
 regions (3), (4), (5), (6), (7), (8), (11), (12), (13), (14), corresponding to real impossible triangles 
 with pure imaginary bisectors, regions (9), (10) with real possible triangles, one bisector being 
 external and two internal, and regions (15), (16) with impossible triangles and real bisectors. 
 
 After the symmetric function transformation these compartments are to be followed to the 
 (y, z) plane. 
 
 The y, z Plane 
 
 The locus a=co is represented by 
 
 4y*—y'-\-gz' — 6y'z — 2yz-^z = o. 
 
 Real points only occur in general for | y | < i|/ 3. There is however a conjugate point at 
 
 y = ^, 2 = ^;. z has a maximum for y = K^ , z- 
 
 conjugate. At the origin there is an inflexion y' = z. 
 
 y = i, 2 = V-- 2 has a maximum for y = ~ — ^, z = — '^ , and a minimum at the 
 
 J' 3, 2. -^ 16 ' 576 
 
72 
 
 THE PROBLEM OF THE ANGI-E-BISECTORS 
 
 The sides and bisectors of the reference triangle transform as in case of the internal prob- 
 lem, the real region being inside the curve A (Fig. 20). The closed curve a =00 lies entirely 
 inside the curve Z),. 
 
 a = o, /8 = o is represented by the parabola 32— 4y-|->' = o. This passes through theorigino, 
 the point F(j, o), the point A (^, ^jV) and cuts the line 4y — 8z — i =o[/3= 00 ] at F and Q(|, i\) 
 and has no contacts with the other curves in the real region {y, z). 
 
 Along Z?, ()8 = 00 , a = 4) is represented. 3 = represents /3 = 00 and so does /^y — 8z — i=o. 
 The curve D2 is 
 
 i6yH — ^y*+2i6z^-iSoyz^+Soy^z+y^+2,(>z'—i2yz+z = o (27) 
 
 This approximates the semicubical parabola 2 72^+ 2^3 = in the infinite regions, and has 
 an asymptote 323)— 1282 — 5 = 0, the further intersections with which are complex. There is 
 a conjugate point at (^, iV) on the ordinary branch, an inflexion at the origin yi+z=^o, and no 
 
THE TRANSFORMATIONS 
 
 73 
 
 other singularity. It touches Z?, at (J, o) having 43/— 8z— i =0 as the tangent and also touches 
 Z), at the cusp A . From the cusp out to y=cc D^is outside Dj, otherwise inside. 
 
 T is represented by the line y = 4 . 
 
 There is no difficulty in identifying the regions (i) to (16) in the {y,z) diagram. They 
 
 Case n.U.Vcy 
 
 cover the interior of D,. The exterior is divided by the curves into twelve regions which are 
 marked I, II, . . XII as a basis for discussing the transformation to the (a, j3) plane. 
 
 Tke a, /3 Plane, Limits 
 
 Since the transformation from {y, 2) to (a, y8) is not everywhere point for point it is neces- 
 sary to investigate certain limits. 
 
74 
 
 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 
 
 
 
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THE TRANSFORMATIONS 75 
 
 Limits for infinite values of y,z are: 
 Along z = my' 
 
 For tn=(X} : a= i, /8=o 
 
 for w = o : a = 4 but j8 = o is not reached, z = o giving /3 = oo . The point a = 4, /3 = o is attained 
 as nearly as we please in VIII, IX and also in (15), (14). Any positive o<4, ^ = is reached 
 in IV or V and IX and also in I, XI, the parabolas being eventually outside Z>,. 
 
 Along z'=my^ 
 
 o -8 
 
 "~'^''^~w(27w + 4)' 
 
 = 4, )8 any negative is reached in I, XI, V, IX. At w= —^\ the curve eventually falls on £>, 
 but /8= — 00 so that no limitation is set on /3 in XI. 
 
 For the limits at ^(§, jV) writing y = ^+\ z = ^+p we have . 
 
 (5^-9P)' 27(5A-9p)^ 
 
 " i3X»-63Xp + 8ip» '^ 3('^-3P)'+4'^'' 
 
 Using the tangents to the D^ curve and the parabola representing a = o, /? = o as axes 
 
 5A.-9P = X 
 X-3p=F 
 
 and approximating along semicubical parabolas Y' = mX^, we obtain 
 
 For positive w's o< /3< 54. For o>w>— g/Sis positive and the curve lies in regions (i), 
 (2). For »t< —i the curve is in VII, XII. 
 
 Hence VI, VIII cover 0=4 from /8=o to 54, 
 VII, XII cover a = 4 from /3=o to — 00 , 
 (i), (2) cover a = 4 from ^8=54 to +00 . 
 
 AtAii\=pp,a = — ^^~^^^l ,, and j8=o. 
 '^'^' i3-63/>+8i/)' 
 
 For /> = 3, a = 4 and for /> = !, a = o. 
 
 Hence values of a between o and 4 are found paired in VII and VIII and also in XII and VI. 
 
 Limits at F(i, o) are investigated by writing y = \-\-^, z = p. 
 
 We have 
 
 I6(3P-X)' (3P-^y ' 
 "■ = — ; P — — 7a v; • 
 
 X— 2p p(X— 2p)' 
 
 Using (X— 2p) the tangent to 0=00 and j8=oo as X axis and X— 3p the tangent to a = o, ^ = 
 
 as Y axis. 
 
 i6X» „ X^ 
 
 'Y'(Y-X)' 
 
 For Y = mX' : a='^^ , /8=oo, for F=wX : a=o, /3=--r-^ r . 
 
 m m \m — \) 
 
76 THE PROBLEM OF THE ANGXE-BISECTORS 
 
 and we have 
 
 »i>i : j8>o the region VIII, ->3, 
 
 P 
 
 |<OT< I : /8<o the region IX, -<o, 
 
 P 
 
 o<w<| : /8<o the region X, o<-< 2. 
 
 P 
 
 For - between 2 and 3, m is negative and the regions are the real sheets (i), (2), (3), (5), 
 
 (6), (7), (12), (11), (10), (9). 
 
 Limits at g(f,i>e)- 
 
 Here 4y — 8z — 1=0, (/3 = o) and 32 — 4^^+^ = 0, (n = o, j8 = o) cut. 
 
 If y = j+K z=A+p:'i = o, ^=i6(2A-3p)3--2i7(X-2p), 
 
 and cubical parabolas give a = o, /3 any. 
 
 /3 is positive in V, VII and negative in IV, VI. 
 Limits at 0(o, o). 
 
 »-=(y+3zy-^{z-y'), /3=(y+32)3H-(42-y")z 
 
 are proper approximations. 
 
 Along z=my' : o=— , /8=co . 
 pt 
 
 These parabolas fall in III for w>j : /8>o, a<4 and do not fall in II but in I, which is 
 limited by a = 4. Hence I and III are joined at/8=oo,o<a<4. 
 
 Along y^ = 4l3z' : a = o. These curves all fall outside A, and join I to (13) for a=oo at 
 O is y'-\-z = o and is closer to the axis than all the semicubical parabolas. 
 
 The point /(f, i\) is a conjugate point on D. and ^ = 5 is a sixfold root for a = 3, 18=27. 
 Two sheets from the real y, z regions are VI and VIII : the other four sheets are complex in the 
 neighborhood. 
 
 In transforming to the (a, /3) plane we notice first that for the whole of the regions outside 
 
 A, 0<a<4. 
 
 P' 
 For a— 4 = -— where P' is the discriminant of the equation whose roots are the sides and 
 
 is negative, while Da the denominator of a only vanishes on a closed curve inside Di and is 
 positive outside: while a= (32— 4/+;y)^-hZ}a is positive (3) (II, i). 
 
 The a, p Plane 
 In this plane we trace: 
 
 D{a, /3) as in case I. 
 
 r(a,/3) = (a-4)(^+27)+8l=0 (28) 
 
 P(a,^) = /3^(a-4)(a-9) + S4^(a-l)(2a-9) + 729a(a-l) = o (29) 
 
 a = o, a=i, = 4, /3 = 0. 
 
 These are all the finite discriminantal lines. 
 
 The locus P = o has no representative in the y, z plane but effects there a pairing of points 
 which come together in the a,p plane forming a nodal line in the y,"-,^ surface. The locus 
 
V}— ■^ S^fO^ XJJiJ 
 
78 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 a=i is of the same character while = 4 and y8 = o, which are infinite cuspidal edges on the 
 surface, partake of this property. 
 
 The regions (i), (2) being within D^ and outside a=oo in the y,z plane have a>4 /3?r54 
 and reach 0=4 at A for any /8^S4, and a= 00 , /3= 00 at F. Being divided by A they fold 
 on D and on a = 4 and cover the region inside D's cusp and above = 4. 
 
 The regions (7), (8) are within Z), and have a<o, /8>o and join on D^. 
 
 In (a, /3) they fold on D in the fourth quadrant reaching a = o, /8 = o and a= — 00 , ^= 00 . 
 
 The regions (3), (4) are separated by T in (^,2;), are within Dj and outside a= 00 . Hence 
 a>4, ;8<o. Theyreacha=4, ;8=oo on r and Z), also a=oo, ;8=— 27 on r and a=oo {y = \, 
 
 The regions (5), (6) also double on T but here a<o, /8<o. 
 
 The regions (15), (16) have a>4, /3>o and fold on D. They reach a = 4, ^ = 54 along 
 Di at infinity. A is here approximately 2']z'-\-2y^. (15) reaches a = 4, P= <» along z = o, and 
 (16) attains the same values along I>,. The values a = 4, 54<)8< 00 are reached for various 
 
 w's along z' = wy3 whose limits are a = 4, /3= ; , — r . The A value m=—-hf being a 
 
 ^ -^ w(27w+4) 
 
 turning-point for /3's denominator. 
 
 (9), (10) behave in a similar manner and also fall on the upper cusp region in (a, /3). 
 
 (11), (14) are divided by A, reach a = 4 |8= 00 but /3<o throughout. They fold on D\ 
 hyperbohc branch in the second quadrant. 
 
 (12), (13) similarly fold on D in the third quadrant. They reach a= 00 , |8= co at F and 
 respectively and so fall on the negative side of B. They reach a = o,^ = o with (7), (8). 
 
 Of the sheets I, II, . . XII with real iy,z) but complex {a,b,c) which all lie between a = o 
 and a = 4; 
 
 I reaches a = 4, ;8<o along z'' = my^ for m< —^S, reaches j8=oo , o<a<4 at alongz = W3»» 
 for m>l, reaches a = o, o>^> — 00 at o along y^ = 4^z', reaches /3 = o, o<a<4 along z = my' 
 for 2='», o<w< I, and thus covers the rectangle o<a<4, /8<o. 
 
 II and III are continuous over y = o which is not discriminantal. They reach a = o, 
 o</8< 00 at o along y^=4fiz'. The values are here paired with those in (8) since Di is closer 
 to the axis than any of the parabolas. The sheet reaches ^ = 0, o<a<i for z = my'', 2=00. 
 It does not reach a > i but is folded with IV along T. 
 
 IV reaches /3=oo, o<a<4 along QZ: a = o, o</8<oo at Q on cubic parabolas; joins 
 VII along j8=oo,o<a<4; reaches ;8 = o, o<a<i ioi z = my^ pairing with II, III. 
 
 V covers the whole rectangle P<o, o<a<4. It reaches a = o, /3<o at Q: <x = 4, P<o on 
 semicubical parabolas at infinity: o<a<4, /3 = o for z = my^ atz=<» :o<a<4, ^=00 along 
 4y— 8z — I =0 from Q to 00 , a increasing monotonously. 
 
 VI reaches a = o, /3>o at Q the cubical parabolas pairing the values with IV; a = 4) 
 o</3<54at^: o<a<4,/8=oo with V along 4)) — 8z — 1=0. It contains I>2 and 7 which goes 
 to the cusp and is folded with VIII along D. 
 
 VII covers the rectangle ^<o, o<a<4. It reaches a = 4, o>^> — 00 at A: a=o, 
 o>/3> — 00 a.tQ: /8= 00 , o<a< 4 along QZ joining IV; and /8 = o, o<o<4 at .4. 
 
 VIII covers the same region as VI folding symmetrically on D". _ 
 DC contains T and ^<o. A (0 = 0, i3 = o) is not reached. It reaches a = o, /3= — ^ on T 
 
 aXF: a = 4,l3<oonz'—my^ioTZ=M,m>o. j8=a3, o<a<4 is reached with VIII along z = o 
 f rom F to y = =0 : and ;8 = o, i<a<4 on z = w)''forz=oo, wj<o. 
 
THE TRANSFORMATIONS 79 
 
 X and XI cover the same region as IX folding on T. 
 
 XII covers the negative rectangle. It reaches a = o, /3 any, at F joining (12) : = 4, /3 any, 
 with VII at ^ : /3 = o, o< a< 4 with VI at A and j8= 00 , o< a< 4 at F for y = inx', m >\. 
 
 The nodal curve P = o does not affect the reaUty of the roots but causes the sheets to cross. 
 It has asymptotes = 4 . = 9, /3=27(— 2±i 3). There are no real points for i<a<3, nor 
 for \\<^<2'j. At 1 = 3, /3=27 is a cusp which falls on the cusp of D with coincident tangent, 
 but the P curve includes the D curve up to a = 4, /3 = 54 where they cross, and up to = 4, ;8= 00 
 where they have contact. The curve P meets T only at a=i, /3 = o and two complex points 
 (i4±i ^) «_ 27(9 + v^^) 
 
 = 
 
 , j8= '^ , the positive signs corresponding. P and D meet at the 
 
 22 83 
 
 cusp and at = 4, j8= 00 and also at the origin where the tangent to P is 30—2/8 = 0. 
 
 The curves have also two intersections for a= 00 , ^= — V. 
 
 Since y is given as a rational function of (a, 13) it is real for all real points, but the z's may 
 form a complex pair. The condition is 
 
 12(1— a)(4)/»— y)+a(6y— 1)'<0. 
 
 This is satisfied for the branch which reaches the origin in the third quadrant and for the 
 branch between T and D in the second quadrant, and in the region round the cusp of D up to 
 the crossings. 
 
 To determine which compartments of the y, z plane are paired by P we consider first the 
 region inside the cusp of D with a>4. The sheets involved are (i), (2), (9), (10), (15), (16). 
 
 At j8= 00 for positive approach the roots are o • j, J, , _ , , — - , _ . — . 
 
 Taking a = 6 as a typical case where P acts they are 
 
 o, i, i, -^, ±-g-. 
 
 Now (i), (2), (9) can all reach y = i but (2) is the only region reaching y>l, y8= <» . Hence 
 (2) has the root —r- and (i) = (9) has J. (16) reaches y = o but (15) cannot. (10) and (15) 
 
 o 
 
 have then the negative values. At the point a = 6 on D = o we have (i) = (2), (9) = (10), (16) = 
 (15) and as only one crossing occurs between this value of /3 and /8=oo it must be (10), (16) 
 which cross. 
 
 The cross-section of the surface has the arrangement of the diagram (Fig. 22). 
 
 Passing across /3= 00 (4) and (11) are paired by P for a>g. 
 
 Next consider the region /3<o, o<a< i. Where P crosses a = o three roots are equal to j. 
 These are in X, IX, XII since VIII has l3>o. P can only pair IX and XII since the y's must 
 be the same and a = o for P gives y = z- 
 
 The region /3>o, o<a< i. Q approached on 43;— 8z— 1=0 has )8=oo , a = o. For small 
 o's we have roots in IV, VI and y8>o. IV can only reach fi=o by moving' to infinity along 
 z = my', while VI must reach A. The y, z diagram (Fig. 20) shows that this entails the crossing 
 of the projections on the y-axis. 
 
 The branches of P at the cusp join complex z's. Using the continuity of the real surface as 
 a guide we see that the sheets (10), (16) crossing on P for a >4 represent two roots which become 
 equal and infinite at 0=4 and then complex, and so the nodal line is as usual continued as a 
 
<*. <. 
 
 oc =3 
 
 I Co n II* 
 
 
 it Nod^l Points 
 
 Infinite At>t^^oacltf^ 5^»jmi/oIizCii ' 
 
 3. Y 
 
 gfc 
 
 Fov 3i. = o Ihercwti aw , 'h ,'Ih, '/u , '/i ,Vs 
 
 ^TCclit ^OV 3 = 
 
 <<wi <: 1 
 
 __JL--^ 
 
 01=1 
 / 1 
 
 VI . - 
 
 VII. IX 
 
 ivcuiir 
 
 XII — — - 
 
 — ^^ 
 
 
 (Ol.-t)=-0 PcYiijirimiU ?o»itii'Yis 
 
 1 «^ot it 3 
 
 
 3 ■<• ot «: 4 
 
 F«v Ol a ^ tkt ^0 or. are /i '/j 
 
 4 < a <; p 
 
 oC > g 
 
 Fiq 2 2 d sectiam of FCvj , x, (J>) t 
 
DETERMINATION OF THE SIDES 8l 
 
 real isolated nodal line whose a, /3 projection is the curve P and whose y values run from oo to 
 5 at the cusp and back to <» at the end of the asymptotic branch. 
 
 The small arch of P between the origin and a= i, )8 = o pairs IV and VI. From the origin 
 along the first vertical asymptotic branch (12) and (13) are paired; crossing a =00 the sheets 
 are (11), (14) and the nodal line is isolated in both these parts. 
 
 From 0=1, /3=o moving to /8< o the paired sheets are IX, XII until a = is crossed, when 
 (6), (12) take their places. Passing a=oo (4), (u) are paired. For the last two parts the 
 nodal line is ordinary. 
 
 In a similar way the equalities further marked in the diagram may be established. The 
 result though a compendious collection of information is not, as remarked in case I, a complete 
 and consistent statement of all the facts. In particular as special defects the representation of 
 the points a = 0, j8 = o : a=i,j8 = o : a = 4, j8= 00 , for which the equation becomes indeterminate, 
 and the surface has a line parallel to the y axis, is omitted. 
 
 A set of diagrammatic cross-sections of the surface F(y,a,^) — o is given in Fig. 22. 
 
 II. THE DETERMINATION OF THE SIDES 
 
 The side of a triangle with given bisectors is a root of a sextic equation whose coefficients 
 are rational and unsymmetric in the bisectors. 
 To determine these sides we write 
 
 a—b = \a,b—c=Ka 
 whence 
 
 6 = a(i — A,), c=a(i — A.— k) (30) 
 
 The fundamental formulas for external bisectors then give 
 
 ---.=<i^i^- (3.) 
 
 and ii k= pK 
 
 p(i-2pK-K)=^{i-pK){i-K){i+py (33) 
 
 qil~2pK-K)={l-pK-K){l + K)p^ {^^') 
 
 From these quadratics « can be eliminated and we have 
 
 p'(p+iy+pP<P+i)-qpip+iy-p'p'+pqp{p+i)+q'{p+iy = o, (34) 
 
 and K is given rationally in p by 
 
 pp-q{p+i)-p(p+j)_ 
 
 pp+q{p+i)-p{p+i){2p+i) 
 
 (35) 
 
 The explicit sextic for k has coefficients of the third order in p, q and only five coefficients 
 vanish. It is not as convenient for computation as the chain above. 
 
 Since only the ratios of bisectors are used in this we may choose the scale so that {b—c)= i. 
 The isosceles triangles do not enter in the equation so that this is permissible. We then have 
 
 I 
 
 a=~ . 
 
 K 
 
 As in case I we may conclude that a is the root of an irreducible sextic of the symmetric 
 groupG„„(I, § 15), 
 
82 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 12. THE CASE OF EQUAL BISECTORS (EXTERNAl) 
 
 The group-theory argument of I, § 20 may be repeated in this case with the same result. 
 The sextic for the side a remains a sextic when K^L and then gives three a's and three 6's. 
 The sextic for c reduces to the square of a cubic but the rational relation fails and it is necessary 
 to solve a quadratic also. 
 
 The explicit determination of a which involves the solution of a sextic solvable by a chain 
 of a cubic and quadratic is conveniently performed by a special method. 
 
 If k — l=i and m be taken as the single parameter of the problem and we write a+b=t 
 ab = u the fundamental equations (11) give easily 
 
 {a-by m{t+a-iy . m{t+b-iY 
 
 {i-2a.){i-2b)ab i2t-i)(i-t){i-2a)a (2/- i)(i-/) (1-26)6 
 
 (37) 
 
 Equating the first fraction to the arithmetic mean of the second and third 
 
 2{P-4u)ii-t)i2t-i) = m[-2l^-4u''+st^+ut-4P+t] (38) 
 
 and the equation Di{a,b) = o is 
 
 2<3-l-4M/-5/2-3M-f4/-i = o (39) 
 
 From these we have the cubic 
 
 mit-i){2t-iy=(4t-3)(4i'-5t+2) (40) 
 
 giving t, while (39) gives « as a rational function of t and the sides are obtained by the quadratic 
 and linear equations 
 
 a-|-6 = /, a6 = M, and c=i — a— 6. 
 
Ill 
 
 I. THE MIXED PROBLEM 
 
 Given two bisectors at one vertex and another. 
 
 In this case the symmetry being destroyed a special unsymmetric method is usfed. The 
 cases where the third bisector is internal or external are reached by a change of sign in a side 
 and can be treated in one set of equations. 
 
 As data we take 
 
 If A, B, C, are the angles of the triangle the fundamental formulas give 
 
 *=tan , o=sm B cos ^sm A cos . (2) 
 
 2 ^ 2 2 
 
 If the angles are determined the remainder of the problem is an affair of ruler and compass. 
 The problem of determining the trigonometric functions of the angles involves an irration- 
 ality of the tenth degree, the root of an equation whose group is the symmetric group. 
 We write 
 
 B-C=6, B=2M,qcos- = H, (3) 
 
 and treating H, 9 as assigned M as required 
 
 H sin(43/-(9)-sin 2M • sin {2,M-e) = o. (4) 
 
 For the purpose of determining the group we write 
 
 e'"=x, 2Hi=k, e-"^=h, 
 
 and obtain the algebraic equation 
 
 kx{ho^— i)—{x*— i){hx^— i) = o, which is obviously irreducible. (5) 
 
 2. THE MONODROMIE GROUP 
 
 For h — osix roots of the equation are infinite, the other four being given by 
 
 x^—kx~i = o. 
 
 This has double roots for four values of k, y. 
 
 44 
 
 k^= , T,x^-\-i = o, k = 4x'. 
 
 27 
 
 For k = o the roots are +1, +i, —i, —i and may be named from this position in this order 
 
 (7), (8), (9), (o). 
 
 83 
 
84 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 If k = pe and p decrease from o to -, — tj the roots (7), (8) approach and (9), (o) depart 
 
 from the origin in the y plane. The path is for y = re'^. 
 
 ^•(1 + 2 sin 26)= I 
 in rectangular co-ordinates 
 
 {x^+y^) {x^+4xy+y'') = i . 
 
 At the extreme value mentioned for p, (7) and (8) are equal and give rise to a two-cycle. 
 By moving k along th«^ same radius to the corresponding positive value of p, (9) and (o) become 
 equal. 
 
 By using the perpendicular radius as a path for k, (o), (8) and (9), (7) respectively become 
 equal in the two directions, the path being 
 
 r*(i — 2 sin 26) = I. 
 
 Hence we get all interchanges on (7), (8), (9), (o). 
 
 Returning to the six infinite roots we find that they form a six-cycle iorh = o for all finite k's. 
 
 For ^=cc of (7), (8), (9), (o) three become infinite and one approaches zero. This is 
 obviously the negative root of x*—kx— 1 = and was named (o) in the original position. 
 
 For the approach ^ = 00 followed by A = o nine of the roots are infinite and eight form a 
 cycle. Since every interchange of (7), (8), (9), (o) is allowed we take this cycle as 
 
 [(i), (2), (3), (4), (S)> (6), (7). (8)] in some order. 
 
 Transforming [(o), (7)] by the powers of this substitution we have with [(o), (9)] every 
 [(o), («)] and so the symmetric group. 
 
 3. THE EQUATION FOR THE TANGENT OF A HALF-ANGLE 
 
 Writing y= tan M, t= tan 0, and k=2Hi the equation (4) becomes 
 
 H'{y'+i)(,ty^+4f-6yH-4y+ty-4y'iy3-syH-sy+ty = o (6) 
 
 The coefficients are rational in {k, h, i) and y is rational in {x, i). The group is also the 
 symmetric group. 
 
 For every y there are two x's but ten belong to F{x, —k,h) = o. These in the real cases 
 correspond to values of M increased by ir, and lead to the same triangles. So for a change of 
 sign mp,t and y change also and the same triangles occur, and without loss of generality we may 
 take p, q as positive. 
 
 By differentiation and elimination of H' we obtain as a discriminantal equation 
 
 /y9-6//^- i5/y+ (/'- 2o)/+2i>'5- 2i/^y4- 2i/y+ (3/2- i2)y^-\-()y- 1=0 (7) 
 
 Treated as a quadratic in t the discriminant is 
 
 (>''+i)V-36/-i2). 
 
 Hence real roots only occur for | y | < 6 . 02 . . . 
 
 The real discriminantal curve is then determined rationally in y and a square root of a 
 function of y. It may be traced by assigning to y all real values outside the critical values and 
 calculating I, H and so p, q. The curve has quadrantal symmetry in the {p, q) plane. The 
 critical values correspond to p= . 1456 . . q= . 2851 ... 
 
EQUATION FOR TANGENT OF A HALF-ANGLE 
 
 8S 
 
 From this value t= 2974. . one value of t increases monotonously to 00 and the other 
 decreases monotonously to zero. The p, q values tend each monotonously to (1,0) and (o, ^) 
 respectively. 
 
 . There is a cusp at (o, 5) where < oc -,/>«: — , and q—^ «: ~ . At (i, o) the proper 
 
 approximation is a pair of parabolas. 
 
 FltZ3 
 
 Under [lermutiitions o\ (^,<J.K) 
 tht|30(nT5 1.2,5 corT«s)3ont( 
 -3.(|i. 3, C[= 10) lias Svfiil.Yoots 
 
 The curve is continued past p= i by negative values of y and t, which double change of 
 sign leaves H' unchanged while (/>, q) are transformed to ( - , | j, the same transformation as 
 
 that effected by the interchange of two of the assigned bisectors, or say of (/, g) where 
 
 III 
 I : p : q : : ~ : - : t. 
 f g ft 
 
 In addition to the real branches thus traced there is a conjugate point ^= 00 ,9 = (Fig. 23). 
 
86 
 
 The graph of 
 
 THE PROBLEM OF THE ANGLE-BISECTORS 
 4. THE SOLUTION OF THE SEAL PROBLEM 
 
 6 
 
 H = q cos - = sin 2M sin {i,M—0)-^sm {4M—O) 
 
 is of the same general character for any 0. The diagram is for 0= - (Fig. 24). 
 
 4 
 
 In general the zeros are : o, - , tt - tt independent of d, and — | , m=i, 2, . .0. 
 
 22 33 
 
 The infinities are — | , m=i, 2, 
 
 4 4 
 
 8. 
 
 r.^u 
 
 XV h 
 
 As p ranges from o to co , ranges from o to ir, and these values coalesce only for 6= 
 f\ 
 
 - when - + -=- . This corresponds to p=i and two zeros coalesce but remain real on 
 2332 
 
 passing the value, (3) and (4) being interchanged. 
 
 For given H the roots are either 10 real or 8 real. The critical values can be found from 
 
 the derivative vanishing at the roots of 
 
 2 cot 2^—4 cot (43/— ^) +3 cot (3JW— ^) = o. 
 These values can be found without much trouble from the table of natural cotangents. 
 
CHARACTER OF THE SOLUTIONS 87 
 
 The values of the roots which are then entirely separated can be found either by Horner's 
 method from the equation (6) or by trial from the table of logarithmic sines, and 
 
 log H=log sin 2Af+log sin (33/— ^) — log sin {4M—O) (8) 
 
 The triangles are all real and possible il y or M is real, for in all cases A + B+C=Tr. The 
 values oi A,B,C are not always positive and the results are subject to an interpretation by 
 way of interchanging internal and external bisectors. 
 
 5. THE CHARACTER OF THE SOLUTIONS 
 
 First take o</'< I, i.e. o<0<- . 
 
 '^ 2 
 
 For the root (i) 
 
 o<M<^ : o<B<- : -e<C<-^ : -e<B+C<o. 
 42 2 
 
 Since sin yl=sin {B+C) we have fl<o, 6>o, c<o and the solution refers to K, K, L in 
 place of K, K, L. 
 For the root (2) 
 
 '-<M<'V- : ^-^5<-V : -'<C<--' : '<B+C<.. 
 3 4 4« 3 223 223 
 
 Hence a>o,6>o, and c is of doubtful sign. Fot q<pjj<o. 
 
 So for q<p (2) gives a,b,c : +, +,— and refers to K, K, L and for q>p, a,b,c : +, +, 
 + and refers to K, K L that is to the original verbal statement of the problem^ _ 
 The roots (3), (4) are alike and have a, b, c : — , +, + and refer to K, K, L. . 
 For the root (5) 
 
 e , TT ,, d ,2Tt 
 
 '+-<M<-^ . 
 
 42 3 3 
 
 Here a is always positive, b negative, and c changes sign when C = tt or when ?= i. 
 The other cases are of invariable class and the results may be collected: 
 I. The original case a,b, c : -\- , +, -{- occurs for (6) and (2) if q>p. 
 II. The case a,b, c : -\-, - , - occurs for (3), (4), (8), (10) and (5) if q<x. 
 
 III. The case a,b,c : +, -, + occurs for (5) i{ q> 1 and for (i). 
 
 IV. The cased, 6, c : +, +, - occurs for (7), (9). 
 
 The cases where p>i can be included by noticing the transformation (/'» t) combined 
 
 with iq, -) which entails (B,Tr—6) and if also we interchange (Af, ir—M) the original equa- 
 tion is unchanged. 
 
 This interchange however takes sin B to — sin B and leaves sin yl, sin C invariant. The 
 triangles are unchanged but the internal and external bisectors at A are interchanged. The 
 classes of solutions (I, III) and (II, IV) are interchanged in the pairing given. 
 
88 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 _ The whole transformation is equivalent to an interchange of the fundamental quantities K, 
 K,oriiK:K:L::f:g:h to {f,g). 
 
 Under (g, h) which involves {p, q), however, a new problem arises, and so under (/, h) 
 
 which replaces * by - and o by - . 
 
 q q 
 
 Of the six permutations of (/, g, h) three sets of two lead to distinct sets of triangles. 
 Namely for this problem 
 
 (/, g,h) = {g,f,h) corresponds to {p,q) = (^,^ . 
 
 ih,f,g)=(f,h,e) (q,p) = (^i,t'j. 
 
 6. THE CASE or EQUAL BISECTORS 
 
 For p = q=i the equation for a:=tan — becomes 
 
 {x'+i){x*-6x^+i)-Sx'(sx^-iy = o, 
 which reduces to 
 
 (x^— i)(x''— i4r'-|-i)(a;''-(-4a;^-|-i) = o (9) 
 
 The first factor gives in two ways the triangle A=B = - ,C = o. 
 
 The second factor gives in four ways the triangle ^=5 = -,C= — . 
 
 The third factor gives in four ways the triangle determined by 
 
 B=2 tan-'I/(i/5-2) 
 
 or approximately when the angles are taken positively and internal, 
 
 ^ = 13° 40', 5=128° 10', C = 3&° 10'. 
 
 In this case the B bisector is external, in the second case it is internal, and in the first the 
 words internal and external have no proper distinction. The case oi p=i,q any, has an equa- 
 tion containing only even powers of x. If the problem were solved in terms of the sides the 
 locus p=i would be discriminantal, but for this equation, although the group reduces so that 
 the equation may be solved by solving a quintic and quadratic, there are no equal roots, the 
 roots merely referring to the same five triangles in pairs. In the former cases we had the 
 phenomenon of a discriminantal locus in one solution corresponding to a locus of reducibility 
 for another; here we have it corresponding to a locus of group reduction. 
 
IV 
 
 I. THE GENERAL PROBLEM FOR REAL DATA 
 
 If three real numbers are assigned as the lengths of any three bisectors the problem of deter- 
 mining the triangle is to be solved by successive application of the methods of the three cases. 
 The number of real solutions depends on the data, and the character of the dependence is 
 revealed by considering the discriminants of the three cases simultaneously. 
 
 In cases I and II any three assigned real numbers cause the (a, /3) point to fall in the region 
 within the cusp of D{a, fi). For a>4 I has 8 real solutions with possible triangles; II has 4. 
 For a<4 the numbers are 7 and o respectively. 
 
 The condition = 4 is expressed in (/>, q) as the vanishing of the product 
 
 {p+q+l){p^q-l){p-q+i){~p+q+l). 
 
 On account of the symmetry of the discriminant of III it is only necessary to consider one 
 quadrant of the (/>, q) plane (Fig. 24). 
 
 Taking the first quadrant for p+q— i >o we have a<4. There are then three regions and 
 three classes of the general problem: 
 
 Class A : a<4 and A3<o. 
 Class B: a>4 and ^i<o 
 Class C: o>4 and A3>o 
 
 For class A, I has 7, II has o, and III has 3 permutations, each of which has 8 solutions. 
 The permutations of if,g,h) leave the square (0 = 4) invariant. 
 
 The total for class A is then 33 real solutions with proper triangles. For class B the per- 
 mutations of (/, g, h) do not carry the representative point across the discriminantal curve. 
 Ill has then three sets of 8 solutions, I has 8, and II has 4, the total being 36. For class C q<^ 
 and its reciprocal occurring as a 9 under {h,f) is outside A3. The other transform is inside Aj. 
 
 For this class two sets in III have 10 real solutions and I has 8, and II 4 solutions : in 
 all, 40. This is the greatest number and occurs, for example, if / : g : /; : : 3 : 30 : 10 (Fig. 23). 
 
 This case has been taken for the triangles in the illustration (Figs. 25, 26, 27). 
 
 2. THE PROBLEM WHEN A RIGHT ANGLE AND TWO BISECTORS ARE GIVEN 
 
 Taking the right angle as C the sides b and c are rational functions of the side a. 
 
 2a— I — 2a^-f2a— I , , , 
 
 b = —, -s, c= -. r — , a+o+c=i 
 
 2(a— i) 2(a— i) 
 
 (i) 
 
 By interchange of (a, b) and by changing the signs of sides the fifteen pairs of the six bisec- 
 tors can be reduced to three cases. 
 
 89 
 
WHEN A RIGHT ANGLE AND TWO BISECTORS ARE GIVEN 91 
 
 Case I. Given K,L. 
 
 The ratio -^^ becomes a perfect square in (a, j 2) namely 
 
 -=;=2]/2a(a— 1)^-^(1 — 2a)^ (2) 
 
 If Z, H- 2 j/2 A" is plotted against a the same curve which occurred in the internal problem (I, § Sj 
 is given. 
 
 For -p>o there is then one real solution of the cubic. This gives a real triangle with all 
 K. 
 
 positive sides and is the solution of the problem as stated, namely K, L are internal bisectors. 
 
 For t;<o there are three solutions: 
 K 
 
 i) a>i, b>i, c<o. The range of c has a maximum at c= — | 2— i corresponding to 
 
 the right-angled isosceles triangle and c tends to — 00 as either a or 6 tends to <x> . 
 
 The bisectors are both external. 
 
 2) i>a>5, — <» >b>o, <x> >c>^. The A bisector is external, the B bisector internal. 
 
 3) o>a>— 00, 5<6<i, 5<c<oo. The ^ bisector internal, the 5 bisector external. 
 
 In all there is a single solution iot each arrangement of the bisectors as internal or external. 
 The discriminant of the cubic is 
 
 L{ 8i/2L'+i3LK+8i/2K') , . 
 
 The double points are at L = o, ^=0 and t^ = ^~^^ ,^^}- ^ . 
 
 Case II. Given K,M. 
 
 The equation for the side a is a sextic; obviously irreducible. 
 
 SA'Ca- iya'-M'{2a''- 1)^(20'- 20+ 1) = (4) 
 
 8K^ 
 Plotting -jrTj=/> against a we have the real graph (Fig. 28). At^ = o a double transposition can 
 
 be effected (1,2) (3,4) and at /»= 00 a two-cycle and a four-cycle which must separate 5, 6 which 
 are conjugate complex for the real approach. It may properly be denoted by (2,3) (1,5,4,6). 
 Approaching /)= 00 from the negative side no double point is encountered between p = o and 
 /)= — CO since the double points are at p = o, 00 and the four complex roots of the remaining 
 factor of the discriminant : 
 
 i6/)4- i52/)3-|-93/)^-f-5i2/)+32768. 
 
 The corresponding values of a being given by 
 
 6a''— 8a3+ 'ja'—4a+ 1 = 0. 
 
 Since the conjugate pairing must be kept the cycle at /»= °o must be for this approach either 
 (1,2) (3,5,4,6) or (3,4) (1,5,2,6) or (5,6) (1,3,2,4). 
 
 Since the complex double points are distinct and the second derivative does not vanish at 
 them, a single transposition occurs as an element of the monodromie group. It is then easy 
 to see that the monodromie group and therefore the algebraic group is the symmetric group, 
 
 It 
 
FH.Z6 
 
SPECIAL CASES OF ISOSCELES TRIANGLES 93 
 
 however the cycle for />= — <» be named. The discriminant not being a square, adjunction 
 of \/p does not reduce the group. 
 
 To discuss the character of the solutions we divide into classes by /)<8 that is A'<M and 
 by the values for a when p = 8. 
 
 Class la. K<M,a< =, i>b>—-^, c<i. Af is external. ^ 
 
 y 2 1 2 
 
 ClassIIa. K<M, -.2516. . >a> ^, .6005 . . <6<— -, .6511, <c<i. M is 
 
 ]/ 2 ] 2 
 
 external and the general character of the figure is the same as in la. 
 
 Class Ilia. K<M, -^>a>h ^<6<o, i>c>^. A' and M are both external. 
 
 V2 ) 2 
 
 Class IVa. K<M, -^<o< .7600 ,., --^>6>-i .083 .., i<c<i. 323 .. ,Aand 
 V 2 ) 2 
 
 M both external. 
 
 Class 16. A:>Af, — .2Si6<a<o, .6oo5>J>i, .65ii>c>i. Af is external, A" internal. 
 
 Class lib. K>M, o<a<^, ^>b>o, c has the value \ at each end of the range and 
 decreases from these values each way to a minimum 1 2— i which belongs to the right-angled 
 isosceles triangle. 
 
 K and M are both internal and this case is the only one solving the problem verbally 
 expressed for internal bisectors. 
 
 Class III6. K>M, .7600. . <a<i, -i .o83>6>- <» , 1323. . <c<oo. Both bisec- 
 tors are external. 
 
 Class INb. K>M, i<a<oo, co >b>i, c has the value — 00 at each end of the range 
 and reaches a maximum — | 2— i for the case of the isosceles triangle. K is external. 
 
 The approximate values for a are the roots of the equation for equal bisectors. For this 
 case only two real non-trivial solutions exist : 
 
 a : b : c :: — .2516 . . : .6005 . . : .6511 . . The angle A about 22°4o', K internal, M 
 external. 
 
 a: b : c :: .7600 . . : —1.083. > '■ 1-323 •   , A about 3S°4', K and M both external. 
 Case III. Given two bisectors at one vertex and the right angle. 
 
 The tangent of half the difference of two angles, and one angle are given, hence the prob- 
 lem is one for ruler and compass. 
 
 3. SPECIAL CASES OF ISOSCELES TRIANGLES 
 
 The general method of Case II leaves the construction of an isosceles triangle given an 
 external bisector at the base indeterminate. 
 Other conditions must be given. 
 If the base a and the external bisector L are given we have 
 
 j,_ {-a+2b)a'b 
 
 ^- (b-ay • 
 
 To determine the angles write == p = -. — ^ where <^ is half the external angle at the base. 
 ^ X sm 2</) 
 
I. CIO 
 

 IDENTICAL RELATIONS AMONG THE SIX BISECTORS 
 
 95 
 
 The solution is 
 
 cos <i>- 
 
 />±1 (j)'+4) 
 
 The problem can be solved by ruler and compass and is an extension of Euclid's decagon 
 problem (/>= — i). 
 
 The sign of p does not determine any representable difference of configuration, but for 
 
 ^ < I /) I < - one triangle has the bisector internal : below the lower value two solutions 
 
 V/'2 2 
 
 have external bisectors, above the higher one triangle is complex. 
 
 If the sides b = c and L be given the problem requires the solution of a cubic equation. 
 
 Namely \i-j-=K, and t = ■^, 
 
 X3+ ((C- 2)A^— 2k\-\-k=o. 
 
 F,ff.Z«. 
 
 The discriminant is k(4k^+ 13(0+32), and k=<x is also discriminantal. 
 
 For K>o there are three real roots, one negative referring to an internal bisector and 
 
 two positive referring to external bisectors. For k= i the angle ^ is — and the bisector inter- 
 nal, or vl is — or - and the bisector external. 
 
 7 7 
 
 4. THE IDENTICAL RELATIONS AMONG THE SIX BISECTORS 
 
 IT 
 
 I. We have = = tan 
 K 
 whence 
 
 {'~-^)^ 
 
 ^iE\^u'^ 
 
 (s) 
 
 II. The length of the line joining the extremities of the two bisectors from A is r^ 
 
 Hence 
 
 „ I 
 
 2abc 
 J' 
 
 y\K'+K') 
 
 -=o 
 
 (6) 
 
96 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 III. The altitude of the right triangle included by the bisectors at A and the line joining 
 
 their extremities is 
 
 K • K 9 
 
 //a= „   and as altitude of the triangle Ha= — , 
 
 2' {K'+K') 2a 
 
 where 5' is the area of the triangle. 
 From this equality 
 
 a = S . UKl + K^ 
 
 K-K " 
 
 while K'' s   {s-a){b-cy 
 
 K^ {s-b){s-c){b+cy 
 
 By substituting for a, b, c from (7) in 2=^ a third relation is obtained. 
 This is for convenience expressed by writing 
 
 K K 
 
 and 
 
 when it takes the form 
 
 jjk'^ {^ p)m i p-qy 
 
 K U(p+q-r)nip+qy ^^^ 
 
 It is to be noticed that p : q : r : : a : b : c. 
 
 The independence of the conditions I and II is obvious. For III the set of values 
 
 satisfy I and II but not III. 
 
 5. THE INDIRECT PROOF 
 
 In a series of papers in Phil. Mag., IV (1852), the problem of the triangle with two equal 
 angle-bisectors is made the text (with some other elementary problems) of a discussion as to 
 the necessity of the reductio ad absurdum in geometry. The chief parts were taken by Sylves- 
 ter and De Morgan. 
 
 De Morgan claims to see "identity in 'Every A hB and every not B is not ^"' by a pro- 
 cess of thought prior to syllogism; and so denies the necessity of an indirect proof in any case. 
 
 Sylvester surmised that "The reductio ad absurdum not only is of necessity to be employed, 
 but moreover in propositions of an affirmative character, need never be employed except when 
 the analytic demonstration is founded on the impossibility or inadmissibility of certain roots 
 due to the degree of the equation implied in the conditions of the question. If this surmise 
 turn out to be correct we are furnished with a universal criterion for determining when the use 
 of the indirect method of geometrical proof should be considered valid and admissible and when not." 
 
 It is difficult to deny De Morgan's general proposition though his immediate application 
 is a little unfortunate. The problem being, as stated by Sylvester, "To prove that if from the 
 
1 > ■> 1 
 
 1 1 ^ 
 
 THE INDIRECT PROOF 97 
 
 middle of a circular arc two chords be drawn, and the remoter segments of these chords cut off 
 by the line joining the end of the arc be equal, the nearer segments are equal." The doubtful 
 word is of course "remoter." If this word means every point of which is remoter, then De 
 Morgan's contention that "proving that the inequality of the nearer segments makes the 
 inequality of the remoter ones follow, the equality of the remoter ones makes the equality of 
 the nearer ones follow " is a proper special case of his general argument, can be made good. This 
 interpretation has however a disadvantage from the geometric point of view. It is not appli- 
 cable to the allied problem where the analytic geometer would say the chord has complex points 
 of intersection with the circle. Yet for this problem an entirely analogous theorem is true and 
 it is desirable to so state the problem that both cases are included. 
 
 If the problem be stated: "A line of given length has one extremity on a straight line, the 
 other on a circle, and the line passes through a cut of the circle and the perpendicular on the 
 given line from the center of the circle which is not separated from the foot by the second cut; 
 then if the length of the line be less than the distance from foot to second cut, and in case the 
 foot is outside the circle greater than the mean proportional between twice the distance from 
 foot to second cut and the distance from foot to first cut, and greater than the mean propor- 
 tional between four times the diameter of the circle and the distance from foot to first cut, 
 four positions are possible for the line, and these have symmetry in pairs, and for each sym- 
 metrical pair the segments cut from the line by the circle are equal" — then it is possible that 
 justice has been done to the facts. However, in the general case the segments by the circle are 
 neither nearer nor remoter and from the inequality of the circle segments the inequality of the 
 circle line segments does not follow without a specification of the pairing. The syllogist's 
 difficulty lies in the definition of the classes, and in this special case the class is at least not 
 conveniently defined by equations alone. 
 
 Turning to Sylvester's view we note that the proof that equal internal bisectors implies 
 isoscelism falls very neatly in his scheme but the corresponding problem for external bisectors 
 presents a new difficulty. Sylvester with the proper mathematical instinct generalized the 
 problem before solving it:' namely, he said divide the internal angle in a given ratio instead of 
 bisect. This generalization unfortunately does not include external bisection. To compare 
 the two cases we write the equation 
 
 K—L c a+6-f-cr c b{a-\-b-\-2c) a 
 
 K+L b+c 
 
 [_c b{a+b+2c) a "I 
 a+c (a+cY b+c] 
 
 This holds from the fundamental equations for internal bisectors, and the spirit of Sylvester's 
 method is to say — 
 
 The right-hand side is essentially positive for a-positive non-trivial triangle, and is more- 
 over expressed in products and sums of products of ratios each geometrically interpretable. 
 If then K — L = o, b — a = o or the axiom of Archimedes fails. 
 
 In the external case, however, 
 
 K — L a+b—c c rc3—{a+b)c''+sabc—abib+ay 
 
 t ci- {a+b)c'+ sabc- abib+an 
 {b-c){c-ay' ' ] 
 
 b-a K+L ib-c)\ 
 
 the last factor in the numerator may vanish for positive non-trivial triangles. It is in fact 
 
   Blichfeldt, Annals of Malh^., II, 4.22, gives the same generalization. His proof is valid also for non- 
 Euclidean space. 
 

 98 THE PROBLEM OF THE ANGLE-BISECTORS 
 
 — Diia, b) and the curve I>2 = o actually enters the region of proper triangles in the (a, b, c) plane 
 (Fig. 18). 
 
 In this case, however, the non-isosceles triangles with equal external bisectors have the 
 bisectors oppositely directed so that further specification of the conditions of the problem 
 which must again presumably be by means of inequalities and not equations will permit the 
 proof as above by Archimedes' axiom. 
 
 It would appear that in general, though the difficulties of expression may be great, any 
 theorem true analytically for a properly restricted class might conceivably be thrown into a 
 form similar to the above and further a direct geometric proof might be given by Archimedes' 
 axiom and adequate restrictions based on order postulates. 
 
 6. GENERALIZATIONS OF THE PROBLEM 
 
 Sylvester's generalization {loc. cit.) which substitutes division of the angle in a given ratio 
 for bisection does not include the external case as well as the internal under the same general 
 formulas. The same thing is true if for bisectors which meet sides in points dividing them in 
 the ratio of adjacent sides we substitute lines through the vertices dividing opposite sides in a 
 ratio compounded of the ratio of adjacent sides and the ratio of a corresponding pair of three 
 assigned numbers. 
 
 To Professor E. H. Moore is due a generalization embracing both the internal and external 
 cases in one set of formulas. 
 
 He introduces three parameters u, v, w and defines the given quantities by 
 
 ,^j , , ,_{au+bv+cw){—au+bv+cw)bcvw 
 
 Ka{u,v,w; a,b,c) = ^1— r- 
 
 {bv+cwy 
 
 kI{u,v,w; a,b,c) =Ka{u,v,w; b,c,a) 
 
 Kc{u,v,w; a,b,c) =Ka{u,v,w; c,a,b) 
 
 Then for {u,v,'w)= {1,1,1) the internal formulas are given and for {u,v,w)={i, i, — i) the exter- 
 nal formulas. 
 
 The problem for the spherical triangle, the formulas are 
 
 ,, 4 sin s sin is— a) sin b sin c 
 ^r\^K = ^   \i,/ N , etc. 
 
 It may be noted that the dual spherical problem reduces as the sphere becomes a plane 
 to a ruler and compass problem. Given the angles which the medians make with the sides to 
 construct the triangle. 
 
: : •••:*• 
 
 VITA 
 
 Richard Philip Baker was born February 3, 1866, at Condover, Shropshire, England, and 
 was educated at Clifton College, Bristol (1877-84), and at Balliol College, Oxford (1884-S7). 
 He graduated with the degree of B.Sc. at the University of London in 1887. 
 
 In 1888 leaving England for the United States he studied law and was admitted to the 
 Texas bar in 1890. In 1895 he became a graduate student of mathematics at the University- 
 of Chicago where he attended courses by Professors Young, Laves, Maschke, Bolza, Moore, 
 Dickson, and Wilczynski. After several years of teaching in secondary schools he became in 
 1905 instructor in mathematics at the State University of Iowa. Here he studied physics with 
 Professors Guthe and Stewart. In 1910 he was appointed assistant professor of mathematics 
 in the same university. 
 
 99