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 On the Cardioids 
 
 FULFILLING 
 
 Certain Assigned Conditions 
 
 By 
 
 SisTEH Mary Gervase, m.a. 
 
 of 
 
 THE SISTERS OF CHAHITY, H 4HPAX, N. S. 
 
 A DISSERTATION 
 
 Submitted to the Catholic Sisters College of the Catholic 
 
 University of America in Partial Fulfillment 
 
 of the Requirements for the Degree 
 
 Doctor of Philosophy 
 
 Washington, D. C. 
 June, 1917 
 
"if 
 
On the Cardioids 
 
 FULFILLING 
 
 Certain Assigned Conditions 
 
 Sister Mary Gervase, M.A. 
 
 of 
 
 THE SISTERS OF CHARITY, HALIFAX, N. S. 
 
 A DISSERTATION 
 
 Submitted to the Catholic Sisters College of the Catholic 
 
 University of A merica in Partial Fulfillment 
 
 of the Requirements for the Degree 
 
 Doctor of Philosophy 
 
 Washington, D. C. 
 June, 1917 
 
NATIONAL CAPITAL PRESS, INC., WASHINGTON, 0. C. 
 
Contents 
 
 PAGE 
 
 Introductory 5 
 
 The Simpler Cases: 
 
 I. Given 3 Conditions 9 
 
 II. Given 4 Conditions 15 
 
 More Difficult Problems: 
 
 I. Given 3 Conditions 24 
 
 II. Given 4 Conditions 35 
 
 362099 
 
Introductory 
 
 Tlie triciispidal, bicircular qiiartic of the third class defined 
 
 hy the 
 
 Cartesian equation (.i'-+?/-+a.r)'" = a-(.r'^+?/") 
 polar equation p = a ( 1 — cos 6) 
 
 and commonly known as the Cardioid, has for many years been 
 the object of mathematical investigation. It has lately been 
 studied by Raymond Clare Archibald in his Inaugtiral-Dissertation 
 "The Cardioide and Some of Its Related Curves" (Strassburg, 
 1900), which work contains an historical sketch of the curve and 
 a presentation of results prior to the year of its publication. 
 Since then, the only work on the subject of considerable length is 
 Professor Archibald's paper, "The Cardioid and Tricuspid : Quar- 
 tics with Three Cusps."* Besides this there have appeared a few 
 detached problemsj and contributions in periodicals treating the 
 curve from either a metric or a projective standpoint. 
 
 The chief characteristic of former research along this line seems 
 to be the examination of the cardioid as a fixed curve and the 
 consideration of its properties as such. The present investigation 
 starts from a different point of view, which we may outline as 
 follows : 
 
 In general, a curve of the fourth degree is capable of satisfying 
 14 conditions. The cardioid, however, having 3 cusps, two of 
 wjiich are at the fixed (circular) points / and J, can be subjected 
 to only 4 conditions.! If, then, 3 conditions be imposed, there 
 are oc ^ curves satisfying them; therefor^, the special elements 
 (cusp, focus, double tangent) describe definite loci. If 4 conditions 
 are given, there are a finite number of curves satisfying them. It 
 is our purpose to obtain the loci generated in the first case; and, 
 in the second, to determine the number and (where possible) the 
 reality of cardioids for various kinds of assigned conditions. 
 
 The co-ordinate system which lends itself most readily to such 
 an investigation is the system of conjugate (also called circular) 
 co-ordinates, in which a point is named by a vector which has its 
 
 * AnnaU- of Mathematics: M S. 4. 190^2-3, pp. 9off. 
 
 t Cf., for example, Questions 14,4^)5; 1().^2G(); 10.^}9"2. Ed. Times (London ) . 
 i Cusps at I and J count for 4 conditions each. The third cusp counts for 
 2 more conditions. Thus, the 14 conditions reduce to 14 — (2X4 + 2) -=4. 
 
6 CanUoids FulfUliiKj Crifdin Assigned CoiuUtions 
 
 initial extremity at some fixed point, called the origin. Denoting 
 a vector by z, its projections on the real and the imaginary axes 
 are designated by x, ?/, respectively; and 
 
 z = x-\-iy.'^ 
 
 With a vector z is associated its conjugate z' . This may be geo- 
 metrically defined as its reflection in the real axis. Accordingly, 
 
 z' = x — iy. 
 
 All vectors may be considered as obtained from the standard 
 unit vector by means of a stretch and a turn. Turns are regularly 
 designated by the letter /; fixed turns, by /i, ^2, ^3 • • • ; variable 
 turns, by ^, ^^ r . . . f 
 
 Regarding the cardioid as the epicycloid generated by equal 
 circles, the map-equation of the curve is 
 
 The centre of the fixed circle is at 0, which point, we shall, with 
 Professor Morley,| call the centre of the cardioid. This point is 
 also the singular focus of the curve. The cusp is at z=^r (r being 
 on the real axis). 
 
 This map-equation involves its conjugate. 
 
 '"('-;.)■ 
 
 These two equations taken together may be considered the 
 parametric equations of the curve. 
 
 A cardioid with centre at Zo and any orientation has for its 
 map-equation : 
 
 z — Zo = 2at — a'f 
 
 z = Zo-]-— gives the cusp. 
 
 The angle made by the axis of the curve with the axis of reals is 0, 
 where 
 
 ,^ie 
 
 {:)■■ 
 
 * The symbol i denotes, as usual, V — 1. 
 
 f Cf. Morley, "On Reflexive Geometry," Transcicfions of the American 
 Mdfhematical Society, 1907, Vol. 8, p. 14. 
 
 J "Metric Geometry of the Plane X-Liiic"" T ran.sncllonn of the American 
 Mathematical Society, 1900, Vol. 1, p. 10.5. 
 
Cardioids I'ldfiWuif/ Certain Assigned Conditions 7 
 
 Any tangent is given by 
 
 {z-Zo)-{z'-z'o)t'-3at+Sa't^ = 0; 
 the double tangent, by 
 
 a''{z-Zn)+a'(z'-z'o)=Sa'a". 
 
 The kinds of data we sliall consider are: point, line, centre, cusp, 
 or double tangent given, to solve a specified problem. The last 
 three are equivalent to 2 conditions each. The problems arising 
 naturally fall into 2 classes: 
 
 I. Given 3 conditions, find specified loci; as, 
 
 (1) Given the centre and a line; find the locus of cusps. 
 
 (2) Given the centre and a ])oint; find the locus of cusps. 
 (8) Given the cusp and a line; find the locus of centres. 
 
 (4) (iiven the cusp and a ])oint; find the locus of centres. 
 
 (5) Given the double tangent and a line; find the locus of cusps. 
 (Jiven the double tangent and a line; find the locus of centres. 
 
 (0) (iivcMi the double tangent and a point; find the locus of cusps. 
 Given the double tangent and a i)()int; find the locus of centres. 
 
 (7) Given 8 hues, find the locus of centres. 
 
 (8) Given "i lines and 1 point, find the locus of centres. 
 
 (9) Given 1 line and 2 points, find the locus of centres. 
 
 (10) (liven 3 points, find the locus of cus})s. 
 
 II. Given 4 conditions, find how many solutions there 
 
 are; as, 
 
 (a) (liven the centre and 2 lines, how many cardioids are there.^ 
 
 TIow many are real? 
 (})) (liven the centre, a point and a line. Apj)ly the same ques- 
 tions. 
 
 (c) Given the centre and 2 ])oints. 
 
 (d) (iiven the cusp and 2 lines. 
 
 (e) (jiven the (tusj), 1 line and 1 })oint. 
 
 (f) Given the cusp and 2 ])()ints. 
 
 (g) Given the double tangent and 2 lines. 
 
 (h) (liven the double tangent, 1 line jind 1 point. 
 
 (i) Given the dou})le tangent and 2 i)oints. 
 
 (j) (liven 4 lines. 
 
 (k) Given .'J lines and I point. 
 
 (1) Given 2 lines and 2 j)()ints. 
 
8 Vaidiohh Fulfilling Certain Assigned Conditions 
 
 (ill) Given 1 line and 3 points. 
 (n) Given 4 points. 
 
 Auain, tliere is an evident division of the problems into simpler 
 and more difficult cases. It is not the intrinsic value of some 
 of the simple cases which authorizes their appearance in this 
 work, but rather their importance in the solution of some of the 
 more difficult problems. 
 
 Before taking up the problems, it will be well to indicate a 
 guiding ])rinciple which will be found of great importance for 
 cases where the cardioids are to touch several lines. It may be 
 stated thus: 
 
 If d be the angle between 2 lines which touch a cardioid, the 
 angle between the cuspidal rays to the points of tangency has 
 
 some one of the values %, f ((9+360°), ^((9+7^20"). 
 
 For the case of parallel tangents, this specializes to the well- 
 known theorem: 
 
 The points of contact of any 8 i:)arallel tangents lo a cardioid 
 subtend angles of 120° at the cusp. 
 
 With these considerations premised, we shall proceed to a 
 treatment of the simpler cases. 
 
The Simpler Cases 
 
 I. Given 3 Conditions. 
 PROBLEM (1): Given the center and a line; find the locus of 
 cusps. 
 
 Let us take the centre at 0; the line, as 2+2' = 2. 
 The map-equation of the cardioid is z = 2at—a'f 
 
 The cusp: z — —r 
 a 
 
 Any tangent: z-Sat-i-SaT-z't^ = 
 
 If the line 2+2' = 2 is to touch the cardioid, the following iden- 
 tity must exist: 
 
 z-Sat+Sa'f-z't^^ z+z'-2: 
 
 whence, ^^ = — 1 (1) 
 
 and -Sat-\-Sa'f= -2 (2) 
 
 From (2) we get t= —1, -co, or — oj^* 
 
 It will be sufficient to take one of these values; for, since the 
 
 expression for the cusp I -, I does not occur rationally in (2), the 
 
 rationalized form of the equation will be the same, no matter 
 which value is selected for t. Cubing (2) and substituting the 
 value ^= — 1, we obtain 
 
 27(a3+a'3) -54aa'+8 = 0. 
 Now the cusp is given by 2 =-7, and we seek the cusp locus. 
 Therefore, if we call — -„ k [whence result the equations: 
 
 a 
 
 aa' = kk' 
 a^^a'^ = kk'{k+k')l 
 
 the equation of the locus becomes 
 
 nkk'{k+k')-54^kk'-]-S = 0. 
 
 This is a rational cubic with its double point at A;=- and vertex 
 
 *co and co^ are the imaginary cube roots of unity. 
 
10 Cardioids FulfiUing Certain Assi(/nr(J CotiditioH, 
 
 at A: = -- (Fig. I). The line 2+2' = 2 is an asymptote to the curve. 
 
 Making the equation homogeneous, it becomes 
 27kk\k-\-k'-2iv) +8w^ = 0, 
 which is of the form 
 
 In this form it is easily seen that the curve has points of inflection 
 at the intersections of A: = and w = 0, and of A'' = and w? = 0. 
 
 Fig. I. 
 
 respectively; i. e., at the circular points, / and J. Moreover, 
 A; = is the flex tangent at /; k^ = 0, the flex tangent at J. 
 If the given line be taken to be 
 
 z = z'ti-\-au 
 the cusp locus becomes the circular cubic 
 
 ^7tikk\k-k'ti-ai)-ai^ = 0, 
 which also has points of inflection at / and J. 
 
 PROBLEM (2): Given the centre and a point; find the locus of 
 cusps. 
 
 Let us take the centre at 0; the point, as z= L 
 Map-equation of the cardioid: z = ^at — a'f. 
 Since the cardioid is to be on the point z= 1, we have the equation 
 
 * Cf. Salmon, Analytische Geomeirie der hoheren ebenen Kurven, Leipzig, 
 1882 (Zweite Auflage). p. 50. 
 
Cardioids Fulfilling Certain Assigned Conditions 11 
 
 which involves its conjugate 
 
 •t ^2-1- 
 From these two equations it is necessary to eliminate t and express 
 the result in terms of -7, which, as before, we shall designate as k. 
 The elimination of t gives 
 
 3aV2+6aa'-4(a3+a'3) = 1. 
 
 In terms of k, k' the locus is 
 
 Sk%'^+6k¥-^kk\k-{-k') = 1. 
 
 This bicircular quartic (see Fig, IV, p. 20) has its vertex at — - 
 
 and Cusp at 1. Expressed in homogeneous form 
 
 kk\Skk'-4>kw-Ww-\-6w^)-iv' = 0. 
 
 it is easy to see that the curve intersects k = 0, k' = in 4 coincident 
 points. The form 
 
 ¥%{Sk-^w)-^w{Qk¥w-^k''k'-w'') = 
 
 shows that the curve goes through /, the tangents thereat being 
 A* = and Sk — 'iw = 0. But since there are 4 coincident points at /, 
 k = (i must be a flex tangent; that is, / is a flecnode. Similarly, 
 the tangents at J are A:' = and 3A*' — 4w' = 0, the former being 
 a flex tangent. 
 
 When J) is the given point, the cusp locus is the bicircular quartic 
 
 S¥k'^-^kk\kp'^k'j))-\-Qpp'kk'-p^p''' = (), 
 
 which has flecnodal points at / and J with A; = and A:' = as flex 
 tangents. 
 
 PROBLEM (3): Given the cusp and a line; find the locus of 
 centres. 
 
 Let the cusp be at 0; the line, 2+2' = 2. 
 
 The equation of a cardioid with cusp at 6 is 
 
 z-\-^, = ^at-a't^\ 
 a 
 
 {a-a'ty 
 I.e., z=-- r-^ 
 
 a 
 
 The centre is given by 2= — , 
 
12 Cardioids Fulfilling Certain Assigned Conditions 
 Any tangent to this is 
 
 z-z'P-Sat+3a't^-^-^+ , = 0. 
 a a 
 
 If 2+2^ = 2 is to be tangent, 
 
 '2^3 2 
 
 z-z't'-Sat+Sa'f-^-^+-, =z-\-z'-2. 
 a a 
 
 From this identity, it follows that 
 and 
 
 
 (1) 
 (2) 
 
 From (1), /= — 1, —CO, or — oj^ 
 
 Cubing (2) and substituting the value t= —1, we obtain 
 
 27[a3+a'3+3aa'(a+aO]= -^ -—-2. 
 Calling }, c; , c', the equation of the locus reduces to 
 
 (c+c'-2)3+54cc' = 0. 
 
 Fig. II. 
 This is a nodal cubic, known as the Tschirnhausen Cubic,* 
 
 with vertex at - and double point at —2 (Fig. II). 
 4 
 
 If the given line is z = z^ti-\-aiy the centre locus becomes the 
 
 nodal cubic (c — c'ti — ai)^ — 27cc'aiti = 0. 
 
 PROBLEM (4): Given the cusp and a point; find the locus of 
 centres. 
 
 Let the cusp be at 0; the point, at z=l. 
 
 * It is also called the Cubique de I'Hopital. Cf. Archibald, op. cit., p, 19, 
 where he states that "the locus of the vertices of co-cuspidal cardioides 
 tangent to a given line is a Tschirnhausen Cubic." 
 
Cardioids Fulfilling Certain Assigned Conditions 13 
 
 The equation of a cardioid with cusp at is 
 
 (a-a'ty 
 
 z= —■ 
 
 (centre : c = r\ 
 a' ) 
 
 a 
 The conditions that 2; = 1 be on the cardioid are expressed by 
 
 2at-a'f--, =1 
 
 2^'-l-^ =1 
 t V- a 
 
 EHminating t from these equations and expressing the resulting 
 equation in terms of c and c', we obtain as the sought centre locus: 
 
 4cc'=(c+c'-l)2 
 
 This is a parabola with vertex at - and focus at 0. 
 
 When p is the given point, the locus is the parabola confocal 
 with the preceding: 
 
 ^cc'pp' = (cp'-\-c'p—pp'y. 
 
 PROBLEM (5): Given the double tangent and a line; find the 
 locus of centres. 
 
 Let the given line be z = tiz'; the double tangent, 2+2' = 2. 
 The equation of the cardioid may be taken to be 
 
 z—Zo = r{2t — i'^), where r is real. 
 
 That the given conditions be fulfilled, the following identities 
 must exist: 
 
 (z-zo) - {z' -z'o)t^-Srt(l-t) ^z-z'ti (1) 
 
 (z-Zo) + (z' -z'o) -Sr ^ z-\-z' -2 (2) 
 
 From (1) P = ti [whence, t = T, cor, coV, where r = ^V#i] 
 
 and Zo-^z'o-{-3r = 2. 
 
 Combining these conditions and eliminating r and ty we find the 
 
 locus breaks up into the 3 lines given by the equations : 
 
 Ml-r) 
 
 Zo = Z o T 
 
 Zo = z'oOJT-{ 
 
 1-T + t2 
 
 2coT(a)r— 1) 
 
 Zo = Z^oCxi^T-\ 
 
 l+coV^ — cor 
 2cot{t — c*)) 
 
 These three lines pass through the point of intersection of z = z^ti 
 
34 Cardioids Fulfilling Certain Assigned Conditions 
 
 and z+z' = 2. If 6 be the inclination of the given line, z = z'tu 
 the first line of the centre locus is inclined to the axis of reals at an 
 
 angle equal to -; the others at angles — - — , — - — , respectively. 
 
 The locus of cusps of cardioids which have z-\-z' = 2 as double 
 tangent and touch the line z — z'ti = consists of 3 lines through 
 the intersection of 2+2'= 2 and 2 — z'^i = such that if a be the 
 angle the centre locus makes with the double tangent, and /8 be 
 the angle the cusp locus makes with the same tangent, then 
 
 /3 = tan 
 
 (Jtanay 
 
 PROBLEM (6): Given the double tangent and a point; find the 
 locus of cusps. 
 
 Let us take the point at the origin, and the equation of the 
 cardioid as: 
 
 z = Zo-\-r — r-\-r{^t — t^)y where r is real, 
 
 = k-r(l-ty ■ 
 
 For the fulfillment of the given conditions, the following identities 
 must exist: 
 
 {z-k)-\-(z'-k')-r z 2+2'-2 (1) 
 
 k-r{l-ty = -...(2) 
 
 k'-r{ l-]y = (3) 
 
 From (1) k-\-k'+r = 2 (4) 
 
 Eliminating t and r from (2), (3), (4), we obtain as the required 
 locus : 
 
 4{^2-k-kykk'=[(-k-k'){^-k-k')+kkj 
 
 As an aid in the discussion of the curve, let us transform it to 
 Cartesian co-ordinates by means of the equations 
 
 k = x-{-iy] 
 k'=^x—iyy 
 
 The equation becomes 
 
 9a:^-6a:V+2/^+24j-?/2-8a:3- 162/2 = 
 
 This is of the form 
 
 r/V+a:3(9x-8) = 0, 
 
 which indicates a cusp at the origin with 2/ = as the cusp tangent; 
 the curve has also a branch cutting the X-axis at the point I q» 1. 
 
Cardioifh FulfiUhif/ Certain Assigned Conditions 15 
 
 Noting the behavior of the curve at infinity, we observe that it 
 has 2 paraboHc branches with directions determined hy y= =tV3x; 
 i. e., the infinite branches tend towards angles of ±120° with the 
 X-axis. 
 
 Where p is the given point, the resulting locus takes the form: 
 
 4{^-k-kyik'-p'){k-p) = [(2-k-k'){p'-k'+p-k) 
 
 +{p'-k'){p-k)]\ 
 
 which has one cusp at p and 2 parabolic branches with directions 
 
 . k 1 ±V3i 
 
 aetermmed by , / = — - — . 
 
 The locus of centres of cardioids which have z-\-z' = 2 for double 
 tangent and pass through the origin is the quartic: 
 
 3x^-lSxY-\-27y'+ Wx^-^ lUxy^+24>x^-72y^- 16 = 0. 
 This curve has 2 parabolic branches tending towards angles of 
 ±150° with the X-axis. From the form 
 
 2/2(27!/2-18x2+144a!-72) + (.r+2)3(3x-2) = 0, 
 
 it is easily seen that the curve has a cusp at the point ( — 2, 0) 
 with y = as the cusp tangent, as well as a branch cutting the 
 
 X-axis at the point I q' ^ )• 
 
 The foregoing complete the simpler cases when 3 conditions are 
 given. Let us now proceed to the cases arising from them. 
 
 II. Given 4 Conditions. 
 
 In these problems we shall first endeavor to find the number of 
 cardioids which fulfil 4 conditions as variously assigned. In the 
 cases here treated this number is found by the intersections of 
 the various loci already obtained. Such a solution is, however, 
 the maximum number and includes the imaginary curves, if there 
 are any. It will therefore be our next problem to separate these 
 imaginary solutions and thus determine the number of real 
 cardioids for each case. 
 
 PROBLEM (a) : Given the centre and 2 lines. 
 
 Let the given centre be at 0; the 2 lines, z+z' = 2 and z = z'ti-\-ai. 
 We seek the number of cardioids which have for centre and 
 touch each of the two given lines. 
 
 The cardioids which have for centre and touch the line z-\-z' = 2 
 have their cusps on the circular cubic 
 
 27kk\k-\-k') -54kk' -\-S = 0. 
 
16 Cardioids Fulfilling Certain Assigned Conditions 
 
 Similarly, the cardioids having for centre and touching the line 
 z = z'ti-{-ai have their cusps on the circular cubic 
 
 27tikk'{k-k'ti-ai)-ai^ = 0. 
 
 Thus, the cusps of cardioids which have for centre and touch 
 both of the given lines are given by the points of intersection 
 of these two circular loci. These cubics have 9 intersections. 
 Therefore, there must be 9 cusps of cardioids, and hence (since 
 cusp and centre uniquely determine a cardioid) 9 cardioids, 
 which fulfill the required conditions. But, since the" loci have 
 points of inflection at the circular points and the same tangents 
 thereat, 3 of their intersections are at each of these imaginary 
 points. Thus, 6 of the 9 cardioids are imaginary. The other 3 
 are always real. For, considering the two given lines, either 
 
 I. They will be equidistant from the given centre; or, 
 
 II. One will lie nearer that point than the other. Now the 
 circle with radius equal to the minimum radius vector of either 
 cubic lies wholly within the loop of that cubic. The vertex of the 
 second cubic will, then, lie either on the circumference of this 
 circle (Case I) or within it (Case II). In either case, this point 
 lies within the loop of the first cubic. The second curve, in tending 
 towards its asymptote, must intersect the loop of the first in 2 
 real points, and only in 2. But if there are but 3 possible real 
 intersections, and two have been shown real, the third must 
 necessarily be real. Thus, there are 3 real cardioids which have 
 a given centre and touch 2 given lines. (See Fig. X, p. 44.) 
 
 PROBLEM (b) : The centre, a point and a line given. 
 
 Take the origin as centre; z-\-z' = 2 as the given line and /> as 
 the given point. 
 
 The intersections of cusp-loci will, as before, give the number of 
 solutions. The cusp-locus for cardioids with centre and touching 
 z-\-z' = 2 is the circular cubic 
 
 27kk'(k-\-k')-5ikk'-\-S = 0; 
 
 the cusp-locus for cardioids with centre and passing through p 
 is the bicircular quartic 
 
 Sk''k'^-6pp'kk'-^kk\kp'-\-k'p)-py^ = 0. 
 
 These two curves have 12 common intersections; i. e., there should 
 be 12 cardioids satisfying the required conditions. But, since 
 
Cardioids Fulfilling Certain Assigned Conditions 17 
 
 the two curves have 4 points in common at each of the circular 
 points, there can be, at most, 4 real cardioids. However, there 
 are not always 4 real. Therefore, let us determine the regions of 
 the plane where there are 4, or only 2, or no real cardioids satis- 
 fying the given conditions. 
 
 The map equation of any cardioid with centre at is 
 
 z = ^at-aT , 
 
 Since 2+2' = 2 is to be tangent to this, 
 
 z-Sat+Sa'f-z't^ -s+z'-^ 
 
 Employing the value t= —1, we have for a, a\ any one of the 8 
 
 equations : 
 
 3a+3a/=-2 
 3aco+3aV=-2 
 3aco2+3a'a;=-2 
 
 It will be sufficient to take the first equation. Since p is the 
 given point, we have 
 
 ^at-a't^ = p 
 
 whence, eliminating a, a', we have for t the quartic 
 
 3 3-2 
 
 2/ -f" p 
 
 1 2 
 
 -t^ t ^ 
 
 = 0; 
 
 i.e., p'^4_|_2y/3_^2^2_|.2^^_|_^ = 0. 
 
 This shows, as before determined, that there are at most 4 cardioids 
 satisfying the specified conditions. We shall now seek the regions 
 of the plane where there are 4 real, 2 real, or none. 
 
 Now, for a quartic (a, 6, c, 6?, e){x, \Y with realc oefficients, 
 A = indicates the coincidence of 2 of the 4 roots; 
 A<0 gives 2 real and 2 imaginary roots; 
 A >0 gives or 4 real roots; 
 
 A > and, besides, 62_ac>Oand 12(62— ac) 2- (ae—46<Z+3c2)a2>0 
 gives 4 real roots.* 
 
 The locus A = will, then, separate the different regions of the 
 
 * Cf. Halphen, Trait e des Fonctions Elliptiques et Leurs Applications, Paris, 
 1886, Premiere Partie, p. 123. 
 
18 , Cardioids Fulfilling Certain Assigned Conditions 
 
 plane which we seek. In order to apply these criteria, transform 
 the unit circle into the real axis by the transformation 
 
 x-\-r 
 
 by which the quartic becomes 
 
 The discriminant of this quartic is 
 
 A=-256pp\p-j-p'-2)[27pp\p-\-p'-2)+8] 
 
 12ff2_/a2 = ^(2-.s'0- [(2-3^1)2-4] 
 
 o 
 
 The discriminant, equated to and geometrically interpreted, 
 breaks up into the lines 0/, OJ (which, being imaginary, may be 
 disregarded), the line p-\-p'—2 = and the circular cubic 
 27pp'{p-\-p' — 2)-\-8 = 0. Plotting these, we can easily mark the 
 sought regions of the plane. 
 
 For points on the cubic and its asymptote, A = 0; i. e., there are 
 2 of the 4 cardioids coincident. Note that p given in such a 
 position would be a cusp (on the cubic) or a point of tangency (on 
 the line). For points not on these curves, A is either >0 or <0; 
 hence, these curves mark off the regions of the plane whei*e there 
 are 4, 2, or real curves. 
 
 Let us test points in the various sections. For point _, A>0 
 
 and 12H^ — Ia^<0. Therefore, for points in the region where 
 
 is (i. e., in the loop), there are no real cardioids. This was to be 
 expected, for is the given centre and the cubic is the cusp-locus 
 for cardioids touching z-\-z' = 2. Therefore, p taken between the 
 centre and the cusp would certainly lead to no real cardioid. 
 
 For point 2, A < 0, which indicates that p in the region exterior 
 to the circular cubic and its asymptote gives 2 real and 2 imaginary 
 cardioids. 
 
 For point 1 A>0, b^-ac>Osind 12H^-Ia^>0; therefore, in the 
 region between the cubic and its asymptote, p gives 4 real cardioids. 
 
Cardioids Fulfilling Certain Assigned Conditions 19 
 
 Thus, p will determine the number of real cardioids according 
 to its position in the various regions marked in Fig. III. 
 
 It is worthy of remark that, taking as the given point, z-{-z' = 2 
 as the given line and c as the centre, the discriminant of the 
 resulting quartic in t is the parabolic cubic 
 
 which forms the centre-locus for cardioids with cusp and line 
 2+2^ = 2. Centres in the loop give imaginary solutions; between 
 the infinite branches there are 4 real cardioids; centres , at other 
 points in the plane yield 2 real and 2 imaginary curves. 
 
 Fig. hi. 
 
 PROBLEM (c) : Given the centre and 2 points. 
 
 Take the centre at the origin; the points as 1 and p. 
 
 The cusp-locus for cardioids with centre and passing through 
 1 is 
 
 Sn'^+6kk' -4^kk'{k-\-k') -1 = 0. 
 The cusp-locus for cardioids with centre and passing through p 
 is 
 
 3k^k''+6kk'pp' - ^kk'{kp'+k'p) - pV^ = 0. 
 These two curves have 16 common intersections; i. e., there should 
 be 16 cardioids satisfying the given conditions. But, since the 
 two curves have double points at I and J with one branch of each 
 having a flex thereat, and the flex tangents also being common, 
 there are 12 intersections at / and /. Therefore, there can be, at 
 most, 4 real cardioids. Let us now determine the regions of the 
 
20 Cardioids Fulfilling Certain Assigned Conditions 
 
 plane where there are 4, 2, or curves satisfying the required 
 conditions. 
 
 From the conditions that the sought curves have centre and 
 pass through 1 and p, we obtain the equations : 
 
 2at-a'f = p] 2aT-aV=l^ 
 
 From these a, a', r must be eHminated. The eHmination of 
 these quantities leads to the quartic in t^ : 
 
 4p'^{p' -i)t'^+HSp'^-5pp''+Qpp"' -9p'^)i^ +s(npY^ - Wpy 
 
 - l6pp'^+S0pp'-9)t^-\-^(Sp^-5p'p^-i-6pY-9p^)t^ 
 -\-4p^{p -1) = 0. 
 
 Fig. IV. 
 For this the invariants are 
 
 41 =S^{S2-l){S2^-\-2lS2''-S2SiS2 + 51S2-9) 
 
 8J = S^(-S2^-\-ms2^-H0s2^Si-Ss2^-\-4>8s2hi+108s2^+585s2^ 
 
 -Q24>SiS2''+nSSih2^+U4>SiS2-270S2 + 27) 
 
 ' - 26.925l3 + 2%l2( - 252^+ \0S2' 
 
 + 12^2-1) 
 + 225i(-52^+ 18^2^-5452^- 128502 
 -3352+6) 
 
 -h(352«-3052^+1752' + 30052' + 21352' 
 + 1852-9) 
 = 3352^(45i + 52--652-3)3 (-45152 + 352^ + 652- 1). 
 
 451+522 — 652 — 3 = is the cardioid with cusp at 1, centre at 0; 
 — 45152+352^+652- 1 = is the bicircular quartic with cusp at 1 
 which is the locus of cusps of cardioids with centre at and 
 passing through 1. It would seem that there are never more 
 
 A = 3^52^451 + 522-652-3) 
 
Cardioids Fulfilling Certain Assigned Conditions 21 
 
 than 2 cardioids real, for if the second point lies either out^de the 
 cardioid 
 
 451 + 52^-652-3 = 0, 
 
 or inside the quartic 
 
 -45152 + 3522 + 652-1 = 0, 
 
 there will be none real; if it lies between these curves, there 
 
 will be 2 real. 
 
 PROBLEM (d) : Given the cusp and 2 lines. 
 
 Take the cusp at 0; the lines as 2+2' = 2 and 2 = 2:'^i+ai: 
 
 The centre-locus for cardioids with cusp at and touching the 
 line 2+z' = 2 is the parabolic cubic 
 
 (c+c'-2)3+54cc' = 0. 
 The centre-locus for cardioids with cusp at and touching the 
 line z = z'ti-\-ai\s the parabolic cubic 
 
 (c-c'^i+ai)3-27cc'ai^i = 0. 
 These two curves have 9 common intersections. By a repetition 
 of the argument used in Problem (a), it may be shown that 3 of 
 these are always real. And, as a matter of fact, these loci have 
 but 3 real intersections; i. e., there are 3 cardioids with a given 
 cusp and touching 2 lines. See Fig. IX, p. 43, where the 3 cardioids 
 are shown. 
 PROBLEM (e) : Given the cusp, a line and a point. 
 
 Take the cusp at 0; the line as z+z' = 2, and the point as p. 
 
 The locus of centres of cardioids with cusp at the origin and 
 touching the line 2+2' = 2 is the nodal cubic 
 (c+c'-2)3+54cc' = 0. 
 The centre-locus for cardioids with for cusp and on the point j) 
 is the parabola 
 
 ^cc'fj)' = {cp'-]-c'p — pp'y. 
 These confocal curves have 6 common intersections, 2 of which 
 seem to be always imaginary; i. e., there seem to be at most 4 real 
 cardioids having a given cusp, passing through a given point and 
 touching a given line. 
 PROBLEM (f) : Given the cusp and 2 points. 
 
 Take the cusp at 0; the points as 1 and p. 
 
 The centre-locus for cardioids with cusp at and passing through 
 
 1 is the parabola 
 
 4cc'=(c+c'-l)2. 
 
22 Cardioids FuipUng Certain Assigned Conditions 
 
 The ceptre-locus for cardioids with cusp at and passing through 
 p is the parabola 
 
 4!cc'pp' =(cp'-\-c'p — pp'y. 
 These conies have 4 intersections in common; but being confocal 
 parabolas, only 2 of these intersections are real.* There are, 
 then, only 2 real cardioids with a given cusp and 2 given points 
 (Fig. XI). 
 
 PROBLEM (g) : Given the double tangent and 2 lines. 
 
 Take the double tangent as z-{-z' = 2; the lines as z = tiz' and 
 z = t2z'-\-a2. 
 
 The centre-locus for cardioids with double tangent z-\-z' = 2 and 
 touching the Hue 2 = 2^/1 consists of three lines through the inter- 
 section of 2H-2' = 2 and z = z'ti such that if ti = e^^^, the inclinations 
 
 of the lines to the axis of reals are ^, —^7—, — - — , respectively. 
 
 Similarly, the centre-locus for cardioids with 2+2' = 2 as double 
 tangent and touching z = t2z'-{-a2 consists of three lines through 
 the intersection of z-\-z' = '2 and z = 2^^2+02 with inclinations 
 
 o. ~~^' ""^ — » respectively, where t2 = e^i<P. These loci intersect 
 
 in 9 real points; so there are, in general, 9 cardioids with a given 
 double tangent and 2 lines. If, however, the 2 given lines are 
 parallel, there are only 6 proper cardioids; and if 1 line is parallel 
 to the double tangent, there are but 3. 
 
 PROBLEM (h) : Given the double tangent, a point and a line. 
 
 Let the double tangent be 2+2' = 2, the given point, and 
 2= 2'<iH-ai the given line. 
 
 The centre-locus for cardioids on and touching 2+ 2' = 2 as a 
 double tangent is the quartic: 
 
 (2-2-20^ + 4(2-2-20' (2+20 + 18(2-2-20222'-272V2 = O. 
 
 The centre-locus for cardioids touching 2 = 2'/i+ai and having 
 2+2' = 2 as a double tangent consists of the 3 lines as described on 
 page 13. Since a double tangent and a centre determine a 
 cardioid, we should expect the number of cardioids fulfilling the 
 required conditions to be 12, as given by the intersections of the 
 quartic and the -3 lines. However, because of the inclination of 
 the parabolic branches of the quartic and the 60°-inclination of 
 
 * Cf. Charlotte Scott, Modern Analytical Geometry, N. Y., 1894, p. 77. 
 
Cardioids Fulfilling Certain Assigned Conditions 23 
 
 the lines towards each other, there are always at least 2, and 
 possibly 4, of the intersections imaginary. Hence, there are at 
 most 10, and possibly only 8, real cardioids with a given point, 
 a given double tangent and touching a given line. 
 
 PROBLEM (i) : Given the double tangent and 2 points. 
 
 As a preliminary statement, let us remark that the 2 points 
 must be both on the same side of the double tangent; otherwise, 
 there would be no real solution. 
 
 Let 2+2;' = 2 be the given double tangent; and p the given 
 points. 
 
 The cusp-locus for cardioids with z-\-z' = 2 as double tangent 
 and going through the origin is the quartic 
 
 The cusp-locus for cardioids with double tangent 2+z' = 2 and 
 passing through p is the quartic 
 
 ^2-k-k'y{k-p){k'-f) = [i2-k-k'){p'-k'+p-k) 
 
 +{p-k){p'-k')Y 
 
 These two curves have doubly parabolic branches which are 
 respectively parallel. Moreover, they each consist of a cuspidal 
 and a non-cuspidal part, which we may designate for the two 
 curves (Q, Q^) as K, N, K^, N^, respectively. 
 
 Since a double tangent and a cusp uniquely determine a cardioid, 
 the number of cardioids satisfying the required conditions is 
 given by the number of intersections of the quartics Q and Q^. 
 There are 16 such intersections. Of these, however, 4 lead to 
 degenerate curves the cusps of which are on the line at infinity at 
 the points of intersection of the two pairs of parallel parabolic 
 branches. Let us now determine how many of the 12 remaining 
 intersections can be real. 
 
 N and iV\ K and K^ can each intersect but once; A^ and K^, K 
 and N^ may have 0, 1, or 2 intersections. Moreover, p may be so 
 placed as to combine the maximum number of intersections of the 
 cuspidal and non-cuspidal parts of the respective curves (see 
 Fig. V, p. 43). Thus, there are, at most, 6 real proper cardioids 
 going through 2 points and touching a given double tangent. 
 
 Here conclude the simpler problems, in each of which the data 
 include one of the fixed elements of the cardioid. We shall now 
 consider a few more difficult problems in which lines and points 
 form the data. 
 
More Difficult Problems. 
 
 I. When 3 Conditions Are Given. 
 PROBLEM (7) : Given 3 lines; find the locus of centres. 
 
 Let the 3 given lines be z = z'ti-\-ai {i= 1,2.3). 
 Since each of the lines is to be tangent to the cardioid, we have 
 the identities : 
 
 z-z'P-zo-Sat-\-Sa'f-}-z'ot^--z-z%-a, 
 whence, t^ = ti (1) 
 
 and Zo-\-Sat-a'f-z'ot^ = ai (2) 
 
 From (1), we get / = ^V U = Ti, cor,, coV,-. 
 
 Substituting the value t = Ti in (2), we get the three equations: 
 
 Zo-^3aTi — Sa'Ti^ — z'oTi^ = ai 
 
 zo+3aT2 — 3aV 
 
 Zo-{-SaT2 — Sa'Tz^ — z'oTs^ = a3 
 Eliminating a, a' from these 3 equations, we obtain the line 
 
 = 0, as part of the 
 
 locus. Since we can choose each of the 3 values oft{t = Ti, cor,, co're) 
 in 3 ways, there are 27 different combinations or 27 different sets 
 of 3 equations in zo, z'o, a, a'. But these lead to only 9 distinct 
 lines, arising thus : 
 
 Zo—z'oTi^ — ai 
 
 Tl 
 
 Tl' 
 
 Zo—z'oT'^ — ai 
 
 T2 
 
 T2' 
 
 Zo — z'oTz' — Ciz 
 
 T3 
 
 T3^ 
 
 [ti, r2, T; 
 
 {coTiy cor 2, CO 
 
 The combinations ^cori, cor 2, cors > give the same line, which 
 
 [coVi, C0V2, C0V3J 
 
 we shall designate as Zi. 
 
 coVi, r2, cor 3 j- give Zj 
 n, cor2, C0V3J 
 
 T2y OiTs 
 
 0)Tu 
 
 cor 2, 
 
 C0V3 
 
 give Z4; 
 
 coVi, 
 
 C0V2, 
 
 r3 J 
 
 
 TU 
 
 cor2. 
 
 T3 1 
 
 
 cori. 
 
 WV2, 
 
 cor3 
 
 ■ give Ze; 
 
 coVi, 
 
 T2y 
 
 C0V3 
 
 
 cori, 
 
 T2y 
 
 T3 
 
 
 coVi, 
 
 cor2, 
 
 cor3 
 
 > giveZg; 
 
 Tl, 
 
 C0V2, 
 
 C0V3 
 
 
 24 
 
 
 
 
Cardioids Fulfilling Certain Assigned Conditions 25 
 
 The directions of, these lines are determined by the ratio of the 
 coefficient of Zo to the coefficient of z'o in the respective equations. 
 The coefficient of zo for Zi is 
 
 1 ri 
 
 1 T2 
 
 1 rs 
 
 = (ri — r2)(T2 — r3)(T3 — n). 
 
 The coefficient of z'o for the same Hne is 
 
 73^ T3 
 
 = —TiT2T3(ti—T2){t2—T3){t3 — Ti). 
 
 The ratio of the coefficients of Zo and z'o is 1: — 0-3. Likewise, it 
 will be found for L2 and Z3 that the ratio of the coefficients 
 of Zo : and z'o is 1: — 0-3; i. e., Zi, Z2, -L3 are parallel. 
 
 Similarly, Li, Ze, Ls are parallel with directions determined 
 by the ratio 1 : — cocs; and Z5, Lj, Z9 are also parallel with directions 
 determined by 1: — C0V3. Now, if 6, cp, \f/ be the angles these 
 t hree sets of lines make with the real axis, then 
 
 Similarly, 0—4/==^=- 
 
 o 
 
 and <p—^==^^\ 
 
 i. e., the sets of parallel lines are inclined at angles of 60° towards 
 one another. 
 
 Thus, the locus of centres of cardioids touching 3 given lines 
 consists of 3 sets of parallel lines, inclined to each other at angles 
 of 60°. (See Fig. VI.) 
 
 Let us endeavor to interpret this locus geometrically. Desig- 
 nating the 3 given lines as h, h, h, respectively, of which Zi and Zj 
 intersect at an angle a, let us consider any one cardioid touching 
 these given lines. This cardioid possesses a cusp which may 
 have any one of three positions relative to Zi and Z2 (its relation to 
 Zs being, for the present, left out of account) according as the 
 
 cuspidal rays to the points of tangency make an angle of a, 
 
 •1 o 
 
2G Cardioids Fiil filling Certain Assigtied Conditions 
 
 2 2 
 
 Qa + 120°, or ^aH- 240°. With each of these three cusps is asso- 
 
 o o 
 
 ciated a definite centre which, as the variable cardioid assumes 
 positions fulfilHng the required conditions, traces out one of the 
 nine lines found, as the centre locus. Moreover, each of these 
 three lines forms an angle of 120° (or 60°) with the others. Further, 
 combining each of these three positions relative to Zi, 1 2 with the 
 three possible relations with Zs, we obtain the complete locus of 9 
 lines as above. 
 
 PROBLEM (8): Given 2 lines and a point; find the locus of 
 centres. 
 
 Let \ z' \he the given lines, and p the given point. 
 
 1^=7? J 
 
 The identification of these lines with a tangent to the cardioid 
 leads to the equations : 
 
 111 
 
 /'3=,/^2. t' = t^\ Ojti\ Oi%^ (1) 
 
 Zo-z'ot''-\-Sai'-Sar'~ = (2) 
 
 T' = ~-,;r = ti ^a^/l \icHi 3 (3^ 
 
 2o-2V+3ar-3aV = (4) 
 
 We have, also, the equations : 
 
 p-zo-^at+a't^ = (5) 
 
 (z'o-py^+^a't-a^O (6) 
 
 The elimination of Z, Z', r, a, a' from these six equations will give 
 the required locus. 
 
 From (2), (4), we obtain 
 
 — ZoSi — z'oS'^ 
 
 3a = - 
 
 From (5), (6): 
 
 where r^ = ;;+^ 
 
 — Z0 — Z0S1S2 \S2 = tT 
 
 a' —2a p — zo 
 
 a' —2a v — zo 
 
 z'o-p' 2a' -a 
 
 Q z'o-p' 2a' -a 
 
 0. 
 
Cardioids Fulfilling Certain Assigned Conditions 27 
 
 Substitution of the values of a, a' in this determinant furnishes a 
 quartic as part of the sought locus. Since there are 3 values each 
 for t' and r, there seem to be 9 quartics in Zo, z'o, ^i, ^2. But these 
 9 reduce to 8; for the combinations of t\ r, divide off in sets of 3, 
 producing only 3 distinct developments of the determinant. 
 These combinations are, in fact, the following: 
 
 t\ r 1 Oil', coV ] oiH\ 
 
 a)t', cor \ , coH', r > , t'. 
 
 H\ coV r, o)T cor, 
 
 cor 
 coV 
 
 r 
 
 Now the only terms in the development involving a or a' are those 
 in a^, a'^ aa\ a^a'^. The effect of replacing t\ r by cat', cor, respec- 
 tively, in the values of a and a' is to multiply the equation by 
 co^( = l). Similarly, the substitution of co^^', coV for t' and r, 
 respectively, multiplies the equation by 1. Thus, the combina- 
 tions in the first set lead to the same line. Likewise, those in the 
 other 2 sets lead to but 2 other distinct lines. 
 
 These 3 sets of combinations lead to the following values of ^i, ^2: 
 
 2 
 
 Sl = 
 
 = 2 cos 
 
 i« 
 
 Si- 
 
 = 1 
 
 
 Sl= 
 
 = 2 cos 
 
 (l-H 
 
 52 = 
 
 = 1 
 
 
 5i = 
 
 = 2 cos 
 
 (M') 
 
 52 = 
 
 = 1 
 
 where ti = e . Hence, 52 may always be taken equal to 1, and 
 the development is 
 
 iZo + z'oSiy{ZoSi + z'oy-4[{Zo-p){Zo + z'Siy-{-{z'o-p'}(ZoSi + z'oy] 
 
 -\-18{zo-p)iz'o-p')izo-\-z'oSi)(zoSi-\-z'o)-27{zo-pylz'o-p'y = 0, 
 where Si has one of the values above. This equation can be put 
 in the form 
 
 (si''-4^)W-SiZoz'o-\-z'o^y+f(zo, z'o, p, py = 0, 
 which indicates 2 parabolic branches. These branches are inclined 
 
 at angles ±-, =^ I 0+^ )' ^ ( q"^ q I ^^^ ^^^ three quartics, respec- 
 tively. Moreover, the locus has a cusp at the point where the 
 variable cardioid comes into the position where the fixed point 
 
28 Cardioids Fulfilling Certain Assigned Conditions 
 
 is its cusp. Thus, the centre locus consists of 3 quartics with 
 properties as described above. The curves are of the form shown 
 in Fig. VI. 
 
 Geometrically interpreted, it appears that the 3 quartics are 
 traced out by the centres of those cardioids (touching the given 
 lines and on the given point) of which the cuspidal rays to the 
 
 points of tangency are inclined towards each other at angles -9y 
 
 o 
 
 2 2 
 
 -^+120°, -^+240°, respectively, where is the angle between the 
 
 o o 
 
 given lines. 
 
 The specialization of this problem, arising when p is the point 
 of tangency on one of the given lines, proves instructive. Let uS 
 first view the problem from a geometric standpoint and see what 
 a priori information we can derive therefrom. Consider a case 
 where 3 lines, lu h, h, are given, of which h and I2 are fixed in 
 position, while /s rotates about a fixed point p on h. For any 
 position of the variable line, the centre-locus consists of 3 sets of 
 
 3 lines as found in Problem (7). The limiting position of h is 
 that in which it comes into coincidence with li. Then p becomes 
 the point of tangency on /i, and the conditions become equivalent 
 to the data of this specialized problem. Thus, we may expect 
 sets of lines in the centre-locus. Will the 3 sets appear.^ To 
 answer this, recall the fact that the 3 sets of lines in the centre- 
 locus of Problem (7) are associated with cardioids such that the 
 cuspidal rays to the points of tangency make angles equal to 
 
 2 2 2 
 
 -dy -^+120°, -^-f 240°, respectively, where is the angle between 
 
 the tangents. In the case under consideration 6, the angle between 
 li and /s, is 0; and the cuspidal rays to the point of tangency make 
 an angle 0° only. Thus, the 3 sets of 3 lines seem to reduce to 
 one set of 3. What happens to the other 2 sets of lines .^ They 
 are evidently associated with cardioids such that the cuspidal 
 rays to the points of tangency make angles of 120° and 240°, 
 respectively; i. e., with cardioids of which the tangents make 
 angles of 180° or 360°, that is, with cardioids with parallel tangents. 
 Thus, the other 2 sets of lines belong, not to the case where Iz 
 and li are coincident, but to the case where they are parallel.* 
 
 * Cf. the theorem indicated on p. 8. 
 
Cardioids Fulfilling Certain Assigned Conditions 29 
 Let us now consider the problem analytically. 
 
 liCt us take ] „_^_ r as the given lines, and p as the point on 
 
 [ ti" I 
 z = t^z'. Let us find the locus of centres. 
 
 The identification of the given lines with a tangent leads to 
 equations (1), (2), (3), (4) of Problem (8). Since p is the point 
 of tangency on z = tiz' and is also on the variable cardioid, we have 
 
 p = Zo^%at'-a't'^ (5) 
 
 The elimination of a, a' from (2), (4) and (5) gives the line 
 
 Zo-z'or^ St 3r2 
 
 Zo-z'qI'^ ' St' St"" =0 
 
 Zo-p n' t"" 
 
 as the required locus. 
 
 By development of the determinant, this line is found to be 
 
 Zo{^t'\ - i'T"- - 1'^) - z'o t'^-ri^i'^ - t'T"- - 1'^) - Spt'rif - r) = 0. 
 
 Although there are 9 possible combinations of t' and r, there are 
 
 not the same number of lines in the locus. In fact, since t' = -^ 
 
 T 
 2 
 
 ", or - , and the combinations of t' and r are all of the third 
 
 r T 
 
 2 2 t_ 
 
 degree, there are only 3 distinct lines with clinants ^i^, co^l^ a)^<l^ 
 respectively; i. e., the 3 lines are inclined towards each other at 
 angles of 60° (or 120°). 
 
 PROBLEM (9): Given 2 points and 1 line; find the locus of 
 centres. 
 
 Take the line as z-\-z' = %, the points as and p. Let the equa- 
 tion of the cardioid be 
 
 z = zo-\-^at — a't^. 
 Since the cardioid touches 2+2;' = 2, we have the identity: 
 z-zo- iz' -z' o)t^-Sat-Sa't'^ ~ z-\-z' -'l\ 
 
 whence, /^= — 1; ^= — 1, -co, — co^ (1) 
 
 zo-z'ot^-^Sat-Sa'f = % (2) 
 
 Since p and are points on the cardioid, we have 
 
 p = Zo+^at'-a't'^ . (3) 
 
 f , , ^a' a , 
 
 V =^o+Y ~j2 W 
 
-a' 
 
 2a 
 
 Zo — p 
 
 
 
 
 
 -a' 
 
 2a 
 
 Zo 
 
 p'-z'o 
 
 ^^a' 
 
 a 
 
 
 
 
 
 v'-z'o 
 
 -2a' 
 
 a 
 
 30 Cardioids Fulfilling Certain Assigned Conditions 
 
 = 2o+2ar-aV2 (5) 
 
 o=,'„+?£'_«^.. ..(e) 
 
 The elimination of t, t\ r, a, a' from these 6 equations will give 
 the required centre-locus. 
 
 Eliminating t' from (3), (4), we obtain the equation: 
 
 = 0. 
 
 Developed, this is 
 
 'Sahi'^-^a\z'o-p')-^a'\zo-v)-Qaa'{zo-v){z'o-p') 
 
 -i-{zo-py(z'o-pr = (7) 
 
 The elimination of r from (5), (6) leads to 
 
 3a2a'2-4a32o-4a'Vo-6aa'zo2'o+2oVo2 = (8) 
 
 From (1), employing the value ^= — 1, 
 
 2o+2'o-3a-3a' = 2 (9) 
 
 From (7), (8), (9), a, a' have still to be ehminated. Let us 
 transform to Cartesian co-ordinates by replacing a by a + i6, 
 z by x-\-iyy p by p-\-iq, with a corresponding substitution for their 
 conjugates. Equation (8) becomes 
 
 Equation (7) becomes a similar equation in which x — py y — q 
 
 X— 1 
 replace x and i/, respectively. Equation (9) gives a = — — . 
 
 o 
 
 Substituting the value of a in (7) and (8), we obtain the quartics 
 
 in 6: 
 
 2764-7263?/-M862(4;r2-9i/2-2x-n)+7262/(x- 1)2+/(.T, ?/)4 = 
 
 2764-726"X2/-^) + 186y(x, y, p, qy-\-72b{y-q){x- ly 
 -\-^{x,y,p,q)'^ = 
 
 The elimination of b leads to an 8-row determinant, which, devel- 
 oped, is an equation of the 12th degree in x and y. Thus, the 
 centre-locus is a 12-ic. We shall not attempt to discuss the 
 singularities of this locus. 
 
 Let us now find the centre-locus for the case where one of the 
 points is on the line; that is, let the data be the line 2+2' = 2 and 
 the points and p, of which p is on the given line. 
 
Cardioids Fulftlling Certain As^gned Conditions 31 
 
 From p. 29, we have / = — 1, -co, — co^ (1) 
 
 Zo-z'ot^-\-Sat-3a't'^ = 2 (2) 
 
 SaW^-^a^zo-W^z' o-Qaa'zoz' o +zo^z'o^ = (3) 
 
 Since p is the point of tangency on the cardioid, 
 
 j) = Zo-\-^at-a'f .(4) 
 
 From (2) and (4), 
 
 3a/ = 3j9-22o+2V_2 
 
 ^' = 3p'-2.^+f3-2 
 
 Since a, a', occur in (3) only in the combinations a^a'^, a\ a'^ aa\ 
 any of the values /= — 1, — co, or — oj^ will lead to the same result. 
 Employing any one of these values, we obtain a quartic in Zo, z'o 
 as the sought centre-locus. 
 
 We might have expected such a result, for the data from the 
 limiting case for 2 lines and a point,* where the lines coincide and 
 their point of intersection becomes the point of tangency. More- 
 over, we can account for the reduction in the number of quartics 
 in the locus by recognizing that the 2 quartics which do not 
 appear here belong, not to the case where the 2 lines coincide, 
 but to the case where they are parallel. 
 
 PROBLEM (10): Given 3 points; find the locus of cusps. 
 
 Let the given points be 0, p, q, and the map-equation of the 
 cardioid be 
 
 z = h-a{l-t)\ 
 
 If /i be the value of ^ at 0, 
 
 k-a{\-tiY = 0\ 
 
 whence, {k¥-a'k-aky = 4>aa'kk' (1) 
 
 Similarly, since the cardioid is on ^, 
 
 [(k-p){k'-p')-a\k-p)-a{k'-p')Y = 4>aa\k-p)(k'-p').{2) 
 Since g- is a third point on the cardioid, we have 
 
 [{k-q){k'-q')-a'ik-q)-a{k'-q')]^ = 4>aa\k-q)(k'-q')..{S) 
 We have to eliminate a, a\ from (1), (2), (3). 
 
 * Cf. Problem (8), p. 26- 
 
32 Cardioids Fulfilling Certain Assigned Conditions 
 
 Transforming to Cartesian co-ordinates and letting 
 
 k = X-\-iY 
 a = A-^iB 
 
 q=U-^iV 
 the equations become 
 
 (Z2+F2-2^Z-25F)2 = 4 (^2_^52)(Z2+r») (1) 
 
 [{X-PY+{Y-QY-<iA{X-P)-W {Y-Q)f 
 
 = 4(.42+52) [(X-P)2+(F-Q)2]. . .(2) 
 [{X-UY-V{Y-VY-%A{X-U)-W {Y-V)f 
 
 = 4(^2+^2) i(^x-UY+{Y-'Vy]. . .(3) 
 
 Interpreting A and B as Cartesian co-ordinates, these equations 
 represent parabolas with as focus. We seek the condition that 
 the three curves have a common point. 
 
 Since (1) and (2) [also (1) and (3)] are confocal parabolas, 
 they have 3 known common lines, of which two are imaginary (the 
 lines from I and J); and the third (the line at infinity), real. 
 Their fourth common line is, therefore, rationally obtainable and, 
 therefore, their self -con jugate triangle. This triangle will have 
 1 real vertex and 2 imaginary ones; and the one real pair of common 
 chords of the two parabolas will pass through the real vertex. 
 If we find the equations of the line-pairs common to parabolas ( 1) 
 and (2), (1) and (3), respectively, and require that, with properly 
 chosen sign, they be simultaneously true, their solution will yield 
 the co-ordinates of the point which is to be common to the 3 
 parabolas. The substitution of these values in the equation of 
 one of the parabolas will give the cusp-locus sought. 
 
 Parabola (1) in line co-ordinates (using ^, ?;, f corresponding 
 to a, a'y w) \^ 
 
 V^k'+nk+^vkk' = 0. 
 Parabola (2) is 
 
 Vnk'-p')+mk-p)+^rj{k-p){k'-p') = 0. 
 
 Their fourth common line is 
 
 ^ ^ I = 
 
 kp\k-p) k'p{k'-p') kp'-k'p 
 
 The one real diagonal point of the 4-line is [kp'{k — p)y —k'p{k'—p'), 
 kp'-k'p]. We can now find the equation of the sought line-pair 
 by writing the pencil C7+XF=0 built up on the 2 parabolas, and 
 
Cardioids Fulfilling Certain Assigned Conditions 33 
 
 determine X so that the resulting curve shall contain this point. 
 We easily find as the equation of the line-pair 
 
 {k-v){k'-p'){W-ay-a'kY = kk\{k-p){k'-'p')-a{¥-'p') 
 
 -a\k-j>)? 
 
 Using Cartesians, the line-pair is 
 
 [(X-P)2+(F-Q)2 [(X2-f-F2-2.4X-25F)2 
 
 = (Z2+F2)[X-P +F-e -^AX-P-WY-Q]\ 
 
 X2-fF2 = p2 ' ] 
 
 Now, if we let ^ 2 I' ,where p, g are taken to be 
 
 X-P-\-Y-Q=a'] 
 
 the positive square roots, the 2 lines are 
 
 (r(p2-2ylX-25F)=±p[(72-2^(X-P)-2B(F-Q)]....(4) 
 Similarly, the line-pair common to the parabolas (1) and (3) is 
 
 T(p2-2^X-2PF) = ^p[r2-2.4(Z-C/)-2P(F-F)]....(5) 
 where r = +^I{X-UY + {Y -Vy 
 
 If the three curves are to have a common point, equations (4) and 
 (5), with properly chosen sign, must be simultaneously true. 
 In terms of A and B, these equations are 
 
 2^[Pp-Z(a-+p)]-f2P[Qp-F(a+p)] + crp((7+p) = 0. . . . (4) 
 ~ ^A[Up-X{T+p)] + ^B[Vp-Y{r+p)]^Tp{r+p) = Q. . ..(5) 
 
 The solution of these two equations and the substitution of the 
 resulting values of A and B in the equation of one of the parabolas 
 will yield the cusp-locus sought. 
 From (4), (5) 
 
 o"(o-+p) 
 T{r+p) 
 
 [Qp-Y{cT+p)] 
 [Vp-Y{r+p)] 
 
 [Pp-X{a-\-p)] [Qp-F(cr+p)[- 
 [Up-X{r-\-p)] [Vp-Y{r-^p)] 
 
 B 
 
 (t{(T-\-p) 
 r{r-\-p) 
 
 [Pp-X{a^p)] 
 [Up-X{r+p)] 
 
 [Pp-X(cr-l-p)] [Qp-Y{cr-\-p)] 
 [Up-X{r+p)] [Vp-Y{T^-p)] 
 
M Cardioids Fulfilling Certain Assigned Conditions 
 
 Designating the determinants in the numerators as Di and D2, 
 respectively, and the denominator determinant as Do, 
 
 r (/)i^+/)2^) 
 
 AX-\-Br = 
 
 4 Do^ 
 
 2 Do 
 
 Di^+D2' = pV{p+,anU' + V^) + {a-Tnp-{-a)Hp + ry 
 
 + r'{p + rnp-^+Q^)-2aT{p+<T){p + T)(PU-hQV)] 
 -2p((r+p)(pH-r)(cr-T)[(T(p+o-)(rZ+FF)-r(p+r)(PX+QF)] 
 
 Z)lZ + 2>2F = p -(7(p + (7) 
 
 X Y 
 U V 
 
 + r(p-{-T) 
 
 X 
 
 p 
 
 Do may be expressed thus 
 
 P Q 
 u V 
 
 p\p 
 
 X Y \ 
 
 X Y 
 
 P Q 
 
 The substitution of these values in 
 
 [X'-\-Y'-2(AX-\-BY)Y = 4(A'-\-B^){X'+Y^) 
 
 gives a result which, after division by p-, is of the form 
 
 E-\-Fpa + GpT+HaT = 0, 
 
 where E is of the sixth degree in X, F; and F, G, and H are of the 
 fourth degree in the same variables. Rationalizing: 
 
 {E+H(TTy=iFpa-\-GpTy 
 
 E^-\-H^(t^t''-F'p^(t''-GVt^=(^FGp'-- 2EH)(tt 
 
 Squaring again, 
 
 E^+FY(T'+GYr'+H'(T^T^-2lE^F^(TV-\-E''GVT^-{-H^GYa^T' 
 +E2HVt2+F202pV2t2+F277VVV] +SEFGHp^a^T^ = 
 
 Thus, it appears that the required locus is of the 24th degree. 
 We shall not attempt to discuss the singularities of the curve 
 beyond remarking that, since there are 4 cardioids with a given 
 cusp and on 2 given points, it would seem that this locus has 
 quadruple points at each of the 3 given points. 
 
 Further, we may reduce the unrationalized equation to a con- 
 venient form -thus : 
 X^+Y'-2(AX-\-BY) = 
 
 p {p' 
 
 P Q 
 V V 
 
 -ip'-"') 
 
 X Y 
 U V 
 
 + {P'-T^) 
 
 X Y 
 
 P Q 
 
 p 
 
 P Q 
 
 U V 
 
 -(p+o) 
 
 X Y 
 
 U V 
 
 + {p+r) 
 
 X Y 
 
 P Q 
 
Cardioids Fulfilling Certain Assigned Conditions 35 
 
 This numerator can be reduced to 
 
 -P 
 
 — P 
 
 
 
 P' 
 p'+q' 
 
 M2 + r2 
 X 
 
 V 
 u 
 
 
 , which, put in the form 
 
 and equated to 0, is the product of 
 
 the equations of the circle through the three given points, 0, p^ q, 
 and the hues 0/, OJ. Let the numerator be called —pC. Notin a 
 that p, 0-, r are factors of A'^-j-B^, this expression may be designated 
 as p(7tK. Thus, the required locus may be put in the form 
 
 -C = ^P(ttKDo, 
 
 which shows that the locus intersects each of the lines joining the 
 given points with I and J only at these circular points. 
 
 II. When 4 Conditions Are Given. 
 
 We may remark that in the problems which follow, the inter- 
 sections of loci do not furnish the exact number of solutions as 
 is the case in the simpler problems. In these latter, the elements 
 common to curves on both loci are such as to uniquely determine a 
 cardioid; while in the cases about to be considered, the common 
 elements do not so determine a single cardioid. 
 
 PROBLEM (j) : Given 4 tangents. 
 
 Let the 4 given lines be z = z'U-\-at {i=l, 2, 3, 4). The 
 centre-locus for cardioids touching z = z'ti-{-ai (i=l, 2, 3) con- 
 sists of the 9 lines as described on page 25. The centre-locus 
 for cardioids touching z = z'ti-{-ai (i=l, 2, 4) consists, similarly, 
 of 9 lines which divide off into 3 sets of parallel lines with directions 
 60° apart. 
 
 It might at first sight seem that there are 81 cardioids touching 
 the four given lines. But, recalling the fact that the locus is in 
 each case generated by the centres of 3 variable cardioids so 
 related to the lines h, h that the angles between the cuspidal rays 
 
 2 2 2 
 
 to the points of tangency are 6, _^+120°, -^+240°, respectively, 
 
 it is readily seen that only the intersections of corresponding lines 
 of the two loci give a common centre for the 4 lines. The lines 
 
36 Cardioids Fulfilling Certain Assigned Conditions 
 
 correspond in sets of three. Thus, there are 27* cardioids touching 
 4 given lines. 
 
 As a confirmation of this result, we may note that, although 
 the centre-loci for cardioids touching /i, 1 2, h and h, 1 2, h intersect 
 in 81 points, these intersections will not give 81 cardioids unless 
 there is but a single cardioid with a given centre and touching 2 
 given lines. There are actually 3 such cardioids, which suggests 
 the reduction of the number of cardioids touching 4 given lines 
 to 27. 
 
 PROBLEM (k) : Given 3 lines and a point. 
 
 Let the given lines be z = z'ti-{-ai (z = l, 2, 3), where 1^ = - 
 
 ti 
 
 and ai = a2 = 0; take p as the given point. 
 
 The centre-locus for cardioids touching z = z'ti-]-ai (z=l, 2, 3) 
 consists of three sets of parallel lines similar to those described 
 in Problem (7). The centre-locus for cardioids touching 
 z=z'ti-{-ai (i= 1, 2) and passing through p consists of the 3 quar- 
 tics discussed on page 27. 
 
 Although these loci intersect in 108 points, there are, by no 
 means, that many cardioids. For, with each centre on these loci 
 is associated a definite cusp; and it is only the intersections of 
 centre-loci corresponding to the same cusp that give a common 
 centre for a cardioid on the given point and touching the given 
 lines. Now these loci pair off in such a way that to each quartic 
 correspond 3 of the 9 lines ;t so that there are 36 centres of cardioids, 
 and, therefore, 36 cardioids satisfying the given conditions. Fig. VI 
 illustrates the case where the given lines form an equilateral 
 triangle. 
 
 When the point is on one of the lines, there are but 9 cardioids. 
 For, designating the lines as lu I2, h, with ^ on Zi, the centre-locus 
 for cardioids on Zi, pi, I2 consists of 3 lines;t likewise, the centre- 
 locus for cardioids onlu pu h consists of 3 lines. These loci have 
 9 common intersections, which yield centres for cardioids on 
 lu hy h and p. 
 
 * Professor Morley has indicated the number as 2^. Cf. Tram. Amtr. 
 Math. Soc, 1900, Vol. 1, p. 114. As above shown, it would seem that there 
 are SK 
 
 t Cf. p. 24. 
 
 t Cf. Problem (8), p. 26. 
 
Cardioids Fulfilling Certain Assigned Conditions 37 
 
 Fig. VI.— The loci marked 
 
 are generated by the centers of 
 
 cardioids of which the cuspidal rays to the points of tangency make angles 
 of 40°, 160°, and 280°, respectively. 3 of the 36 possible cardioids are here 
 shown. For this particular case, there are actually 12 of the cardioids real, 
 as shown by the marked centers. % 
 
 PROBLEM (1) : Given 2 lines and 2 points. 
 
 Let the given lines be z = t^z' and z = — ; the given points, p and q. 
 
 The centre-locus for cardioids on p and touching z = t^z\ z = —^ 
 
38 Cardioids' Fulfilling Certain A.^f^igned Conditions 
 
 consists of the 3 quartics discussed on page 27. The centre- 
 locus for cardioids on q and touching the given lines consists of 
 3 quartics similar to the above, in the equations of which q and q' 
 replace p and /)', respectively. 
 
 These quartics, which we may designate as Qu Q2, Qs, Q\, Q't, 
 Q's for the two loci respectively, pair off in such a way that every 
 intersection of Qi and Q'i gives a cardioid on the given lines and 
 points. There are 48 intersections. But Qi and Q'» are both 
 doubly parabolic with the same points at infinity. This accounts 
 for 4X3 intersections, which are to be regarded as improper. 
 This leaves 36 proper intersections; i. e., there are 36 cardioids 
 which touch 2 given lines and pass through 2 given points. 
 
 If one of the points is on one of the lines, this number reduces to 
 12. For the centre-locus for cardioids on /i, h, p\ (on U) consists 
 of 3 lines; the centre-locus for cardioids on /i, j>i, 7? 2 is a quartic. 
 These loci have 12 common intersections, which yield 12 centres 
 for cardioids on /i, h, pi, />2- 
 
 Further, if />2 is on 1 2, the number reduces to 6. For the centre- 
 locus for cardioids on /i, pi, 1 2, consists of 3 lines, as does that for 
 cardioids on /i, I2, P2- Although these loci have 9 common inter- 
 sections, only 6 are in the finite plane; for since the clinants of the 
 lines in both loci are determined by the angle of intersection of 
 li and /o,* 3 6f the intersections are at infinity. 
 
 PROBLEM (m) : Given 1 line and 3 points. 
 
 Let the given line be 2+z' = 2; the given points, 0, p, q. The 
 centre-locus for cardioids on and p and touching 2-f z' = 2 is 
 a curve of the 12th degree; the centre-locus for cardioids on 
 0, qy and the given line is, likewise, a curve of the 12th degree. 
 These curves have 144 common intersections; so there cannot 
 be more than 144 cardioids which touch the given line and 
 pass through 0, p, and q. There will likely be many less than 
 144; and the fact that there are 4 cardioids with a given 
 centre, point and line suggests that this number may reduce to 36. 
 
 If one of the points is on the given line, there is a further reduc- 
 tion of the number to 12. For, the centres of cardioids on /i, 0, 
 and p (where p is on h) describe a quartic; the centres of cardioids 
 on /i, p, q, describe a quartic. These loci have 16 common inter- 
 sections, only 12 of which yield centres of non-degenerate curves. 
 
 » Cf. p. 29. 
 
Cardioids Fulfilling Certain Assigned Conditions 39 
 
 PROBLEM (n) : Given 4 points. 
 
 Let 0, p, q, r be the given points. 
 
 The cusp-locus for cardioids on the points 0, p, q is a curve of the 
 24th degree; similarly, the cusp-locus for cardioids on 0, p, r is a 
 curve of the same degree. Although these two loci intersect in 
 24^ points, there will, by no means, be 24^ cardioids on the 4 given 
 points. While we shall hot attempt to ascertain the exact number, 
 we remark that the fact that there are 4 cardioids with a given 
 cusp and on 2 given points will greatly reduce this 24^. 
 
 We shall conclude with a brief discussion of a problem related 
 to those we have been treating, the solution of which would prob- 
 ably prove of value in the determination of the number of cardioids 
 on 4 points. It has to do with the number of real equilateral 
 triangles which can be inscribed in a given cardioid when 1 vertex 
 is fixed. 
 
 Take the given cardioid with its cusp at 0, and at the right of 
 the figure. Its map-equation is z= —{l — ty. Let this be the 
 standard size. 
 
 First, let us consider a triangle of any shape, and let us determine 
 the relations between the vectors to the vertices. Let the vectors 
 to the vertices of a triangle, of which the sides and angles are 
 Pi, P2, ps, <p, ^y 0, respectively, be a, 6, c, the terminal end of vector 
 a being at the apex of the angle <^, of vector b at xj/, and of vector 
 c at ^. Then 
 
 Pi P3 
 
 P3 P3 
 
 Now, let a and b be the points Zi and 22 on the given cardioid. 
 Then 
 
 P3 P3 
 
 If we regard Si as fixed, 22 as variable, the third vertex of the 
 triangle, c, will describe a cardioid with its cusp at the point 
 
 Pl/e+<p)^^_^^y^ of size l+%<^+<^) ! and with orientation- 
 
 P3 P3 I 
 
 angle [l+^V(«+*')l. 
 
 P3 
 
40 Cardioids Fulfilling Certain Assigned Conditions 
 
 For the case of the equilateral triangle, this becomes particularly 
 simple. Then the third vertex, c, travels on a cardioid with map- 
 equation: c = a)(l — tiy-\-cjo^{l — ty. This cardioid is of standard 
 size, with an orientation of 60° and its cusp at coil — tiY; i. e., the 
 cusp is at the same distance from as is Zi, with its vector turned 
 through an angle of — 60°. Moreover, this c-cardioid goes through 
 the point Zi on the original cardioid. Either of two arguments 
 may serve to show this : 
 
 1. When the second vertex is made to approach — (1 — ^i)-, the 
 third vertex does likewise. Thus, the fixed vertex, Zu towards 
 which, in the case of an infinitely small triangle, the other two 
 vertices tend, must be on both cardioids. 
 
 Fig. VII. — The sixth intersection is not shown. 
 
 2. Analytically, if in the equation above t = tu then 
 c=(o;+co2)(l — <i)2= — (1 — ^i)2; i. e., zi is a point on both cardioids. 
 
 If Zi is given, the number of equilateral triangles is the number 
 of intersections of two cardioids less one, since the infinitely small 
 triangle, all the vertices of which are at Zi, should hardly be 
 regarded as a proper triangle. This number is 7, since the common 
 cusps at / and J account for 8 of the 16 intersections. It happens, 
 however, that 2 of these 7 intersections are always imaginary; so 
 that the maximum number of real proper equilateral triangles 
 inscribable in a given cardioid is 5. However, there are not 
 always 5; there may be but 3, or only 1. 
 
('((rdioids Fulfillinf/ Certain Assigned Conditions 
 
 41 
 
 Tliat there may be as many as 5 is easily shown: for, when t 
 is a turn through an angle numerically less than ±30°, for the 
 point 05(1-/1)2 the cusp of the cardioid-locus of the vertex, c, is 
 inside the given cardioid. When ti, then, is a turn through a very 
 small angle, there will be positions where the number of real 
 intersections will be 5. (See Fig. VII, a). For one value of ti, the 
 two curves will be tangent, and 2 of the 5 will coincide (Fig. VII, b) ; 
 w hile for most values for which /i is a turn through an angle greater 
 than 30° (certainly for values of 6 between 30° and 330°), there is 
 })ut 1 real equilateral triangle (Fig. VII, c). 
 
 z-fi 
 
 c=ua-t»)**w*(i-t)= 
 
 Fig. VIII. 
 
 Furthermore, it may be remarked that if we now let Zi be re- 
 garded as variable, the cusp /i- of the cardioid-locus of the third 
 vcrl(>x c is given by k = o)(l — tiy~= — co[— (1 — /i)'^]; i, e., a cardioid 
 willi the same cusp and size as the original, but with an orientation 
 of — ()(r. Hie relations of these curves is shown in Fig. VIII. 
 
 All hough wo have specialized for the case of equilateral triangles, 
 it is to be observed that the number of triangles of any sliape 
 inscril)ab](^ in a given cardioid is 5, 3, or 1, according to the position 
 of ,~i; and that, regarding Zi as variable, the cusp of tlie r-cardioid 
 w ill also describe a cardioid. 
 
 We shall now indicate how these considerations mav })e of 
 
42 Cardioiils Fid filling Certain A.s.signed Conditions 
 
 value in determining the number of real cardioids on 4 points. 
 Let a, by c, d be the vectors to the given points. First, consider 
 the triangle formed by o, 6, c. If a be fixed on the standard cardi- 
 oid and b be allowed to move along the cardioid, c will trace out a 
 second cardioid with relations to the original as indicated on 
 page 39. Similarly, fixing the point a of the triangle formed by 
 rt, by dy on the standard cardioid and allowing b to move along the 
 cardioid, d. also, will trace out a third cardioid with relations to 
 the original as those on page 39, where d. replaces c, p4, ps rei)lace 
 Pi, p2 and a, i3, 5 replace ip, yp, 6, respectively. 
 
 Now a mechanism could be devised by whicli, after having 
 placed the c-cardioid and the cZ-cardioid in relation to the original, 
 2i is allowed to move along the original curve. The cusjjs of the 
 two third-vertex-cardioids will also move on cardioids, as shown 
 on page 39. As Zi moves, the second point, Z2 (6), will not, in 
 general, be the same point for the c- and the f/-curve; but such a 
 coincidence will certainly occur. When this does happen, we 
 have a cardioid on the 4 given points. x\n arrangement could be 
 made by Avhich such a coincidence would be indicated; and a simple 
 registering of these indications will give the number of real cardioids 
 on 4 points. 
 
C a rdioids Fulfilling Certain Assigned Conditions 43 
 
 Vic. V 
 
 Fig. IX. — Showing the 3 real cardioids when the cusp and 2 lines 
 are given. \ 
 
44 CardioUls Fulfill infj Coin in As>iif/Ue(l Conditions 
 
 Fro. X — lu)N\i: >; the 3 vvr.] Ciir('i(i(l.-> wlien \hc ccnlrf jsiul 2 
 lines are > i\en. 
 
 'iG. XL— Sh()\vin>^- the 2 real cardioid- wIhmi the cusp and 2 
 points are given. 
 
VITA 
 
 Sister Mary Gervase Kelley was born in Roxbury, Mass., Sep- 
 tember 8, 1888. She received her elementary education in 
 St. Patrick's Parochial School and was graduated from the High 
 School in 1905. In 1906 she entered the novitiate of the Sisters 
 of Charity, Halifax, Nova Scotia, and there continued her studies 
 in the Novitiate Normal School. From 1908 to 1913 she taught 
 in the Halifax Public Schools. In 1910 she began work with the 
 University of London, from which institution she received the 
 Matriculation and the Intermediate Arts certificates. The four 
 academic years since 1913, with the intervening Summer Sessions, 
 have been spent in residence at the Catholic Sisters College, 
 Catholic University of America, where she received the degree 
 Bachelor of Arts in 1914, and that of Master of Arts in 1915. 
 In her graduate work the principal courses followed have been 
 those under x\ubrey E. Landry, Ph.D., and John B. O'Connor, 
 Ph.D., for the work done under both of whom it is the writer's 
 pleasure to express her appreciation, and in particular to acknowl- 
 edge gratefully the constant interest and kindly encouragement 
 and assistance given by Dr. Landry, not only during the prepara- 
 tion of this dissertation, but during her entire University course. 
 
 45 
 
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