SOME CIRCULAR CURVES GENERATED BY PENCILS OF STELLOIDS AND THEIR POLARS BY CLARENCE MARK HEBBERT B. S. Otterbein College, 1911. M. S. University of Illinois, 1914. THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY IN MATHEMATICS IN THE GRADUATE SCHOOL OF THE UNIVERSITY OF ILLINOIS 1917 SOME CIRCULAR CURVES GENERATED BY PENCILS OF STELLOIDS AND THEIR POLARS BY CLARENCE MARK HEBBERT B. S. Otterbein College, 1911. M. S. University of Illinois, 1914. THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY IN MATHEMATICS IN THE GRADUATE SCHOOL OF THE UNIVERSITY OF ILLINOIS 1917 TABLE OF CON TENTS. I. II. III. IV. Introduction...................,........ *.....,. The transformiation z'=............... 0.. * *~ 1 The transformation z' =1............................ The general transformation z'=- n.......................... f (z Bibliography.............................. V ita.............................................. PAGE. 1 2 8 11 13 15 Extracted from THE TOHOKU MATHEMATICAL JOURNAL, Vol. 13, 1918, edited by TSURUICHI HAYASHI, College of Science, T6hoku Imperial University, Sendai, Japan. THE TOHOKU MATHEMATICAL JOURNAL Some Circular Curves Generated bV Pencils of Stelloids and Their Polars, by CLARENCE MARK HERBERT, Cham-paign, Ill., U.S.A. I. Introduction. It is the purpose of this paper to consider the transformation z'- 1 -which has the three cube roots of unity for double points and with which is connected the pencil of stelloids (cuibics)(' ) throug the three cube r~oots of unity and their,associates. Somie properties of the quintie generated by the pencil of cubi'es and the first polar pencil (equnilateral hyperbolas) will be derived. The more general transformation z'= will also bel studied and the general form of the product of the pencil of stelloidls through the + ith roots of unity and their associates, and thie first and second polar pencils of any point (X/,y' will be determinejd, Some pr'opertics of the asymptotes and foci of these curvres wvifl be derivecd. 'This transformation is simply the contracted form of the general transformation z'= (n + 1)f/ (z), for f(z)=z - 1. The hant sectionx con — f' (Z) siders the general case. 1)A. Emich: 0On covfformial Rational Transformations in a I/o~ne. Re,,dioonti del Circolo, Matematico di Palermo, XXXIV (1912), pp. 1-12. On Slal in geminra1 see: G. L o r ia, Spezielle Algebraische i-Ad Transcendente, Ebene, RurveTn, lbth 1{ap.-Geometrie der Poly-nome, 'Vol. I (1902) pp. 368-80; C. E. Brooks: A Note on thic Or//dc Cubi)'c Curve, Johns Hopkins University Circu'ar (1004), pp. 17-52, and OrtItic? qo'cci, or Aiqe/iraic curves which satisfy Laplace's equation in two dim~enisions, Proceedings of A merle, n Philosophical Society, Vol. XLIII (1901), pp. 291-331. The transformation zf = IIs Studied in. detail in the art/c'e by Professor Emceh, t ~ z Involutoric (circular Transforniations as a Pcarticular Case of tite kstinrai;-7~:n ransformatioo,?,an their Invariant Nets of Cubics, Annals of Mathematics, 2nd bevoies, XIV (1912), pp. 57-71. 2 2 ~~~~CLARENCE MARK HEBBERT: For some of the work, use will be made of the following Theorem I. The product of a pencil of carves and the second polar _pencil of a _point (x', y') is identical with the polar of the product of the,pencil and the first polar pencil of (x', Y~) ) For, let the pencil of curves. be (1) P+2LQ=0, and the first and second polar Pencils (2) AP+~AQ=0, and (3) A2P+MA2Q=0O, respectively. The product of pencils (1) and (2) is (4) P. LXQ -Q..P= 0, whose polar is ZAP.AQ+PA2QAQ.AP-Q.Z2P=O or (5) p.A2QQ.A~:o which is identical with the product( 2) of (1) and (3). II. Transformationz'_-3( -) 1 3z' Z Geometrically, this transformation. represents an inversion, aefeion doubling of the angle and squaring of the absolute value. For it- may be replaced by two transformations, z"= 1- and z 2= Z12, whose propertiesar z well known. Straight lines are reflected on the x-axi's and' their inclinations are doubled. The unit circle corresponds to itself but only the three points (1,0) ( A 1 I /i) — and 4 V'-)are invariant. The three lines joining these three points and the origin are also invariant lines but not point-wise. An equilateral hyperbola, x y= c, goes into the circle, 2 c(x+ y') + y= 0, counted twice. If (x', y') describes a staih lieyh oit(,~) describes a locus of' the fourth order, sincoe ()This curve is called "1Pa-npolare " by St e i ner, who first investigated in a purely synthetic manner some Of its properties in general, Journal f Ur die reine uand angewa-nite Mathematik, Vol. XLVII, pp. 70-8 21. (2 )On products of projective pencils see Cl1 e b s c h, Vorlesungen fiber Geometrie, Vol. I (1876) P. 375. C rem on a, Theorie der ebenen Kurven (German by Curtze, 18335) Para'graph 50, f.f. St u rm, Die Lehre von den geometrischen 'Verwandtschafton, Vol. I (1909) p. 219, fJ.f Euoy. der Math. Wiss. ITI, 2, 3, p. 353, f.f CIRCULAR CURVES GENERATED BY PENCILS OF STELLOIDS. 3.3,the points corresponding to ( Iy) 1) are the base-points of the first polar, pencil of (x', y') with respect to the pencil Of cubics, (stelloids) through the three cube roots of unity and their associates( 2) Since the two transformations z"= and z' = arc conformal around all points -except 0 aud oo, the result of using both of them is conformal, i.e.,. ilpite singularities of curves are preserved in the transformation _' ___ Infinite points, however, are transformled into singularities at the origin. IThe pencil. of enbics is u + 2~ v = 0 where u and v are the real and -imaginary parts, respectively, of z- -1, i.e., (1) m~U+)V=X~3 - 3xyl -1I+ 2,(3x~y- y3) =0. The projective pencil of first polars is.(2) (Xi 2 _2)'r'-2xyy'-1+2j[2xylx'-+-(x'-y2)y']>:O 'pencil of second polars is (3) (cif2-'x - 2x'y y - 1~) [2x''X + (X'2-_y'D2)y]=0. The product of (1) and (2) is, as we should expect from the general,theory, a bicircular quintic.(4) (c yR - i') [RX2 V-2)2 + 2x] +I (I' y)(3 x2 y) 0. The product of (1) and (3) is (5) -2(x2 +y2) (X XI+ yyl)(XIyX yl) + (X'2_y'2) y +2xx'y' M xy' - Mcy + y'=0,,a circular quartic; the first polar of (cI, y') with respect to (4), in agreem-enit with Theorem LI The product of (2) and (3) is the circular cubic (6)(a2-y'2)(xx xy+~'x'2y ~2x'yI(X2 yy'+1yleXX -xyl -XI XI+ X) -2xx'y —y'(x2-y')=O. This cubic belongs to the class discussed by E in coh in the papar referrd to o p. 1, and will not be studedhee The Quintic (4). Since there are no terms of the fourth degree ini equation (4), and,(x y -y' x) is a factor of the fifth degree terms, the line x y -y' x=O ig (1I) L.C r em o n a: Theorie der ebenen Kurven (German by Curtze, 1865) p. 120, Lehrsatz XI. ( 2) A. E'm ch: (ILe. p. 1) pp. 8 and 12. 4 CLARENCE MA.RK HEBBERT: an asymptote. There is a double point at the origin and at each of the circular points, I and J, at infinity. The curve passes through the base-points of (1) and (2), viz., the points (1 0); ( —1 / - -.); -2y_ 2y, -' -iY ~2 V/V,+y'1 E' ~~ 2 V-'+ ' +Y/Z At the first three points above, d- has the values -- 1, -2- 1 xdx x'-1 2x'+1 tind 2Y'+1 /3 2and 2-+ 1, respectively. These show that the tangents at these 32xG + I three points, which are the points representing the three cube roots of unity, pass through the pole (x', y'). This follows directly from the fact that (5), the first polar of (x', y') with respect to (4), passes through these ~xdy + -'-i)l/x-f2 1 ~?2 three points. At the origin - -- -, i.e., the tangent3 to the x + 1/X + y.' curve at the origin are y --, which are orthogonal. If ' 0 is the inclination of either of these tangents, tan 20 =- i-. Hence9 lo construct the tangents to (4) at the origin, join the origin to the point (xt, - y') and bisect the angles made by this line zuith the x-axis. The bisectors are the required tangents. These tangents form the only real degenerate conic of the pencil (2), and are obtained also by putting i=oo in equation (2). This is sufficient to enable us to make a fairly accurate drawing of the curve. (See figure on next page.) Some of the properties of (4) appear more readily if it is put into the polar form (7),o [,p3 (x sin O- y' cos 0)-p, sin 36 + x' sin 20+-t y cos 20] =0. The factor p2 indicates again, that the origin is a double point. If B sin 20+y' cos 20=0, or tan 20-9= ', one value of p is zero. The others are obtained from p2Z(x sin 0-y' cos 0)= sin 30, CIRCULAR CURVES GENERATED BY PENCILS OF STELLOIDS. 5 Graph of the quintic (i). whence p= ---, provided we consider sin 20 negative and cos 21 positive. An interchange of signs would make p imaginary. (The fourth root arises from the fact that the functions of 0 iuvolve the square root.) The curve cuts one of the tangents at the origin in two points equidistant:from the origin. These two points are the real base-points of (2) as may be verified by making use of the coordinates of the base-points as given on p. 4. Since the coefficient of p2 within the bracket is zero, the sum of the three non-vanishing segments on any ray through the origin vanishes. The origin is therefore a center of the curve. More than this, the polar equation (7), gives us a hint as to the form of the equation of the product curve for n>2. This will be discussed later. Quadruple foci of (4). Foci are sect-points of tangents from the circular points to a curve('). The tangents at I and J are of the form y-=ix+b and y=ix+e, re (1) Bassett, Elementary Treatise on Cubic and Quartic Curves, p. 46. Charlotte A. S c o t t, Modern Analytical Geometry, p. 122. Numerous special cases of foci are treated by R. A. Roberts, On Foci and Confocac Plane Curves, Quarterly Journal of Mathematics XXXV (1903-4), pp. 297-384. CLAhE~{NCE MAR~K HEBBERT: spectively. Putting bP + ica, thenr (a, P) is the only real point on the-tangent, i.e., it is the focus. Substituting y~ -i + 6 in equation (4) we have (8) (i~4ibx'-j4b XI>v3 (4i b.)fY-Gb — 2iX'-8b3 XI+2y') x2 ~ (2b x'+ 2i by1- 3i b2-l1 Y'- 5i b4X')X+b3 -bly'+ b5%I-. Tfhe degree reduces to 3 because the circular points are double points. In. order for y 7 -i+b to be tangent to I, the coefficient of x must:also vanish, i.e., Hence the tangents at I are y i ~x~1,.Similarly, the tangents at J are y =i X~4, *,.These intesect in the four points (two of themi real) 2 VXI+ Y/22 lXI'2+ f1 which are the base-points of the pencil (2). We have seen that the orthogonal- tangenits at the origin arc the two 'lines of the real degenerate, equilateral Iiyperbola of the pencil (2). Hfence we may state the Theorem IT. The threce degenerate eq aila teral 'hyp~erbolas of the pencil (2) are the tangents to the 'curtve (4) at the double p.oints, which ar e their vertices. 'The base-p~oints of thre pecncil~ (2) are foci of (4). Single foci of (4). The cjnintic, (4) has three double points and no other singularities., it's cla~ss is therefore 5(5-'1) -3.2 14. Since the circular points are double points we can draw fromn each of them only 10 tangents touching the enrve elsewhere. The 100 intersections of these ten tangents are foci of the curve, but only I10 of these are'real. They belong to the 196 base-points of a pencil of curves of order 14. Each of the 10 tangents from I cuts 'each of~ the two tanogents at f in two coincident polinta (double foci), thus yielding -40 double foci; similarly, the tangents from J determ-ine 40- double foci. The tangents at I,and J determine four quadruple. foci (2 real) considered atbove, counting, for 16 points. Thus we have accounted for, 106+ 404+ 40:+ 16=196 base-point's. To determine, CIRCULAR CURVES GENERATED BY PENCILS OF STELLOIDS. 7 7, the real single foci, irnpcse on equation. (8) the condition that it shall have equal roots, i.e., that the discriminant shall vanish. To obtainl the' diserimi~nant, take the derivative with respect to x and solve the quadratic so obtained for x. Since the double roots of the cubic (8) mUnst also be roots of its 4e'riv'ed equation, we reverse the process and substitutes the roots of the derived equation -in (8). The two expressions thus obtained ~re the' tw fators of the discrimiinant. These, set equal to zero, are b~ix~-~y0=O and (9) 46(x iy)+2T7(xi y)2 -b6(15ix3+117x 2y-45ixy2-39y3 + 54b5 (x - iy) - 54ib4' (x -i~y) - 121i x+i) + 27b6- 27ih2 (x - iy) - 4i (x ~ iy)3 =0. ilenee.,, the line -ix~ ix'-ky" is a tangent to the curve (4). The real point on it is (iny) the, pole, which is therefore a focus. (As is well known, the corresponding value of c is y' -ix'.) By equation (9), the other nine real, single' foci are so situated that the origin, is their centroid and,the product of their distances from the origin, has an absolute value equal to unity. The former follows fromt the fact that the eighth, degree, term is missing; the latter is seen boy dividing through by the coefficient of' b, when the constant term reduces -to i.,Since the inverse of a focus is the focus of' the inverse curve, the prob lem, of finding the foci of (4) reduces' to that of finding the foci of its inverse-with respect to the origin, viz., a circular quartic (10) (X2 py')[2x' xy~y'QxY y')] + y-3x'y+x'y, y'x-0. This does Rot simplify maitters, however. Isotropic Coordinates. The problem of finding foci is much simpler when the equation of the curvef'is expressed in isotropic coordinates.(') Puit z,=x+ijy, ~ =x-iy, ()A. P e r n a, le Equazioni deile Curve -in Coordinate Complesse Coniugate, Rendiconti del Circolo Matematico di Palermo XVII (1903) pp. 65-72. Be ltr amin, JRicerche suilce Geometria. delle forme binarie cubichie, -Memorie dell'Acc. di Bologna X (1870) p. 626. C e s h r o, $'r ice determination des foyers des coniques, KNouvelles Annales des Mathematiques LX (1901) pp. 1-9. G. L e r y, S3ur ila fonction de Green, Annales Scientifi'ques de l'ECole 'Normale Superieure, 1 XXXII (1915) pp.. 49-135. C. B. B ro o ks, (1. a. p. 1.) call's these conjugate coordinates. C ay Icey (Collected Works 'VI, p. 498) uses the name circular coordinates. CLARENCE MARK HEBBERT: or x-=, y= —. Equation (4) becomes 2 2i (11) f [(XI-iy')-l]za-[( +iy)(-l)]z- -z(-iy')+~a3O (12) 3 ') f - 2[(x-)_ 1- 2[(' +iy')(3-1)] z=0. The roots of (12) are = 0, z- 2^+i')(Z3-). Substituting z=0 3(x' - iy')" — 1 in (11), we get,=.-x'-iy', i.e., x', y' is a focus.(1) Substituting the second root of (12) in (11), we have (13) 4(x' + iy') _ ( 3- 27) ' [(x '-iy') 9-1]' + 272 (x _ iy)[( _ -iy/')22 — 1]2 =0. If in equation (13), z is replaced by its equivalent, -ib, the result is identical with equation (9), as it should be. III. Transformation z =z- ( 1) 1 (n + 1)z" ' The pencil of stelloids connected with this transformation is the pencil of curves through the n-tlth roots of unity and their associates. The transformation represents a (1, n) correspondence between the pole (x', y') and the n real base-points of the first polar pencil of (x', y'). To establish the equation of the pencil and first polar in polar coordinates we have +t iv=z + l-l=p= x+l,oS ( + 1) + ip+1 Silt %+) - 1 - 0. The pencil of stelloids is (14) t6+2v=po+l cos (n+l)0-1 +A,o' sinl (n 1) 0=0. The first polar pencil of (x', y') is (15), i, v,-=,o'x'(n+1) cos n)O-y' p'(n -F1) sin nO-(i-+ 1) + (n + 1) ) [', p" sin n 0 +- y' '01 cos n 0 = 0. The product of (14) and (15) is (16) pon p+I (X' sin 0 —y' cos 0)-, sin (, + 1)0 + x-' sin n — + y' cos n0t =0, which. may be written in the form (17),'2(xz'y-y' x) —,'+ sin + 1)_X'?,o'sin n+ y ',o cos O=O 0. (1 ) Lery, (1. c. p. 7) p. 51. B3rooks (1. c. p. 1) p. 309. CIRCULAR CURVES GENERATED BY PENCILS OF PTELLOIDS. 9 This shows that in cartesian coordinates, (xz+y2)n is a factor of the terms containing x and y to degree 2n+1 while the next highest power of x and y is n+1. Hence the Theorem III. The product of the pencil of stelloids determined by the n+ lth roots of unity and their associates as base-points and the first polar pencil of a point (x', y'), is a circular curve (16) having an n-fold point at each of the circular points and at the origin. Also since x'y-y'x is a factor of the highest degree terms, and the terms of order n+2 to 2n are missing, we have Theorem IV. The line x'y-y'x 0O joining the origin and the pole is an asymptote of the product (16), if n>1. The sum of segments on rays through the origin is zero, i.e., the origin is a center. If in the second factor of (16) we put p=O, we get x' sin nO+y' cos n = 0, that is Theorem V. The tangents to (16) at the origin are the lines y=zx tan 9, where tannO = -. These n tangents divide the whole angle about the origin into n equal parts, beginning at y==x tan 4, where ) =are tan ( —Y- =nO + 2kTr. Making use of the values 0- - 2k7 (k =0, 1, 2,..., (n — )) we n n find that the curves cuts the tangents at the origin in the points p (x1 sin 0-y' cos 0) =sin (n+1) 0, or p'- 4-, according as *cosn0 or sinn0 is considered positive, i.e., the curve cuts in other real points all of these tangents if n is odd and cuts only half of them elsewhere if n is even. The points of intersection are the base-points of (15), as may be easily verified by substituting their coordinates in (15). The tangents at the origin constitute the degenerate curve obtained by making A=co in (15). A general theorem( ) states that if two,corresponding curves C" and C? in two projective pencils of curves have a,common multiple point of multiplicities r and s (r<s) respectively, their product K has there a multiple point of order r and the r tangents of K are tangents to Cm. We have here an exanple in which both C" and C" are the real degenerate members of the two pencils. In fact, each (1) Sturm.(l. c. p. 2): Ency. der Math. Wiss. III 2, 3, p. 355. 10 CLARENCE MARK HEBBERT: of themn:ie'cohistsi of straight lines through the origin, the C"' being the i+-1 straight.i liines 'tlirouh the origin obtained by malking 2-) c in (14). Single fo0i of (16)3 Introducing isotropic coordinates z=x+/iy=p(cos 60+i sin 0); =x= i y =0 (Cos 0i sin 0) in equiationi (16), it reduces to (18).f I[(~' - /)Sr' -- 1]}+'- [(,+: l I 1)(%S+l 1)']? +i" +1l(^, i y')zz-0. To find the fci, impose the condition on (23) that it shall have equal roots in (l) To 'do this, we get (19) -( + 1)[('i )E - 12 l C(z ' + i /')(+'-x 1)]j ^ l= 0 If a root of (19) is also a root of (18), it is a double root of (18). Equation (19) has (n-1) roots z 0. In order for z0 to be a root of (18), we must have '1[i- (x'-iy)]=0, -Le., i-.zc' iy' whence the pole (z', y') i a focus.o z0 signifies merely that the origin is a multiple point. The remaining root of (19) is _:. —y(' & To (n, + [ (/ - i)') '( - l] find 'the condition that this shall be a root of (1) it is substituted in (18) giving the condition (20) n [(?/+- i')(",l-'- )1+l -(9 + l ) - s, I (-i Ut? /)] [(.-0;v) l],.=o. Tlhe highest power of ~ in this equation is (n -+1)2 and the next highest power is i2- n -t 1- (n + 1)2 9n. Hence, for ni> 1, the coefficient of the next highest power of iz vanishes and the originl. is the centroidl of the roots of (20), i.e., of the single -foci of (18), or (16). Also o te constant term of (20) arises in the first bracket and has the same coefficient, except for sign, as the highest power of i', i.e., the product of the roots of (20) is ~L1, according as n is even or odd. If in (18) we set the coefficient of z/+t equal to zero, we get at once the double foci viz., the points =-, ch, ahich. are the base-pointts of (15). Hence, Theorem VIL The base-points of t7he first polar pencil (I15) are foci of (16). (1 ) See Lery or B]rooLs (1. c. p. 8), CIRCULAR CURVES GENERATED BY PENCILS OF STELLOIDS, 11. First polar of (16)o The product of (14) and the seccnd polar pencil of (.', jy') is th first polar of (16), viz., (21) pnl tpIo+1 (/-y2) sin 20 - 2.' cos 20] -p2 sin (n + 1) + ('2 - /12) sin (n2 -1) - 2w'y' e os ({n -1) 0. Since the difference in degree of the two highest power of p is n-1 for n >2, the asymptotes are determinied by (22) (12 - (12) sin 260- 2'y' C2 0 = 0O From this, tan 20- 2m - where ml -, Moreover, since tan 26 -21 7l2 X =tan 2(0 +2- it follows that these are the lines joining (r, y) to thet. origin and the line normal to it at the origin. The tangents at the origin are determined by (23) ('2 — 2) sin [( — 1) 0 + 21,.r] + 2,' /y' os [(n- ) 0 + 2 7 C],= since in'(21) this is the condition for a root Op 0. From (23) we,get tan (n- 1) = - x' tan 2 artan( -, ) ) or' (n -1) 62+ m r /2_ 912- tan 2 rctan or (n a = —2A+ r, where A is the inclination of the line joinig the origin and the pole (W',?j). For sinn (n+61)0=O, piT+lcos ( +l1)0, orP p. Hence the curve (21) passes throtgh the (n i-1)th roots of unity. But the curve (16) with respect to which (21) is the first polar of (x', ^-y) also passes through these points.o YV have tlerefore the Theorem VIIo The lines joining the pole (x', ') to the (n + -ltk roots of unity are tangents to the curve (16') o The general itansfmati oR+ -)f (,) f' (,) In the general case(1) f(r )= i+r- ao ((:) 0)o. This- may be thought of as representing $ + I lines( 2) throutgh tlhe circular poilnt, (1) A. Emch (1. c. p. 1): p. 2. ( 2) Compare C. Segre, Le lrappresentaziomn reali delle formze comsples.e e g:i enti iperalgebrici, Math. Annglen XL (1892), pp. 413-167. 12 12CLAIRENCE. MARK HEBBERT: 'The pencil of stelloids is u + Av = 0 and the n + 1 lines are determined iby the value A =. Similarly, for Aii, the first polar pencil u1+ Ar1 =0 represents m. lines through I and the base-points of the first polar pencil. Also u- it' =0 and a,-1iv1=0 represent sets of lines through J By the general theorem regarding multiplicities of products, p. 2Y we then have (assuming that the general theorem applies to imaginary -elements) Theorem VIII. The peoduct u v,-u,v=0 of the projective pencils u ~ A v and u, + A v, has an n-fold point at each of the circular points and-,the n lines nu1+ i'1 =0 are tangents to the product at I and the n lines u- iv, are tangents to J. Since the sect-points of u, + i v1 =0 and u,1i v1 =0 are the base-points,of the pencil U2+Av1 = 0O, and these lines are tangents at I and J, we have Theorem IX. The n' base-points of the first polar pencil u, + A V1 = 0 -are quadruple foci of the product u v,1 -u v = 0. Among these are the -real base-points forming n real foci. Since in the special cases treated the pole is a focus, we might:expect that the pole is also, in general, a focus. This, however, s ct, the case. Equation (27), p. 10 of the article by Eti. e h referred to above is the equation of the product in general, viz., (24) (X -x')(rv-s a) — (y —y9(ru s)=0, where r and s are and - respectively. The form of (24) 91 +1 nt gives us the Theorem X. The product carve is also the product of the pencil of,lines (x- l')-A (y- y')=O, through the pole, and the pencil of circular curves (ru sv) -A(r v-s u)=0. The line x+iy=x'+iy', joining the pole and the circular point 1, lmeets this curve in points of (25) (r'u+ s V) + i (rv - ~ su81) = 0, which is identical with the expression just above equation (4), p. 4 cf that article, where it is shown to be equal to n-id n (26) (a,+ibo) 1 I I1 CIRCULXR CURVEYS GENERATED BY PENCILS OF STELLOIDS. Substituting the value of x=x'-4~iy'-iy, the first twvo sets of factors beome constants and the third one gives n values of y which are (27) (h~, = ln)2 2i Hence two values of y cannot be equal unless two points of F, coincide. In the special cases treated, ~, 0 so that the pole is a focus, except for the case n= 1, which gives only one value of y. In general, two values. of Vcannot be equal in (27) and thc pole is not a focus. This follows directly for gencral positions of the pole, since in this case (25) is independent of (x/, ~'>. From the equation of the tangent line and equation (27) the point of contact is( ) for the case ~, 0=. (The tai'igent hs contact of order n-1 at this point.) This point lies onl the liue -Y= -ixr. In the same way it miay be slhown that the point of cont-act of the tangent joining the pole to the circular point J lies on the line yziX. Hence Theorem XI. In t1e special eases of sections 1Ii and IIJT the:circular _points, the pole, the origin, and the two int o Icotct of the tangets jinin the ole to the circular, points, aie the vetces of a comn pulete quadirilater al, i.e., the p~oints of contact of these tangents are the, -ssociate _points of the p)ole and origin. ACKNOJMLEDGEVIWT I am under great obligations -to Professor Enicek fow. as —,i tame iii. the preparation of this thesis. rio him- and to Professoi r To n Vt),,1i I am also very much indebted for continued encouragemaent ai4vnd ~ C during my work ais a graduate student inl the H-i 7vcisity of on BIBaLIOGr}UMP)817 W. W alton Several papers in Quarterly Journal XlI (1871 ). P, L u c a s GWonagrie des _polyn~ines. Journa.. de i'Eeolc Polytechnicquo. 4Gth Cabicr (1879), pp. 1-33. Geniralisation du thoi'eme de Rolde. Conipe iedsCA(188)2 p. 21. 14 4HEBBERT: CIRCULAR CURVES. P. Luca as: Statiqtue des polyn6mes. Bullo Soc. Math. France XVIIT (1889), pp. 17, ffo Go Fouret: Sur quelqpes proprietes geomen'triqes des stelloides. Comptes Rendus CVI (1888), p. 342. E. Ka sn er: On the Algebraic Potenztial Ctrves. Bull. Amn Matho Soc, VII (1901) pp. 392-399. E. E. Brooks: A Note on the Orthic Gtbic Carve. 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Cesaro Sulli radici delt'hessiana d'IuCa cubica in relazione con quelle dela cubica stessa. Giornale di Battaglini VIII (1901) pp. 24-36. -.-~ Siur la dtinntion de s foyerd s des coniqes. Nouvelles Annales LX (1901), pp. 1-9. R. A. R obe r t s On foci and confocal 2lane curves. Quarterly Journal of Mathematics 35 (1903-4), pp. 297-384. A. Cayley: Collected Works, Vol. VI, pp. 498-9. 1890. 1896-1907. 1.907. 1907-11 1911-12. 1912-15. 1916-17] 1915-16. 1914. VITA. Clarence Mark Hebbert. Born November 6, near Joplin, Misso'uri. Attended public schools at Bloomdatle Ohio. Student in Marietta College Summer SchooL. Student in Ofterbein College. Fellow in Mathematics, Ohio State Utiversityo Assistant in Mathematics, Unsiversitv of Illinois. Fellow in Mathematics, University of Illinois. Electec to Illinois Chapter of Sigma Xio Member of American Mathematical Society and the A.oAoAoS DEGREES. 1911. 1914. B, So Otterbein College. Mo So University of Illinois. PUJBLICATIONS. A. Cardioidograph, Amer- Math. Monthly, 22 (1915) p. 12o The inscribed qand cirumscrib ue ares of a quadrilateral and their sTigni ficance in kcinematic geometry, Annals of Mathiemat'cs 16 (1914-15) pp. 38-42. Properties of four confocal parabolas whose vertical tangents form a equare, Annals Of Mathematics 16 (1914-15) pp. 67-71. THE TOHOKU MATHEMATICAL JOURNAL. The Editor of the Journal, To HAYASHI, College of Science, Tohoku Imperial University, Sendai, Japan, accepts contributions from any person. Contributions should be written legibly in English, French, German, Italian or Japanese and diagrams should be given in separate slips and in proper sizes. The author has the sole and entire scientific responsibility for his work. Every author is entitled to receive gratis 30 separate copies of his memoir; and for more copies to pay actual expenses. All communications intended for the Journal should be addressed to the Editor. 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