# Full text of "Some circular curves generated by pencils of stelloids and their polars"

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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/| BIBLIOGRAPHIC RECORD TARGET Graduate Library University of Michigan Preservation Office Storage Number: ABR1526 ULFMTBRTaBLmT/C DT 09/12/88 R/DT 09/12/88 CC STATmmE/H 010: : I a 18019475 035/1: : | a (RLIN)MIUG86-B50242 035/2: : i a (CaOTULAS)160122853 040: : I a lU I c MiU | d MiU 050/1:0: |aQA603 |b.H45 100:1 : I a Hebbert, Clarence Mark, ] d 1890- 245:10: | a Some circular curves generated by pencils of stelloids and their polars, ] c by Clarence Mark Hebbert. 260: : | a [Urbana, |cl918] 300/1: : |a2p. L., 14 p., 1 L. jbdiagr. ic24cm. 500/1: : | a Vita. 500/2: : j a "Extracted from the Tokohu mathematical journal, vol. 13, 1918, edited by tsuruichi Hayashi." 502/3: : | a Thesis (PH. D.)-University of Illinois, 1917. 504/4: : | a Bibliography: p. 13-14. 650/1: 0: | a Curves 740/1:0 : | a Pencils of stelloids and their polars. 998: : |cWFA |s9124 Scanned by Imagenes Digitales Nogales, AZ On behalf of Preservation Division The University of Michigan Libraries Date work Began: _ Camera Operator: _ y Google SOME CIRCULAB, CTJBVIS GENERATED BY PENCILS or STELLOIDS AND THEIE POLAES CLARENCE MARK HBBBBBT B. S. Otterbein College, 1911. M. S. University of Illinois, 1914. THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OP PHILOSOPHY IN MATHEMATICS ra THE GRADUATE SCHOOL OF THE UNIVERSITY OF ILLINOIS I9I7 y Google y Google SOME CIRCULAR CURVES GENERATED BY PENCILS OF STELLOIDS AND THEIR POLARS 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 y Google y Google TABLE OF CONTENTS, Paqk. I. Introduction 1 II. The transfoni.ation i'-—^ • 2 III. Tlic transformation 2'= — 8 IV. Tlie general transformation !^=z~~ Bibliograpby 13 Vifc. 16 Extracted from THE TOHOKU MATHEMATICAL JOURNAL, Vol. 13, 1918, edited by Tsltruichi Hayashi, College of Science, T61ioliii Imperial University, Sendai, Japan, y Google y Google THE TOHOKTJ MATHEMATICAL JOURNAL Some Circular Curves Generated bv Pencils of Sfelloids and Their Polors, Claeesce :Mark Hbht!Kut, Oluuiipaign, III., U.S.A. I. Introduction. It is the purpose of this paper to coiisitler the transformation s'=— , which has the three ciihe roots of unity for double points and with which is connected the pencil of etelloids (cnbics)(') tlirough the three cube roots of unity and their osaociates. Some propertiei of the quintic generated by the pencil of cubics and tlie first polar pencil («cLuilateral hyperbolas) will be derived. The more general transformation :' = — will also bo studied and the genersd form of the product of the pencil of stelloids through the «+l*'i roots of unity and their associates, and the first and second polar pencils of any point ix' , y') will be determiiio;]. Some pro- perdes of the asymptotes and foci of these cnrves wiil be derived. This transformation is simply the contracted form of the general trans- formation z'=z- i'^ . f.plii). , for f{s) = z"-'^-l. The kst section con- siders the general case. ( 1 ) A. Emoh; On eonfannai Uatlowl Transformations in a Fltms. JReiidioonti del Circolo Matematico di Palermo, XSXIY (1012), pp. 1-13. On Klcllui.lj in ^miralsee: G. Loria, Speaisllo Algebiaisobe und Ttanscendente Ebene Kurveti, \5ib Knp. — Geometrie ■Jet Poljmome, Vol. I (1902) pp. 368-80 ; C. E. Brooks : A A'ofe on th- OrU>.ic Pubic Owve, Jolina HopkiuB Uoiveteity Oirou'ar (1304), pp. 47-52, and Orlhic i itfcen, oc Alg^iraie curoes vihicfi mtUfy Xapidce's equation In two ^imensUws, Proceedings ol Ameciam Philoaopliioal Society, Vol. SLIII (1904), pp. 294-331, The transfoimntion z' = — is studied in detail m the orlie'e Ly Profi'Koc Emoh, InvoMoric Circular Tranirformations /is a FaHicula/r Case of the bkiivrlji I'r.msformalion and tfteir Iwaiiant Nets of Otibks, Aimiila of Malhematios, -hid Sories, SIV (1912), y Google 2 CLARENCE MAEK HEBBEKT : For some of the work, use will be made of the following Theorem I. The product of a pmcU of curves and the seeond polar pencil of a point (a^, y') is identical iinth the polar of the produ-nt of the penoU and the first polar pencil of {x', y'){ ' ). For, let the pencil of curves be (1) P+iq=o, and the first and second polar penc;iU (2) AP+AAQ=0, and (3) £i?P-vk£i?q—0, respectively. Tlie product of pencils (1) and (2) is (4) P-AQ-Q.AF=0, -whose polar is AP-Ag+P.A=Q-A§-AP-Q.A=.P:i.O or (5) P-A=§-§-A=P=0, which is idontlcal with the product( ^ ) of (1) and (3). II. Transformation -ifca are invan- Geometi'Ically, this transformation represents an inversion, a reflexion, doubling of the angle and sriuaring of the absolute value. For it may be replaced by two transformations, z" = — and z' =z"'^, whose properties are well known. Straight lines are reflected on the a^-axis and their inclina- tions are doubled. The unit circle corresponds to itself but only the three points (1,0) (-~, — i/T), and (-y. -y^^^) ant. The three lines joining these three points and the origin are also invariant lines but not point-wise. An equilateral hyperbola, xy=(!, goes into the circle, 2{!(ic''-|-i/^) + i/=0, counted twice. If (ai', y') describes a straight line, the point (^, y) describes a locus of the fourth order, since ( 1 ) This ourre is calletl " Panpolare ' by'tteiner, who first inveiitieafeiJ in a purely ajnthstie mannei; some of its propertie^!! in ^oneml, Jouiaal file die reins und angewandte Mathematik. Vol. XLVII, pp 70-82 ( 2 ) On products of projcctivo pencils see Clebsoh, Vorleoungcn liber Geometric, Vol I (1876) p. 375, Cremona, Iheone der ebenen Evir\eii (Germaaby CurtEe, 1835) Pacagniph 50, t.t. Sturm, Dig L'hro Mu dun gpimetristben VerwandtaoliaEton, Vol. I (1909) p. 249, f t Eiicj. dor Mutli M 133 III, a 3, p 303, t f y Google CIKGULAR CURVES GENEKATED BY ^E^T[LS OF STELL0ID9. 3 the points corresponding to {x', ll')C) are the busc-pointa of the first polar pencil of (a:', y') with respect to the pencil of cubies (stelloids) throngh the three cube roots of unity and their assoL'iates( ^ ). Since the two transformations z" =.— and z'=^'^ are eonformal around all points except and co, the result of using both of tliem is eonformal, i.e,, finite singularities of curves are preserved in the transformation 2'= — - Infinite poiiite, however, are transformed into singularitiea at the origin. The pencil of cubica is m+Aw=0 where u and « are the real and imaginary parts, respectively, of !?—\, i.e., (1) u + l'G=x^-Zxf-\ + l{Zx'y-f) = i>. The projective pencil of first polars is (2) {x'-f)x'-2xy^f-l^}.\_'ixyx' + {x'-f)y'^ = (i ; •pencil of second polars is (3) K^-y^)a,— 2,x-'2,'y-l+A[2xY^+(«'^-j/")rf=0- The product of (1) and (2) is, as we should expac-t from the general theory, a bieireular quintlc (4) C^Jj-!/' »;)[(«» +!,-)-+2«] + (y-y)(33;'-!/')=0. The product of (1) and (3) is (0) %x--^f){xx'+yy'){x'y-XAj) + {s^^-f^)y-'t-2x!)^^ ~Z^y'~Z3?y+f=0, .a circular qnartic; the first polar of {x' ,y') with respect to (4), in agrcii- nient with Theorem I. The product of (2) and (3) is the circular cubic (6) ix''^ -y'%:i?yx' +0^ y'+a?'if -\-xy'y^-\-y) '\-2a/y'{x^yy'+y'y'-xx'y''-3/x'^+x)-2x.v'y~y'(x'''-f)^0. This cubic belongs to the class disaisiod by Enicli in the papar referred to on p. 1, and will not be studied here. The ftuintic (4). Since there are no terms of the fourth degree iu equation (4), and (x'y—y'x) is a fector of the fifth degree terms, the line x'y—y'x=0 is ( 1 ) L. Oreroona : Tlieoiie dec ebenen Kurveu (German liy Cuctze, 1865) p. 120, csata XL ( 2 ) A, Emcli : (I.e. p. 1) pp, 8 and 12, y Google CLAEENCE MAEK HEBBEKT: an asymptote. There is a rloiible point at tlie origin and at each of the circular points, I and J, at infinity. The curve passes through tlie base-points of (1) and (2), viz., the points ^^'"H-hm-'H'-m H \-iy' _^ i l^x'+iy' -+-V:^ - At the first three points above, -^ has the values - and ~ ■ , respectively. Tliese show that the tangonts at these three points, whicli ai-e the points representing the three cube roots of unity, pass through tlie pole (a-', /). This follows directly from the fact that (6), the first polar of (a/, y') with respect to (4), parses through these — i — -^ — —^ — ___J_^ I.e., tlie faugcnti to the y' curve lit the origin are y=^~^=^ — " ■■ ■■ ■ •' — x, which arc oriliogonnl. If 6 ifs tlio inclination of either of these tangents, tan25=^— — . Hcnee, to construct the tangenis to (4) at ilie origin, join the origin to the point (p^t ~y') ^^'^ hisect tlie angles made by this line loith the x-avis. The hi&eclors are the required tanijmis. These tangents fonu tlie only real d^jenerate conic of tlie pencil (2), and are obtained also by patting A=co in equation (2). This is sufficient to enable us to make a fairly accurate drawing of Uie curve. (See figure on nest page.) Some of the properties of (4) appear more readily if it is put into the polar form (7) />^[^^(;t,-'Hine-y'co36')-.o.sin33 + ^'sb2i' + ycos2Si] = 0. The factor (P' indicates again, ihat the origin is a double piiiut. If a/sin 2fl+y cos2^~0, or tan 25^ — ?/- , one value of p is zero, Tlie others are obtained from p^{x' sin O-y' cos C) = sin 3f, y Google CIEOUtAE ODEVES GESEEATED BY PENCILS OF STELLOIDS. 5 whence p=±—. Goiph o£ the iiuiutio [i). =■ provided we consider sii 20 negative and cos 29 positive. An interchange of signs would niiikc p imaginary. (The fourth root arises from the faot that the fimutiona of 9 involve tho square root.) The curve cuts one of the tangents at tJie origin in two points ecLuidistanfe 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 p^ within the bracket is zero, the sum of the throe 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 »i>2. This will be dis- cussed later. Quadruple foci of {4). Foci are sect-poluts of tangents from tlie circular points to a curve( ' ). The tangents at / and / are of the form i/=ja;+6 and y—ix+c, rc- (1 ) B^Bsett, Elementarj Trentise on CuMo and Quarlio Curves, p, 16. Charlotte A. Soott, Modem Analyiical Goometcy, p, 123. numerous special eases of foci ace treated by E. A. Roberts, On Foei oihZ Confocai Flam Garves, Quarterly Jonmal o£ Mathematics XXXV (1903~J), pp, 297-38i. y Google ^ CLA.RESCE MAKK HEISBEET ; spectivdy. Putting b=^+ia, tbeia {a, ^) is the only teal point on the tangent, i.e., it is the focus. Substituting y=—ix+b in equation (4) we have (8) {4:i+'Ub'3;'+Abhj')a?+{iiPy'-6b-2ix'-8b^x'-\-2y')x'^ + (2ba^+2iby'-3ib^~bU/~5ib*x')x+P-bh/+b^x'=0. The degree reduces to 3 because the circular pointe are double points. Itt order for y=—ix-'rb to be tangent to I, the coefficient of x must also vanish, i.e., x'—ly' ' Hence the tangents at I are i/= — if ."K^--- ==^^ ) . Similarly, the tan- eents at J are y=iix± ) . The.^ iutcrf=ect in tbc four points V l^.'i^ + iy' / (two of them real) / 1 y^' — iy '± V'x' + % y' ■ i J^aZ-j-iy' ^^i/'x' — i y' \ \~~2 Px" + y" ' ~Y l^FM^'" J' which are the base-points of the pencil (2). "We have seen that the Mrthogonal tangents at the origin aix) the two lines of the real degenerate equilateral hyperbola of the pencil (2). Hence we may state the Theorem H. The Viree degenei-ate equilateral hyperbolas of the pencil (2) are Hie tangents to the ctirve (4) at the- dovhle points, which are tJieir The base-poiiits of the pencil (2) are foci of (4). Single foci of (4). The quintic (4) hue three double points and no other singularities. Its class is therefore 6(5— 1) — 3.2=14. Since the circular points are double points we can draw from each of them only 10 tangents touching the curve elsewhere. The 100 intersections of these ten tangents are foci of the curve, but only 10 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 eaeh of, the two tangents at J in two eoinciclent points {doubie foci), thus yielding 40 double foci; similarly, the tangents from J determine 40 double foci. The tangents at J and J determine four quadruple foci {2 real) considered above, counting for 16 points. Thus "we have accounted for lC0.-l740-i-^0-!- 16=196 base-points. To determine y Google CIRCULAll CURVES GENEEATED El' PEHCILS 01' HTELLOIDS. ^ the real single foci, impo.so on ecjuation, (8) tlie condition tliat it sball Lavo equal roots, i.e., that the discriminaat shall vanish. To obtain the discruniiiaift, take ,the derivative with respect, to x and solve the quadratic so obtained for ar. Since the double roots of the cubic (8) must also be roots of its derived equation, we reverse the process and substitute the roots of the derived equation in (8), The two expressions thus obtained are the two factors of the discriminant. These, set equal to zero, are b—ix'—y'=0, and + 54b'{a!-iy)-54:ib*ia:-iyy~12b\x+i^f + %1b^~2'Jib\x-iy)-U{x+iyf=0. Hence,. Uie^, line y=—ix+ix'+y'iB a tangent to the curve (4). The real point on it is (a/, y"), the, pole, which is therefore a focus. (Aa is ■well known, the corresponding value of c is y'—ix'.) ]Jy equation (9), the other nine reiil single- foci are so dtuated that the origin is their centroid and the product of their distance from the origin has an ab- solute value equal to unity. The former follows fi-om tlio fact that the eighth degree, term is missing; the latter is seen by dividing through by the coefficient of 6, when the constant term reduces to i. Since the inverse of a focus is the focus of the inverse curve, the problem- of finding the foci of (4) reduot'S to that of finding the foci of its inverse with respect to the origin, viz., a circular quartio (10) {^^+y')\pi'xy+y'{x^-f)]^f-Zx'y-\-x'y~iJx=Q. This docs not simplify matters, however. Isotropic Coordinates. The problem of finding foci is much simpler when the equation of theeurvevis expressed in isotropic coordinates.( ' ) Put z~x-\-iy, z=x—iy, ( ' ) A. Peina, Ze J'guoaiom delfs Curve in Coordiiiate Gormplesse C'omuiiate, Kendicouti del Citcolo Mattmatieo di Palermo XVII (1903) pp. 65-72. Beltrarci, Eictrche sulla Qeoimtria diSe forme biirane milAclie, Memorie dcll'Acc. di Boh^na S (1870) p. 626. Cesiiro, Si"' la^ ^frmination des foyefs des comqws,'Soa\-e\les Annales des Hathemati- qnes LX (1901) pp. 1-9. G, Lery, Bur la foncfion de Gre(n, Annales Scientiliquea de I'Ecole Hormale Snpoiieure, XXXII (1915) pp. il9-135. O. E. Brooks, (I.e. p. 1.) call's Iheso conjugate coordiimtcs. C a y 1 e y (Collected ■Works VI, p. 498) uses the name ciiciilac coordinalea. y Google g CLAREKCE MARK HEBBERT : or x=— — '—, ij=- —. Equation (4) becomes (U) /=[(.V-ij')5'-l]i'-[(,^ + ;s')(?-l)]z'-S<C.V-irt+5'=0, (12) |t=3t(x'-i!,')?-l]2'-2[(a!'+i!/')(?-l)]s=0. Theroolsof (12)«ro z=0, •=-?fe^±32(tli. SnbBtitatmg s = in (11), we get z—^'f —i'lf , i.e., ai', y' is a fociis.{') Substituting tlie second root of (12) in (11), we liave (13) 4(a;'+iy)'(?-l)'-27?[(.t'-iy)?-I]' + 275-(»'-i!/')[(>!'-is'F-l]'=0. If in equation (13), z is replaced by its equivalent, —ih, t!io result is identical with equation (9), as it should be. m. Transformation 3'=e— (^+l)(3"-'-l) ^l Tlie pencil of stelloids connected witli this ti-ansforination is the pjncil of curves through the n-i-l'l' roots of unity and their associates. The transformation represents a (1, 5i) correspondence between the pole (it/, y') and the n real hase-pointa of the first polar pencil of {x', ■tf'). To establisli the equation of the pencil and first polar in polar coordinates \ve have n + ii;=s»^i-l=/>"-''coB(» + l)^+i,o"*'sin(u+l)f-l=^0. The pencil of st^Uoids is (14) u+Xv=p''*^ <m{n+l)S-l-^l(>'"' m\{n-\-i:)e=0. The first polar pencil of {.i/, y') is (15) ^l,^ + kVl=p"x'{H + l)<i'.y&nO-y' i>''{n-\-l)sn\nd~{ii->rl) + (» -f- 1 ) A [a;' /)" sin « fl -H j/ />" cos n S] = 0. The product of (14) and (15) is (16) /»''jp"*'(a/ smd-y' cos &)-,-> sin {a-\-l)d + a/ ^nnS+)/' iio\inO\ =rO, which may be written in the fiirin (U) ,o'"(a:'^-iy' a;) -,'."*' sin K-i-l)(y + a:',o" sin )i6' -Hy ,o" cos mf =0. (1) Lerj, (le. p. 7) p. 51. liroolis (I.e. p. 1) p. 309. y Google CmCULAU CURVE? GKNERATED B1 PENCILS OF STEIXOIDS. 9 Ttiis shows that in cartesian coordinates, (x^+y')^ is a factor of the terms containing x and y to degree 2n + l while the next highest power of X and y is n+1. Hence the Theorem III, T/ie product of tke pencU of stdhids determined by tJie »i+i'^ roots of unity and their associates as hase-poinis aiid the first polar pmcil of a point (a/, y), is a circular curve (16) hiving an n-fold point ai each of the dreular 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 v'y~y'x—0 Joiniiig the origin and the pole is an asymptote of the prodwd (16), if n'>l. The sum 0/ segments on rays ■through tlie origin is zero, i.e., the origin is a center. If in iho second factor of (16) we put p—0, we get x'nmne + y'msne^.O, that is Theorem V. The Icmgenls to (16) at the origin are the lines y=x tan 6, where tannfl=— — . These n tangents divide the whole angle about the origin into n equal parts, beginning at y=x tan ^, where ^=arc tan ( —J~\ = n0 + 2h7i. Making use of the values e^^-^A.'L (fc=0,l,2,... , («-l)) we n n find that the curves cuts the tangents at the origiji in the points />"(3;'sin^— Vcos i5')=sin(ii+l)S, or i'" —±: , according as Vx'''+y"' ■coanB or sinn^ is considered positive, i.e., the curve cuts in otlier real points all of these tangents if n is odd and cuts only half of them else- where if n is even. Tlic points of intersectioii are tlie base-points of (15), as may bo easily verified by substituting their coordinates in (15). The tangents at the origin constitute the degenerate curve obtained by making ^ = 00 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 muitiplicaties r and s (»'<s) respectively, their product K has there a multiple point of order )■ and the r tangents of K are tangents to C"". We liavc here an example in which both C"" and C" arc the real degenerate members of the two pencils. In fact, each ( 1 ) Sturm. (1. 0. p. 2) : Ency. dec Mftlh. 'Wiea. lU 2, 3, p. 355. y Google 10 CLAKENCE MARK HEliBERT : of theoi' 'edli.ai^s' of straiglit lines through tlie origin, the C being the )i+l straight- lines ' through the, origin obtained by iimlviug A — «j in (14). Single foci of (16). lutrodtieing isotropi6 coordinates s:=x+iy=p(cm8+is:h\0) ■ l — x — iy^ picas —lAa 0) ill' 'equation (16), it rediiees to (18) /=[(„'-i!/)r-l],""-[(«' + is')P"-l)]=- To find the foci, impose the condition on (23) tJiat it shiiU Jiiive equal roota in t(^). To Jo thie, we get (19) -^H^^ + i)[(^'-i/)S™~l]."-«[(.-' + ij/)(5->-l)]."-=0. If a root of (19) is alai a root of (18), it is a dutihle root of (IS)., Equation (19) has (i— 1) rootsi z=0. In order for .i;=0 to be a, root of (18), we must have z"\^~{3^ ~-iif'j\=0, i.e., &=x' — iy'^ wiicnee the pole (k*, y') is a fociis, zi^O signifies merely that the origin is a multiple point. The remaining root of (19) is ;^--IIiHjtM^!!lr:i)L . To find tlie condition that this shall be a root of (18) it is substituted in (18) giving tlae condition (20) n"[ix'+iy')i^"^'~l)r'' . \ -(^+l)"-^'g«p_(,v'-i/)][(.V-;i/')^"-l]"=0. The highest power of a in this eijuation ie (ii + l)^ and the next highest power is i^-\-n±l={n-\-\'f~n. Hence, for m>1, the ooefiieient <£ the nest Ingheit power of H vanislies and the origin is the centroid of theroots of<20), i.e., of the single foDi of (18), or (16). Al-jo the eonstanfe t6rm of (20) aiL'se m the fii^t bracket and has thu samo coefficient, except foi sign, m the highest power of z, i.e., the product of tlie roots of (20) is ±1, aucording as ii ji even or odd. If in (18) we set the coeffii-ient of *^ equal to zero, we got at once —^ , which, are tlie base-points of (15). Hence, ■ Theorem VI. The base-points of the first polar pencil {ll)) are foci of m- (1) Sm Jjeiy or Btoolis (1. 0. p. 8). y Google CIECULAE CURVES GENEIiATllD BY PENCir,R OF STELT.OIDS. H Eirst polar of (IS). The .product of (14) suid tht! Hoceatl yohiT pencil iif (x', y') is the first polar of (16), viz., (21) jo''-^i,o"+'[(a;"-y^)sIn2i3~2^yTOs2£']-(i^siu(ti + l)f + {x^^-y'^ mi{n--L)e+2x'y' co&{n-l)e]=0. Since the difference in degree of the two highest jjowcr of p is n — lj for n>2, the asymptotes are det«i-niiiied by (22) (x'^-v") sill '2e-2x'y' ms W^Q. From tills, tau20=— ^!!^ wliere m = ^ . Moroovcr, siiieo tan 25 =tan 21 /e + -^V it follows that thesu are the hiios joining (.V, if) to tJie' origin and the Hue norinttl to it at tiie origin. The tangents at the origin are determined by (23) (^''-^'^)sin[Oi-l)^ + 2it;r] + 2.i:'/<;os[Oi-l)£'+2fc;T] = 0, since in (21) this ia the couditiou for a root />~0. From (23) we gei. tan(M-l)^=— ?^^=taii2raretan('— ^^1 , or (n~l)S-2«) + m7£ = — 2^+5>i7r, where A is the inclination of the line joining tlie origiii and the pole (ii/, ?/'). For sin(ji+l)f— 0, |0"*^=cos(«+l)S, or /) = 1. Hence tlie curve (21) passes fhi-oagh the (ji+l)'^- roots of unity. But the curve (16) witli respect to whioh (21) is the first polar of {a^, y'), also passes through these points. "NYe have therefore the Theorem VII. Tke lines jomhig the pole (x', y') to the (n |- 1)"^ rods of unity are tangents io the curve (iO). TW, The general transformation ^-' — ^- '" j-// , • In tlie guneral caijc(^) /(r;) = «-ri?!— t(o/7((r;— .-:4)=0. This' may be thought of as representing ii + 1 lines(^) through the circular point I. (1) A. Emoh [i. 0, p. 1): p. 2. (2) Compare C. Segre, Le rapprcs&iitailani reati ddle /mine cwiiples.-,e e fl(i eiiti iperalgtbHci, Math. Anmlen XL (1892), pp. 413-167, y Google r[2 CLAEENCE MARK HEBBERT : The pencil of stelloids is u+Xv=0 and the n-{-l lines arc detcrinineil ■by the value X=i. Similarly, for X=i, t!ie first polar pencil Ui + lv-y=0 represents it lines tlirough I and tlic base-points of the first polar pencil. Also u—iv = and Ui~ivi=0 represent sets of lines tlirough J. By the general theorem regarding multiplicities of products, p. 2, we then have (assuming that the genera] theorem applies to imaginary Theorem VIII. The produd %lv^~uiV=Q of the projective psmlh u-^lv and Mi+^Ui has an n-fold pohit at each of tfte circular points mid ilie 11 lines iii-f i«i=0 are taiigenis to the prodttct ai I and the n lines %— ivi are tangerUs io J. Since the aectr points of i^ + ^'^j—O and it,— ivi=;0 are the base-points of the pencil Ui-\-kvi=^0, and these lines arc tangents at Z and J, we bave Theorem IX. The n^ base-points of the first polar pmeil Ui-i-kVi—Q ■are quadruple fotd of the j^-oduct uvi~UiV=0, Among these are the n 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, Tiiis, however, is not tlie case. Equation (27), p. 10 of the article by Era eh refeiTed to above is tlie equation of the protluet in general, viz., (24) (»,-.T')(™-s«)-(!/-/)(r» + «t,)=0, where )■ and s are — ~ — - and — — — resneutivcly. Tlie form of (24) w + 1 «+l ^ ^ gives us the Theorem X. The product etirve is also the prodad of the pencil of lines (x—x')—X(y—y')=0, through the pole, and the penoU of circular curves [ru+8v)—X(rv--sii)=0. The line x+iy=^-\-ii/', joining the pole and the circular point /j meets this curve in points of (25) iru+sv) + i{yv-su,) = 0, which is identical with the expression just above equation (4), p. 4 of that article, where it is shown to be equal to (26) (o, + .:6.)'H(a; + ;!/-.,)i7(:«-;!/=8.)=0. y Google CIKCTTLAE CUKVES GENEEATED BY TENCILS 01" STELl.OIDS. Ig. Substituting the value of x=x'+iy'~iy, the first two sets of factors become constants and tlie third one gives n values of y which are (27) ,j=^^±il=^ (i = l, 2,. ...,«). Hence two values of y cannot be equal luile:^ two points of Bj coinoide. In the special cases treated, ^^—0 so that the polo is a focus, except for the case n=l, which gives only one value of y. In general, two values of y eannot be equal in (27) and the pole is not a focus. This follows directly for general positions of the polo, since in this case (26) is independent of {a/, ^'). From the equation of the tangent line and equition (27) ti e point of contaa i-, (jL. ±iyL ^ 2L±UL\ for the ca^ *=0 (Thf tin^cnt has contact of ordei n~l at thi^ p int ) Ihis point 1 es on the hno j}=—tx In the same w&v it mn\ he -^hoftn th'it the punit of t-onlact of the tangent joining the p 1l to th< tutulai pmit J htb on tlic hue y=i 1 Hence Theorem XI In IJic sjtctal cusu, f & ilun IT «iJ III tit. uieidar points, the pole, tlie oitgin, artH the tuo points nf emit id of the tangents jovmng the pde to the cuB)dai points, aie t/is icrtioes of a com plete quadiikiteia}, le, the poirti, of fonfaa of ihesi. tangents ate tlie Qssociatt I lilt Lj- 11 p poll and mu/m fiCKNOWLEDGSMEl'i*. I am under great obhgifion'> to Professor Emeh fin' aanstance in die prepaiatKn of tins thisH To him and to Professor Townsend I am also \cr> much indebted lin continned encouragement and iiwsi.-itunce during mv wdl^ i w .udiutc i\ dent in. the Univcriiity of Iliiiwif;. ^Y. Walton: Several papei'fi in Quarterly Journal XX (1FI71). li". Lncas: Geonietiie dea polyndmes. Journal de I'Ecolo Poly technique, AQ^ Cahier (1879), pp. 1-33. ' : Generalisation du theorcme de Rolle. Comptcs Eendus CVI (1888), p. 121. yGoosle THE TOHOKU MATHEMATICAL JOURNAL. The Editor of the Journal, T. HAlASHi, College of Science, Tohoku Imperial University, Senditi, Japan, accepts contributions Er(tm any person. Gflitribations should be written legibly in EugKsh, French, German, Italian or Japanese an<J ditigrams should be given in separate slips and in propbr sizes. The autlior lias the sole and entire scientific i-espousibility for his work. 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