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Graduate Library 
University of Michigan 

Preservation Office 

Storage Number: 


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: _ 

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B. S. Otterbein College, 1911. 
M. S. University of Illinois, 1914. 


Submitted in Partial Fulfillment of the Requirements for the 

Degree of 






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B. S. Otterbein College, 1911. 
M. S. University of Illinois, 1914. 


Submitted in Partial Fulfillment of the Requirements for the 

Degree of 







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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 


edited by Tsltruichi Hayashi, College of Science, 

T61ioliii Imperial University, Sendai, Japan, 

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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), 

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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 


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 

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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!)^^ 

.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) 

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, 

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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 



\-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 


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, 

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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. 

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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 

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the real single foci, 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. 

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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. 

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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 


■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. 

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of theoi' '^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). 

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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, 

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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 

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. 

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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 


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. 



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