Curvature theory in plane kinematics

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. O. BOTTE MA

Aan mijn Vrouw Aan mijn Zoon

CONTENTS

CHAPTER 1 - INTRODUCTION

1. Notations and conventions 1

2. Poles . 2

3. Special systems of reference 4

4. Some formulae . 4

5. Curvature of orbits . 5

6. Canonical systems 6

7. The path of the origin 6

8. The inverse motion . 9

9. The curvatures of the fixed and the moving polode at the pole 11 10. The curvature of the second ftxed polode at the second pole 1·1

11. Polar co-ordinates 12

12. The Euler-Savary equation 12

13. Bobillier's construction 13

CHAPTER 2 - THE CIRCLING-POINT CURVE AND BALL'S POINT 1. The circ1ing-point curve

2. The curve

cp

3. l?all's point.

4. The Bl-points of the inverse motion 5. Bl-points with excess

6. Geometrical considerations

7. Construction of the circ1es r_o, rand Ï' . CHAPTER 3 - THE BURMESTER-POINTS

1. The Burmester-points

2. The real intersections at infinity of cp and br

3. Theorems on the Bl- and Br-points of general positions 4. A conic incident with the Br-points

5. The Br-points for the case: (aa - 3)ba =1= 0 6. The Br-points in the case: as =1= 3, ba = 0 7. The Br-points in the case: as = 3, ba =1= 0 8. The Br-points in the case: as

=

3, ba

=

0 9. Br-points with excess

10. A conic incident with the Brcpoints 11. The arguments of the Brcpoints . 12. The arguments of the Bl- and Br-points CHAPTER 4 - SPECIAL MOTIONS

1. Symmetrical motions

2. The Bl-points and the Br-points of asymmetrical motion 3. Motions where a point of v describes a straight line .

15 17 18 20 21 22 24 27 28 31 32 34 36 39 39 40 43 44 44 46 47 50

page

4. The elliptic motion . 51

5. Motions where a line of v passes permanently through a fixed point 52

6. The Scotch-yoke motion . 53

7. The conchoidal motion 54

8. The Bl-and Br-points of a conchoidal motion 56

9. Symmetrical conchoidal motions . 58

10. The polodes of a symmetrical conchoidal motion 60

11. Cycloidal motions 61

CHAPTER 5 - THE FOUR-BAR MOTION 1. Motion where a point describes a circle. 2. The four-bar motion .

3. Folded positions of the four-bar linkage 4. The B~-and Br-points of the folded position

5. Bl-and Br-points with excess in the folded position . 6. Some formulae and notations .

7. Special configurations of the four-bar linkage 8. The Bl-points of the special configurations . 9. The Br-points of the special configurations .

10. The Bl-and the Br-points of the general configuration 11. The Bll-points of the general configuration .

12. The Brl-points of the four-bar motion 13. Symmetrical four-bar motion . 14. Bl2-points of the four-bar motion 15. Bl-point on the coupier .

CHAPTER 6 - THE SLIDER-CRANK MOTION 1. Formulae for the slider-crank motion .

2. Folded positions of the slider-crank mechanism 3. The special configurations

4. The Bl-points of the special configurations 5. The Br-points of the special configurations 6. The Br-points of the general configuration

7. Highest order of symmetry of a slider-crank motion 8. The centric slider-crank motion

CHAPTER 7 - THE OSCILLATING SLIDER-CRANK MOTION

65 67 67 69 70 71 74 76 81 87 90 91 94 97 101 103 103 104 106 108 111 114 115

1. Formulae for the oscillating slider-crank motion 117 2. Folded positions of the oscillating slider-crank mechanism 117

3. The special configurations 118

4. The Bl-points of the special configurations 120 5. The Br-points of the special configurations 123 6. The Br-points of the general configuration 125 7. Highest order of symmetry of the oscillating slider-crank motion 126 8. The centric oscillating slider-crank motion . 126

CHAPTER 8 - T-POSITIONS 1. Tz-positions

2. Some formulae for Tcpositions 3. T1-positions of the fust kind .

4. T l-positions of the second kind

5. Curvature of orbits in a T1-position of the second kind 6. T1-positions of a foilr-bar linkage .

7. T1-positions of a slider-crank mechanism CHAPTER 9 - CONTACT OF MOTIONS

1. The concept of contact of two motions .

2. Cardan-positions

3. Examples of motions having C-positions 4. Four-bar motion in contact with a given motion 5. Cycloidal motion in contact with a given motion

CHAPTER 10 - SYMMETRICAL COUPLER CURVES WITH TWO UNDULATION POINTS page 127 128 129 135 137 142 143 145 147 148 151 152 I. CoupIer curves . 153

2. Symmetrical coupIer curves 154

3. CoupIer curves having undulation points 156

4. Symmetrical coupIer curves with two undulation points obtained by means of the special configurations of fig. 10 . 157 5. Symmetrical coupIer curves with two undulation points obtained

by means of the configurations of fig. 15 and fig. 16a . 160

6. Investigation of the ROBERTs-figure 167

7. Other ways to obtain symmetrical coupIer curves with two undu-lation points by means of special configurations 170 8. Survey of the results concerning the special configurations 174

References 175

Samenvatting 177

Cross-references have been made as follows: section x . y means section y of chapter x; section x means seäion x of the current chapter; formula (x . y . z) means formula (z) of section x . y; formula (x. y) means formula (y) of section x; formula (x) means formula x of the current section.

CHAPTER 1

INTRODUCTION

Summary. This chapter is intended to give a brief account of that part of elementary plane instantaneous kinematics which is continually used in the fust seven chapters of this book. The treatment is analytical and is based upon the concept of instantaneous invariants first introduced bij BOTTEMA [5). Special attention has been given to the locaJ properties of the path of that point of the moving plane coinciding in a given position with the pole of this position.

1. Notations and conventions

We consider two coincident planes V and v moving with respect to each other. In order to give a mathematical description of the motion we take in

V and v Cartesian systems of reference XOY and xoy respectively, having the same orientation. The co-ordinates (with respect to xoy) of a point of v

will be denoted by x and y; this point coincides with a point of V which co-ordinates (with respect to XOY) are denoted by X and Y. Furthermore we denote by a and b the co-ordinates (with respect to XOY) of the point of V coinciding with the point (0, 0) of v. The motion of v with respect to

V can now be described by the two formulae

(1) X = x cos q; - y sin q;

+

a

Y=xsin q;+ycosq;+ b

a, band q; being real funtions of a real parameter t (usually called the time), whereas x and y are independent of t.

Putting

X

+

iY = Z, x

+

iy = z, a

+

ib = c (i2 = - 1), one can write the formulae (1) in the compressed form:

(2) Z

=

zeVp

+

c.

Differentiation of a function with respect to t wiH be denoted by placing a dot above the functionsymbol. In the first seven chapters we only consider such positions of v with respect to V to which there correspond non-zero values of the angular velocity

fjJ.

We may therefore - as far as we are dealing with geometrical properties of the mot ion - take q; = t. This is done through-out the first seven chapters. Accordingly we call a value of q; an instant;

[1-1/2] 1 ntl'oduction

with each instant there corresponds one position of v with respect to V.

The position corresponding to cp

=

0 will be named the zero-position.

Obviously it is concerning its geometrical properties in no way privileged

compared to any other position with cP i= O.

Differentiation with respect to cp will be denoted by primes. The value

for cp

=

0 of the n-th derivative of a function

I

of cp with respect to cp will be denoted by

I"

(n

=

0, 1,2, ... ).

The motion being represented by (1) we restrict ourselves to the case in

which a and b can be expanded into powerseries in cp having a positive radius

of convergence (analytical motions).

2. Poles

From (1.2) we get by differentiating n times with respect to cp:

(1) or

(2) zIn) = i"(Z - c)

+

c(") (n = 0, 1,2, ., .).

The point of v for which

zIn)

=

0 (n ~ 1)

at a given instant is called the n-th pole of the motion for the corresponding position. This point coincides in the position under consideration with a point of V which also will be called the n-th pole. We denote both points by P".

Instead of PI we usually write P; this point is called the pole of the motion for the position on hand.

Evidently P n is determined by

(3)

this being the unique solution of the equation

in(Z - c)

+

c(n)

=

o.

From (1) we get for the value

zp

of

z

corresponding to P,,:

"

(4)

The relations (3) and (4) furnish us with parametric representations (cp

being the parameter) of the locus of P" in V (th~ n-th lixed polode) and of the locus of Pn in v (the n-th moving polode) respectively. In the case n

=

1

Introduction [1-2] we call these loci the fixed poiode and the moving polode for short; we denote these polodes by

P,

and

Pm

respectively.

From (3) and (4) it follows that: (5)

(6)

Z'p,.

=

e'

+ (-

ir+ 2e(n+I), z'p

=

(-

_i)"+3e-"'(e(n)

+

ie(n+l)).

n

If we take n

=

I in (3), (5) and (6) we get: (7)

(8) (9)

Zp

=

e

+

ie', Z'p

=

e'

+

ie",

The last two formulae show that

P,

and

Pm

both consist of only one point if and only if for all cp

c'

+

ie"

=

0, th at is, if

In this case one can write instead of (I . 2) : Z - Zp, 0

=

(z - ie1)e"', 50 that

Hence we infer that in the case under consideration the path of each point of v is a circle, the centre being the (fixed) pole; this means that the motion is a permanent rotation.

Excluding th is case we draw from (8) and (9) the conclusion that P, and

P

...

are in contact, the point of contact being in each position the pole P. The common tangent and normal at P of

P,

and

Pm

are called the pole-tangent and the pole-normal respectively. As in view of (8) and (9)

we see moreover that

P

...

is rolling without slip on p,. For aims of further use we write down the equations (3), (5) and (6) in the special case n

=

2: (10) (li ) ( 12) Z'p. = e'

+

e''', Z' p = - ie-"'(c"

+

ie"').

•

[1-3/4J Introduction 3. Special systems of reference

So far our systems of reference XOY and xoy were arbitrary. We now choose them with coinciding origins in the zero-position; this implies

Co

=

O.

Moreover we take the common origin at the pole of the zero-position; then we have Cl = 0 from (2.7). Taking for the X-axis the pole-tangent of the zero-position, the relation (2.8) yields a₂

=

0, so that (2. 10) gives rise to

showing that the second pole of the zero-position lies on the pole-normal and has b₂ as its ordinate.

Summing up we have: in discussing the motion (I .2) it is possible to choose the systems of reference in V and v in such a way that

(1)

These systems of reference will be called the special systems of reference as contrasted with general systems of reference.

4. Some formulae

With respect to general systems of reference we get from (2. I) :

. (1) X4n

=

X

+

a4n . Y4n

=

Y

+

b4,,' X4n+2 = - X

+

a4n+2 . X4n+2 = - Y

+

b4n+2 ' X4n+1 = - Y + a4n+1 . Y4n+1

=

X

+

b4n+1' X4n+3

=

Y

+

a4n+3 • Y4n+3

= -

X

+

b4n+3 ' (n

=

0, 1,2, ... ).

If we make use of the special systems of reference, the fust four pairs of these formulae take the form:

(2) Xo=x. Yo

=

y' 4 X2

= -

X • Y2

= -

Y

+

b2 ' X3

=

Y

+

a3 Y3

= -

X

+

b₃

Introduction [1-5]

5. Curvature of orbits

The orbit of the point (x, y) of v bas at tbe point (X, Y) of V tbe cur-vature

(1 )

,,=

X'Y" - X"Y'

(X'2

+

Y'2)!

With respect to thc special systems of reference and provided that

(x, y) 0/= (0,0), we get from (1) and (4.2):

Tbis leads us to tbe conc1usion:

11 b2 0/= 0, the locus ol the points ol v which are at an inllexion point ol their

orbit in the zero-position is the circle (2)

7f'ith the exception ol the origin.

This circ1e (the inllexion circle) toucbes the pole-tangent at the pole,

passes tbrough tbe second pole and bas tbe diameter 1 b2

I·

Denoting by (~, '1'/) the centre of curvature of the orbit of a point (x, y)

of v not on the infiexion circ1e and coinciding with the point (X, Y) of V,

one has, as is known:

(3) ~=X- Y'(X'2

+

Y'2) , X'Y" --X"Y'

X'(X'2

+

Y'2)

'1'/=

Y+

.

X'Y" - X"Y'

Hence we find for the zero-position and with respect to the special systems

of reference:

(4)

The point (~, '1'/) determined by (4) will be called tbe centre ol curvature

belonging to (x, y) ; it is situated on the line joining (x, y) to the pole. It coin-cides for any point (x, y) 0/= (0, 0) of v with the pole if and only if b2

=

O.

Therefore b2

=

°

is a necessary and sufficient condition for th!' zero-position

to bave the property that all points of v with the exception of the origin, are

at points of their orbits to which the pole belongs as the common centre of'

discus-[I-Sf7] Introduction

sions we suppose that none of the positions we are dealing with is a

R-position; this means that we suppose b₂ =1= 0 and that the inflexion circle is not a point-circ1e.

6. Canonical systems

We suppo~e that the representation (1.2) of the motion refers to the special systems of reference. Then, thanks to the exc1usion of R-positions, b2 =1= O. Furthermore (2.8) and (2.9) teil us that the rate of change of the pole along

PI

and

P".

has the value -

b

2 in the zero-position. This rate of change is called the pole-velocity. Now we choose the orientation of the X-axis (and therefore that of the x-axis too) in such a way that the positive sense on it is the same as the senst' of the vector of the pole-velocity. Then b₂

<

O. Moreover we take - b2 as the unit of length in both systems of reference.

In this way we have got two systems of reference, unambiguously determined

by the motion. These systems will be cailed the canonical systems of the motion The derivatives a_{k ,} b_k (k

>

2) are geometrical invariants of the zero-position under the group of similitudes. We shall call them af ter

BOTTEMA [5] the instantaneous invariants of the position. Throughout the fust seven chapters we use (with only one exception, mentioned explicitely in section 1.13) permanently the canonical systems.

With respect to these systems we have.

(1)

Furthermore the equation (5.2) of the inflexion circ1e assumes the form:

(2) x2

+

_y2

+

_y

₌

_0, whereas (5.4) reads:

(3) ~

=

xy ,

x2

+

y2

+

y

r;=

x2

+

y2

+

y

.

It may be observed that with respect to canonical systems the pole-velocity has the value 1 in the zero-position.

7. The path of the origin

The above considerations do not give any information about the curvature of the orbit of the point (0, 0) of v coinciding in the zero-position with the pole. The path of this point (called the origin for short) can be represented

Introduelion [1-7]

(with respect to the canonical systems and for sufficiently small values of

I

cp

I)

by:

(1)

00

X

=

~ an cpn, n=3 n!

We distinguish three different cases:

Case 1. a3

*

0. If c is a sufficiently small positive number the orbit

described during the time-interval [- c, eJ possesses a cusp at the pole of the

zero-position, the tangent in this point being the pole-normal. The curvature of the path at this point is infinite. This does not mean that the curve of which the orbit constitutes a part (the carrier-curve of the orbit) has a cusp

at the pole. In the case

a = tcp3 - tcp4, b

= -

tcp2

+

tcp3

for in~tance, the carrier-curve shows a triple point at the pole the tangents

being the pole-normal (counted twice) and the line X

+

y

=

°

(fig. 1). y

fig. 1

Case 2. a₃

=

0, a4

*

0. In a sufficiently small time-interval [- c, eJ

the origin describes a part of its orbit having a ramPhoid cusp at the pole of

the zero-position. We find with the aid of (5. 1) and provided that cp

*

0, for the curvature of the orbit the expression

[1-7] Introduction

For sufficiently small values of

I

cp

I

=1= 0 we can write this as follows: ( ) a4 ('5a4ba a5) (3a4ba

2

a4b4 7a5ba a6) 2

3

ïe=

-

+

-

+-cp+ - - + - +

-

+

-

cp +

3 12 8 8 6 48 30

The curvature of the orbit at the pole is therefore:

(4)

Moreover we obtain from (3): (5)

(6)

It may happen that the two branches of the orbit described during the interval [-e, eJ and forming together the ramphoid cusp coincide. In that case the carrier-curve of the orbit may have an ordinary point at the pole. In order to give an example of this behaviour of the orbit we put

a

= icp4

-

cp5

+

cp6,

b

= -

tcp2

+

cpa. y

x

o~---fig. 2 8

Introduction [1-7/8J The carrier-curve of the path of the origin is the parabola y2

=

X, its tangent at the pole being the pole-normal. Investigating in detail in what manner the origin is moving during the time-interval [- 1, IJ, one finds easily (fig. 2) : at the instant q;

= -

1 the origin coincides with A (2!, - 2i) and moves during the interval [- 1, 0J from A along the parabola towards 0; during the adjacent interval [0,

iJ

it is moving from 0 along the arc OA of the parabola to B(2-3_{3-6 , - 2-}1_{3-3); finally in the interval}

Cl.

_{IJ it moves} from B via 0 towards C(i, i)·

Case 3. aa

=

a4

=

0. For sufficiently small positive values of B the orbit described during the time-interval [-B. 1>1 has at the pole a cusp or a ram-phoid cusp accordingly as the smallest value of n

>

4 with a_n oF

°

is odd or even. In this case we have from (3):

(8)

(9) (10)

(11 )

8. The inverse motion

X_o

=

0, Xl

=

tas,

7asba a6

x2

=--+-,

24 15

The motion of V with respect to v will be called the inverse motion as contrasted with the motion of v with respect to V hitherto considered and which will be called the original motion. The inverse motion is represented by (1 .2) if we regard Z as a constant and z as a variable. Therefore: if (1 .2) refers to the canonical systems, the representation of the inverse motion on the canonical systems of the original motion reads

(1) z

=

(Z - e)e-Up.

We want to know the connection between the canonical systems ot the original and the inverse motion. It is easy to de duce from (1) that the vectors of the pole-velocities of both motions coincide. This means that the canonical systems under consideration have a common positive X-axis.

Furthermore we obtain from (1):

[1-8] Introduction

In view of (6. I) these formulae yield:

Zl

= -

iZ, Z2

= -

Z

+ i,

or:

YI = - X

Y2

= -

Y

+

1.

The equation of the inflexion circle of the inverse motion is therefore with respect to the canonical systems of the original motion

X2

+

y2 - Y

=

O.

With respect to the canonical systems of the inverse motion this circle has the equation

x2

+

_y2

+

_y

₌

_o.

We see therefore that the canonical systems ot the inverse motion are

syrrz,-metrical with those ot the original motion with respect to the pole-tangent.

As a consequence the inverse motion is - with respect to its own canonical

systems - given by:

(2)

ë being the conjugate complex of c.

From (2) it follows that

wh en ce in view of _ëo = ë_l = 0:

c"

= _

n~2

(~)ikën_k'

k=O

.

(3)

This equation gives us (as it should be):

(4)

and furthermore:

Splitting up in real and imaginary parts we obtain:

(5) ~a=-aa+3; ä:,=-a,-4ba . ä:s=-as + 10aa-Sb,-10

ba = ba

b,

= - 4a3 + b4 + 6'

b

_s = - Sa, + b5 - IOba 10

Introduction [1-9/10J

. 9. The curvatures of the fixed and the moving polode at the pole

By differentiating (2.3) k times and afterwards putting cp

=

°

we obtain

(1 ) Zp k

=

c_k

+ (-

_i)"+2CnH (k

=

0, 1,2, ... ).

w

For n

=

1, k

=

1, 2 this yields: Zp,!

=

1, Zp,2

=

-

i

+

ica, so that Xp,!

=

1,

Xp,2 = - b3,

Yp ,!

=

0, Yp,2

=

a3 - 1.

The curvature 'XI of the fixed polode at the point coinciding with the pole of the zero-position is therefore given by

(2)

Proceeding in the same way with (2.4) we get:

Zp".k

=

(-

it+2

~

( : ) ( -i)lcnH_1 (k

=

0,1,2, ... ). 1=0 .

This equation gives for n

=

0, k

=

1,2: zp,!

=

1, Zp,2

=

-

2i

+

ic3. We have therefore:

Xp,!

=

1,

X p ,2

= -

b3 ,

YP,!

=

0,

YP,2

=

a3 - 2.

From these relations we get for the curvature 'Xm of the moving polode: (3)

The equations (2) and (3) not only give us the relation

(4) 'XI - 'Xm

=

1

between the curvatures of

PI

and

Pm>

they also furnish us with an (implicite) geometrical meaning of the derivative as. In this connection it is worth no-ting that by giving the curvature of PI (and therefore that of

Pm

at the same time) we get no information about b3 . This means that the curvatures of

PI

and

P,,.

do not characterize completely the infinitesimal properties of the motion up to the third order.

10. The curvature of the second fixed polode at the second pole

Putting n

=

2, k

=

1,2 into (9.1) we find: Zp !

=

cs, Zp 2

= -

i

+

Ct.

t' 2'

This means:

Yp,,!

=

ba,

[1-10/12] Introduction

We find therefore the following expression for the curvature 'XI. of the

~econd fixed polode at the second pole and in the zero-position:

(1) 'XI

=

asb4 - a4bs - as

,

• (aa2

+

ba2)f

provided that aa and ba are not simultancously zero. As a consequence of

(1) we state:

11 as and b_s are not both zero, (2)

is a necessary and sulficient condition lor the second pole ol the zero-position to be at an inllexion point ol the second lixed polode.

11. Polar co-ordinates

For certain purposes it is convenient to use polar co-ordinates in the planes v and V. These co-ordinates rand

e

of a point A (x, y) of v will be de-fined as follows. First of all they are related to x and y by:

x

=

rcose, y

=

rsine.

With regard to the argument

e

we make the following conventions :

if

y

=

0 but x =1= 0 we put e

=

0; if Y =1= 0 we take on the line joining

A (x,y) to the origin a point B having a positive ordinate, the argument

e

of

A then being the angle (measured in the sense indicated by the orientation

of the system xoy) between the half-lines ox and oB. Thus we have for any point not coinciding with the origin 0 ~

e

<

71:.

The argument thus defined will also be called the argument ol the line

oB, the argument of the x-axis being zero. The argument of the origin is not defined.

If y =1= 0 we define the radius-vector r of A by

if Y

=

0 we put

r = x. 12. The Euler-Savary equation

The equations (6.3) yield for a point of v lying on the pole-tangent of the

zero-position and not coinciding with the origin : ~

=

'YJ

=

O. From this we

infer that the centre of curvature belonging to any point of the pole-tangent

and not coinciding with the pole coincides with the pole.

Introduction [1-12/13]

Using polar eo-ordinates let A(r, 8) be a point of v lying neither on the

pole-tangent nor on the inflexion eircle of the zero-position. We denote by

a((I, 8) the eentre of eurvature belonging to A. Then we have from the

seeond of the equations (6.3):

rsin8

(1= .

r

+

sin 8

This is the EULER-SAVARY equation usually written as

(1)

e

r sin8

Bearing in mind that r

+

sin 8 = 0 represents the inflexion eircIe in polar

eo-ordinates one ean decIare (1) to be valid too for any point on this cirde differing from the origin .

13. Bobillier's construction

We give in this section a simple proof of this well-known eonstruetion<.

Let A and B be two points of v not lying on the pole-tangent and not being

eollinear with the pole P. We denote the eentres of eurvature belonging to

A and B by a and f3 respectively. The pole P is then the point of

inter-section of Aa en Bf3. We denote the polar eo-ordinates of A and B by

(rl> 81) and (r2' 8 2) and those of a and f3 by ((ll> 81) and ((12,82), Henee:

(1)

(Ik r_k sin 8_k

(k

=

1,2).

Taking the lines PA and PB as the u-and w-axis of an oblique eo-ordinate

system (fig. 3) we obtain for the equations of the pole-tangent, the line af3

and the line A B in this order:

[1-13] (2) (3) (4) Introduction u sin

el

+

w sin

e

2

=

0, u

w

-

+

-

=

I,

el

e2

From (3) and (4) we deduce that the line q passing through the pole and

incident with a{3 and A B is represented by

Using (I) we rewrite this equation as

(5) u sin

e

2

+

w sin

el

=

o.

We see by comparing (2) and (5) that the pole-tangent pand the line q

are symmetrie with respect to the bisectors of the angles between PA and

PB. This is in tact the essence of BOBILLIER'S construction [20, p. 219].

CHAPTER 2

THE CIRCLING-POINT CURVE AND BALL'S POINT

Summary. This chapter contains a study of the location and the number of the points of BALL of a given position. The notion of a point of BALL wiek excess is introduced here. For aims of further use some properties are given of rational· circular curves of the third order.

1. The circling-point curve

It is easy to obtain trom (1.5. 1) that the curvature ot a path is station-ary if

(1) (X'2

+

Y'2) (X'Y'" - X"'Y') - 3(X'Y" - X"Y') (X'X"

+

Y'Y")

=

0,

X'2

+

Y'2

*

o.

In the zero-position these conditions take the form:

2) (x2

+

y2) (aax

+

bsY) - 3X(X2

+

y2

+

y)

=

0, (x, y)

*

(0, 0).

The points of v being in the zero-position at a point of their orbits where the curvature is stationary are therefore points of the curve represented by (2). This curve is called the circling-point curve or the curve of stationary

curvature and will be denoted by cp.

The origin has to be excepted trom the locus, this point being according to (1.7.5) at a point of its path with stationary curvature if and only if

(3)

The curve cp is circular and has at the pole a node, the tangents being the pole-tangent and the pole-normal. As cp is a curve of the third order, we deduce that it is rational.

It is easily seen that cp degenerates in the following cases only:

10 _a

3

*

3, ba

=

O. The constituents are the pole-normal and the circle

r

represented by

(4) (a3 - 3) (X2

+

y2) - 3y

=

0

[2-1] The circling-point curve and Ball's point

2° as

=

3, ba =1= O. Now cp splits up into the pole-tangent and the drcle

r

o with

(5)

as its equation.

3° as = 3, ba = O. In this case cp consists of the tangent, the pole-normal and the line at infinity of v.

It may be observed that rand ro are the circles of curvature of cp at its node. This observation leads to a geometrical interpretation of ba: ba equals

t

times the curvature of that branch of cp that touches the pole-normal. A similar geometrical interpretation of aa can be given.

In the case of non-degeneration the line

is the real asymptote of cp.

It is well-known that a focus of a curve is deflned as a point of intersection of two isotropic tangents of the curve. If the curve is a circular one there are isotropic tangents having their points of contact with the curve coinciding with the istropic points. The points of intersection of tangents of this kind are called special foei of the curve.

In the case of the curve cp there are two conjugate irnaginary tangents ha-ving their points of contact with cp at the isotropic points This means that cp has one real special focus F. An easy computation reveals that F has the co-ordinates:

(7)

This result shows that F is a point of cp. We see furthermore that the reflection of Fin P is on the asymptote of cp. If Gis the point on P F such that P F = FG we deduce from (7) that G is a common point of rand

ro.

If the curve cp is irreducible we can obtain a parametric representation of it by putting y = UX. The result reads

(8) 3u 3u

2

x = - - - - y= .

(u2

+

_{1) (bau}

+

_{aa - 3)' (u}2

+

_{1) (bau}

+

_{aa - 3)}

The parameter-values b_a/(3 - as) and (aa - 3)/b_{a correspond to the}

point of intersection 5 of cp with its asymptote and to the focus F respe c-tively. Fig.4 gives a sketch of the curve for the case aa

=

ba

=

2.

The circling-point curve and Balt' spoint

fig.4

In the case as =F 3, b_s

=

0 the equations (8) take the form

3u 3u2

(9) x= , y= -

-(as - 3) (u2

+

_{1) (aa - 3) (u}2

+

₁₎ this being a parametric representation of the circle

r.

If aa

=

3, b_s =F 0 we get from (8) the parametric representation

(10)

2. The curve cp

Eliminating x and y between (1 .6.3) and (1 .2) one obtains

(~2

+

1]2) (as~

+

_bs1]) - 3~1]

=

o.

[2-1/2]

zero-[2-2/3] The circling-point curve and BaU's point

position at a point of their orbits with stationary curvature is therefore the cubic curve given by the equation:

(1)

with the exception of the origin. This curve is called the centring-point curve;

we shall denote it by cp. It is a circular curve having the pole as a node, the

pole-tangent and the pole-normal being the tangents at this point. Obviously

cp

is rational. The only cases in which

cp

is reducible are the following.

10 _as

*

_{0, bs}

₌

_{0; the curve splits up into the pole-normal and the circle}

Ï'

with equation.

(2)

2D a3

=

0, b3

*

0; the constituents are now the pole-tangent and the circle

ro.

The circles

Ï'

and

ro

are the circles of curvature of cp at the pole. This

shows that cp and cp have a three-point contact along the pole-normal and

a two-point contact along the pole-tangent.

30 _a

3

=

ba

=

0; the curve consists of the pole-tangent, the pole-normal and

the line at infinity of v.

Clearly ba =

°

is the necessary and sufficient condition for cp and cp

to degenerate at the same time.

Using (1 . 8 . 5) we may rewrite (1) as:

(3)

As has been explained in section 1.8 we get the equation of cp with

respect to the canonical systems of the inverse motion by replacing y

by - y in (3). The resulting equation

(x2

+

y2) (äax

+

baY) - 3x(x2

+

y2

+

y)

=

°

shows that cp is the circling-point curve of the inverse motion. Of course

cp is the centring-point curve of the inverse motion. At fig.4there is a sketch

of

cp

too. As a consequence of the choice aa

=

ba

=

2 the asymptote of

cp has at infinity a second-order contact with the curve; the inflexion circle

of the inverse motion passes more over through the point of intersection S

of cp with its asymptote.

3. Ball's point

For a given position of v a point of v belng at a point of its orbit where

the curvature of the orbit has the stationary value zero is called a point 18

The circting-point curve and Ball's point [2-3]

ot BaU, a Bl-point for short. In the position under consideration the orbit has a contact of the third order (four-point contact) with its tangent at the said point: the point is at an undulation-point of its path.

Evidently a Bl-point is a point of intersection of the inflexion circ1e and the circ1ing-point curve.

If aa

*

°

the circ1ing-point curve intersects the inflexion circ1e in addition to the pole and the isotropic points, counting together as 5 points of inter-section, in only one point having the co-ordinates

(1)

The pole is no Bl-point if aa

*-

0; this results from section 1.7. We may therefore draw the conc1usion that in the case aa

*-

°

there is in the zero-position only one Bl-point indicated by (1). Herein lies (possibly) the reason that one usually speaks of the Bl-point of a given position. That this need not be correct may be seen by observing that in the case aa

=

0, ba

*

°

the origin is indicated by (1) and we know from section 1.7 that this point is a Bl-point if and only if aa = a4 = a5 = 0. This means that there exist positions not having a Bl-point.

We have such a position if

aa

=

0, ba

*-

0, a42

+

a52

*

0.

On the other hand if aa

=

ba =

°

the circ1ing-point curve splits up into the inflexion circ1e and the pole-normal. In this case any point on the in~ flexion circ1e with the possible exception of the origin is a Bl-point of the zero-position, the origin being a El-point too if a4

=

a5

=

°

at the same time.

We give a survey of the above in the following list:

Condition(s) Bl-point(s) aa

*

°

( aaba aa2

+

ba2 ' a 2 ) - aa2

~

_ba2 aa

=

a4

=

a5

=

0, ba :f=

°

the origin aa

=

0, ba

*-

0, a42

+

·a52

*

°

none aa

=

ba

=

0, a42

+

a s2

*

°

the points on the inflexion circ1e with the exception of the origin aa

=

a4

=

a5

=

ba

=

°

all points of the inflexion circ1e

[2-3/4] The ci1'cling-point CU1've and BaU's poi~t

To conclude this section we observe that if aa

"*

0 the Bl-point of, the

zero-position is in the parametric representation (1 .8) of

cp

indicated by the

parameter-value - aa/ba, this being the tangent of the argument of the line

joining the origin to the El-point. This furnishes us with a second (indirect) geometrical meaning of the derivative ba. We see moreover that the said line

is parallel to the asymptote of

cp.

0'

4. The BI-points of the inverse motion

If _äs

"*

0 the inverse motion has (in the zero-position) one El-point; wlth respect to the canonical systems of this motion it is indicated by

(1)

On the canonical systems of the original motion its co-ordinates are therefore:

(2)

whence it expires in view of (1.7) that the lines joining the pole to the focus

of

cp

and to the El-point of the inverse motion are symmetrie with respect

to the pole-tangent. In fig. 4 the El-points of the original and of the inverse

motion are denoted by Band Ë.

The origin is the only El-point of the inverse motion in the zero-position

if and only if:

(3)

This statement results from (1.8.5) and the survey of the foregoing section.

The inverse motion does not possess any Bl-point in the zero-position if

aa

= 3, ba

=

0 and if at least one of the last two equalities (3) does not hold.

On the other hand any point of the circle x2

+

_{y2 - Y}

₌

_{0 is a El-point of}

the inverse motion if and only if:

The circling-point curve and BaU's point .[2-5]

5. BI-points with excess

If we have for a Bl-point of a given position

" = ,,'

= ..

..

=

,,('+1)

=

0, ,,(,+2) =1=

°

this point will be called a El-point with excess r for the position under

con-sideration. We denote a El-point with excess ~ s as a Els-point. .

In the case as =1=

°

the zero-position has one Bl-point.

This-is a Elrpoint if and only if

XIY4 - X4YI

=

0.

If we denote the said point by _{(x o, Yo)} this condition takes the form. (I)

In virtue of X02

+

Yo2

+

Yo =

°

we may write instead of (1):

(2)

and therefore in view of (3. 1) : (3)

This relation which we met already in section 1.10 represents for the case

as =1=

°

a necessary and sufficient condition for the Bl-point of the

zero-po-sition to be a ElI-POint.

If as

=

a4

=

a5

=

0, ba =1=

°

the origin is the only Bl-point of the

zero-position; we see from (1.7.6) that it is a ElI-point if and only if a₆

=

0.

In the case as

=

b_s

=

0, a42

+

a62 =1=

°

any point of the inflexion circle

with the exception of the origin is a Bl-point of the zero-position. From (2)

it follows that all these points are ElI-points if and only if a4

=

0, b4

=

1,

whereas in the case a4 =1=

°

the only Bll-point of the zero-position is given by:

(4)

If aa = _bs

=

a4

=

0, as =1= 0, b4 =1= 1 there is no Bll-point although there

are an infinity of Bl-points.

In the case as = ba

=

a4

=

ar; =

°

any point of the inflexion circle is

a Bl-point of the zero-position.

If b4

=

1 at the same time, all these points with the exception of the origin

are Elcpoints, the origin being in this case a Bll-point if moreover a6

=

0.

If however b4 =1= 1 there is no Bll-point unless a6

=

°

in which case the

origin is the only ElI-POint of the zero-position.

zero-L2-5/6] The circling-point curve and Ball's point

position has precisely one ElI-point; the conditions listed in the fust column are necessary and sufficient ones.

Conditions Elrpoint aa

=

aab4 - a4ba =1=

°

( aa ba aa2

+

ba2' a 2 ) - aa2

~

_ba2 aa

=

ba

=

0, a4 =1=

°

( a4(b4 - 1) a₄2

+

_(b 4 - 1)2' a 2 ) - a₄2

+

_{(;4- 1)2} aa

=

a4

=

as

=

a6

=

0, _{the origin} ba2

+

(b4 - 1)2 =1=

°

6. Geometrical considerations

Let y be a rational circular curve of the third order having a node with mutually orthogonal tangents. By a judicious choice of the frame of re-ference it is possible to represent y in Cartesian co-ordinates by the equation (1) (ax

+

(Jy) (X2

+

y2) - xy

=

0.

We assume that y is irreducible i.e. a{J =1= 0.

If we take a

=

(aa - 3)j3, {J

=

baj3 and a

=

aaj3, (J

=

baj3 the equation (1) is that of

cp

and

cp

respectively with respect to the canonical systems of the original motion. Putting y = ux we get the following parametric representation of y:

(2) U u

2

x= , y= .

(u2

+

1) ({Ju

+

a) (u2

+

1) ({Ju

+

a)

Clearly the parameter-value - ajp corresponds to the point at infinity of y.

The parametric representation of the circle of curvature ro touchirlg y along the y-axis reads:

(3) x = - - -1 , y = - - - -u

(J(u 2

+

1) (J(u 2

+

1)

and that of the analogous circle

r

_l touching

y

along the x-axis is

(4) U u

2

x= ,y= ,

a(u2

+

_{1) a(u}2

+

_l)

The cil'cling-point curve and Ball's point [2-6]

the Cartesian equations of these circles being (5)

and

(6) a(x2

+

y2) - Y = 0

respectively.

It is easily found that the necessary and sufficient condtion for three points Ai of y with parameter-values ui(i = 1, 2, 3) to be on the same line not passing through the origin reads:

(7)

We infer from (7) that a line parallel to the asymptote of y intersects the curve at two points the parameter-values u~ and

u;

of which are related by

(8) u~u; = 1.

If Al and A2 are two points of y indicated by the parameter-values UI and u 2 respectively the equation of AIA2 is found to be:

Obviously this line passes through the point of intersection of the

mutu-ally orthogonal lines represented by

(10) and (11 )

The distance from the origin 0 to the line A IA 2 is given by

( 12)

If A3 is the point of y with parameter-value - UI we see from (12) that the lines A2AI and A2A3 are at equal distances from O. This means that

A2AI and A2A3 are symmetrical _{with respect to A 2}0. We have therefore:

Theorem J. The segments intercepted by y on two lines passing through 0 and lying symmetrically with respect to the x-axis are seen under equal or sup-plementary angles trom any arbitrary point ot y.

[2-6f7] The circling-point curve and Ball's point

If the line AIA2 represented by (9) is parallel to the asymptote of y its equation simplifies in virtue of (8) to:

The refiection of this line in the origin has the equation (13)

The parameter-values of the points of intersection of (13) with Y are found from:

(UI

+

u2) (a

+

(Ju)u

+

(u2

+

1) (a

+

(Ju)

=

O. The roots of this equation other than -

al

(J satisfy

or in virtue of U IU 2

=

1:

The points of intersection A' I and A' 2 of the line (13) with the curve are therefore indicated by the parameter-values - UI and - u2 respectively.

This proves:

Theorem Il.

11

Al and A 2 are two points ol y such that AIA2 is parallel to the asymptote ol y then the rellection ol AIA2 in the node 0 ol y intersects the curve at two points A'l and A'2 such that OA. and OA'i (i

=

1,2,) aresymme-trical with respect to the tangents at the node.

7. Construction of the circ1es

r

_o•

r

and

Ï'

The centre of curvature belonging to a point of the curve cp with para-meter-value U coincides with the point of

cp

having the same parameter-value u. Let Al and A 2 be two points of cp and al and _{a 2} the centres of cur-vature belonging to these points. If UI and u₂ are the parameter-values of

Al and _{A 2} respectively we know that AIA2 has the equation (6.9) with a

=

(a3 - 3)/3, (J

= b

₃/3 whereas al a2 has the equation (6.9) with a

=

a₃/3,

(J

=

b3/3. Hence we infer that AIA2 as weil as ala2 passes through the point of intersection of the mutually orthogonal lines (6.10) and (6.11). The equation (6.10) represents therefore the line joining the pole P to the point of intersection

Q

of AIA2 and _ala2, whereas (6.11) is the equation of the perpendicular on PQ passing through Q. Substituting the parametric

The circling-point curve and Ball's point [2-7]

presentation (6.3) of

ro

into (6.11) we find for the parameter-values of

the points of intersection of

ro

with the said perpendicular the equation

(1)

having UI and U_{2 as its roots. This means that these points are on PAl}

and PA2.

As a consequence we have the following construction of the circ1e

ro

fig. 5

if we are given two points Al and A 2 of

cp

and the centres of curvature

al _{and a 2 belonging to these points (fig. 5). By means of Bobilliers'}

construc-tion we find the pole-tangent

p

.

The perpendicular through the point of

intersection Q of _{a l a2 and AlA2 on PQ intersects PAl and PA 2 at} 51 and 52

respectively.

ro

is now the circle passing through 51 (or 52) and P and having

its centre on the pole-tangent.

In order to give a sirnilar construction for the circ1e

r

of section 1 we start

[2-7] The circling-point curve and Balt' s po'int

from which it follows tbat its point of intersection R witb AlA2 i.e. with tbe

line (6.9) is on tbe line

(2)

tbis being obviously tbe line tbrough R parallel to PQ. Substituting into (2)

the parametrie representation (6.4) of

r

we find once more for the

para-meter-values of the intersections of (2) with r the equation (I). Tbe said

intersections Tl and T2 are therefore on PAl and PA2 respectively. Tbe

construction of r is now easily performed.

Tbe analogous construction of the circ1e

Ï'

of section 2 is evident. If G is the point of intersection of

ro

and rother tban the pole the

mid-point F of PG is tbe special focus of cp (compare section 1). Tbe special focus

ft

of

cp

can be found in a similar way. Tbe results of this section and of the

preceding section will be used in section 5.10.

CHAPTER 3

THE BURMESTER-POINTS

Summary. This chapter is devoted to a study of the location and the number of BURMESTER-points of a given position. The concept of a BURMESTER-point witl! excess

is introduced. In the course of the treatment it appears useful to subdivide the

posi-tions of the moving plane into general and special positions (section 3.3). A number of

theorems concerning the BURMESTER-points and the point(s) of BALL forms part of this

chapter.

1. The Burmester-points

A real point of v not at infinity being in a given position of v at a point of its orbit where the fust two derivatives of the curvature of the orbit are zero is called a BURMEsTER-point (a Er-point for short) for the position under consideration.

This definition implies that a ElI-point is a Er-point. At a Er-point the orbit has a contact of the order four at least with its circle of curvature. If this is a contact of the order 4

+

r precisely we speak of a Er-point witk the

excess r. A Er-point having an excess of at least s will be denoted as a Ers-point.

We de duce from section 1.7 that the origin is a Er-point of the zero-position if and only if

(1) as = 0, _lOa4bs

+

3a5 = 0, 90aèa2

+

40a4b4

+

3Sa5bs

+

Ba o = 0.

It is a ElI-point if (2)

In order that for an orbit x'

=

x"

=

0, we must have besides (2. 1 . 1) :

(3) (X"Y'" -X"'Y" +X' y(4)_X(4) Y') (X'2+ Y'2)_(X'Y'" -X'''Y') (X'X"

+

Y'Y") + -3(X'Y"-X"Y')(X"2+ Y"2+X'X'"

+

Y'Y"')=O.

For the zero-position this takes the form (in virtue of (1 .4. 1) and (1 .4.2)) : (4) {(a4+4ba)x-(4as-b4-S)y+3}(x2+y2+y)-y{a4x+(b4-1)y}=0.

This circular cubic curve br touches the pole-tangent at the pole.

If cp and br have no common components the complete systems of their intersections consists of 9 points To this system belong the origin (counted

[3-1/2] The Burmester-points

threefold) and the isotropic points of v. These points (taken with their proper multiplicity) being omitted there remain four points of the system forming what will be called the partial system at intersectians.

Any real point of the partial system of intersections is a Er-point of the zero-position provided that it is not at infinity nor does coincide with the origin.

If the origin belongs to the partial system of intersections the question whether it is a Er-point has to be decided by checking (1). Thus we are led to the conclusion that in the case of no common components of cp and br

there are at most four Er-points in a given position.

2. The rea! intersections at infinity of

cp

and

br

We distinguish the following four cases:

I. (as - _3)bs =1= 0; the circling-paint curve is irreducible.

We see from the parametric representation (2.1.8) that cp has one real point at infinity corresponding to the parameter-value

u o = (3 - aa) Iba.

By substituting (2. 1 .8) into the equation (1 .4) we find that the parame-ter-values corresponding to the points of the partial system of intersections of cp and br are the roots of the equation

(1) _ba2_u'

+

_{(3bs - 2asbs)u}S

+

_(sb

S2 - 3as2

+

3aa

+

3b, - 3)u2

+

+

(6asbs - 3bs

+

3a,)u

+

as(aa - 3) = O.

This equation has U_o as one of its roots if

(2) _4as2

- 17aa - aab,

+

a,ba

+

4b_s2

+

3b,

+

15

=

0; moreover U_o is a multiple root of (1) if besides (2) the relation:

(3) 4ass - s7as2

+

192aa

+

4asbs2

+

6asb, - 18b, - 27bs2 - 3aè3 - 171

=

0

holds.

This means that U_{o is a root of multiplicity} ft ~ 2 of the equation (1) if and only if a, and b, are given by:

(4) 28 4as3 _{- 33a} s2

+

90as

+

4asba2 - 3bs2 - 81 a₄

=

-

~~----~----~----~----~---3b_s b _ _ 4aa s _{- 4sas}2

+

_141a s - Isbs2

+

4asbs2 - 126 4 -3(aa - 3)

The Burmester-points [3-2]

If moreover aa and ba are related by (5)

the multiplicity of U_o as a root of (1) is at least 3.

As (5) may be written

aab32

+

_(sa3 - 12) (as - 3)2

=

0

this fact can only occur if 0

<

as

<

I

r

The assurnption I-l

=

4 leads, as is easily seen, to as

=

t

>

152 • The multiplicity of U_o as a root of (1) is therefore 3 at most.

If I-l = 1, 2 or 3 the remaining roots of (1) are obtained from (6) _b3(aS - 3)u3 - 3(a3 - 2) (aa - 3)u2

+

(Sasbs - 3bs

+

3a4)u

+

+

as (as - 3)

=

0,

(7) b_{S(a3 - 3)u2 - (as -} 3) _{(4as -} 9)u

+

a_3b3

=

0

and (8)

respectively.

If as

=

0, b3

*

0 the equation (1) takes the form

(1 *) bS2U4

+

3bsu3

+

(sbS2

+

3b4 - 3)u2

+

3(a4 - bs)u

=

0 and (2) and (4) read

(2*) (4*) bS2

+

27 b_s b _ _ sbS2

+

42 4 - 3

In this case there is no b_{s satisfying (5)}; the multiplicity of U_o

=

3/b_s as a root of (1 *) is therefore 2 at most.

For as

=

0 the equations (6) and (7) assume the form: (6*)

and (7*)

[3-2] The Burmester-points

11. a3

=

3, ba ~ 0; the curve cp degenerates into the pole-tangent and the

circle represented by (2. 1 . 10).

The partial system of intersections of br and cp is obtained from

and

There is a common real point at infinity if and only if

(11 )

the point in question being in that case the point at infinity of the pole-tangent.

lIl. aa =1= 3, ba

=

0; the circling-point curve brealls up into the pole-normal and the circle represented by (2.1.9).

The partial system of intersections of cp and br is obtained from

(12) x

=

_{0, (4aa} - b4 - 5)y2

+

(4a3 - 9)y - 3

=

0

and

There is one (real) point of intersection at infinity of the pole-normal if

(14)

this point is counting twice if

(15)

IV. aa

=

3, b₃

=

0; the curve cp consists ol the tangent, the pole-normal and the line at inlinity ol v.

If a4(b4 - 7) =1=

°

the line at infinity and br interseet at a point being represented (in homogeneous Cartesian co-ordinates) by

( 16)

The partial system of intersections of cp and br consists of this point and the points obtained from

(17) (18)

30

a4x

+

3

=

0, Y

=

0,

The Burmester-points [3-2/3]

If a4

=

0, b4

*

7 and also if a4

*

0, b4 = 7 the curves cp and br have two common points at infinity both coinciding with (16). In the case a₄

=

0, . b₄

=

7 the curve br breaks up into the line at infinity and the hyperbola x 2 - y2

+

y

=

0. The only finite point of the partial system of intersections is (0, 1).

3. Theorems on the Bl- and Br-points of genera! positions

A position of v for which a'"b'''(a''' - 3)

*

°

wiil be cailed a general

position; any other position will be denoted as a special position. We have

a general position if aSb3(aS - 3)

*

0. In this case there is one El-point not coinciding with the origin; moreover the parameter-values of the

Br-points are found from (2.1). The El-point is a ElI-point if and only if its parameter-value - a_S

/b

₃ is a root of the equation (2. I).1t is easy to verify (by direct substitution) that this is the case if and only if

(1)

aresult already obtained in section 2.5. If we denote the remaining roots of (2.1) by uI> u2 and u3 we must have

and therefore:

(2) _{U I U 2U 3}

= - - - - .

as - 3

b3

Keeping in mind that the Br-points are on cp and applying the result (7) of section 2.6, we draw the conclusion that the Br-points other than the Bli-point under consideration are collinear. This proves the following theorem already obtained in quite another way by MÜLLER [18]:

Theorem lIl. It the Bl-point ot a general position is a ElI-point, the re-maining Br-points are collinear.

In the next section we show by geometrical considerations that this theorem is true for any position having a unique Elcpoint.

The condition (2.6.7) teils us th at two points of cp indicated by the para-meter-values UI and _{U 2} are collinear with the Bl-point if and only if

(3) a3 - 3

U IU2

= - - - .

aa

[3-3/4J The Burmester-points

This gives us:

Theorem IV.

11

the zero-position is a general one the Bl-point is on the

line joining two Br-points il and only il the parameter-values UI and _{U 2} ol

these Br-points satisly the relation (3).

If (3) holds we find from the equation (2.1) that the parameter-values

u3 and U 4 of the two remaining Br-points are related by

(4)

Now suppose that a general position has a Blrpoint being collineair with two of the remaining Br-points with parameter-values say UI and u2•

Then the above discussion teachesus that this situation occurs if and only if the equation (2. I) has _. a₃

/b

₃ as a root with multiplicity two. We have therefore proved:

Theorem V. A general position has a Rtl-point lying on the line joining two ol its Br-points il and only il the said Rtl-point considered as a Br-point has the multiPlicity two.

In this case the curves cp and br have a contact of the fust order at the

BI-point of the given position. It may be observed that the parameter

-values of the Br-points mentioned in Theorem Vare the roots of the equation.

(5)

Moreover it is easy to verify that - _{as/bs is a double root of the equation} (2. I) if the relations (I) and

(6)

hold simultaneously.

4. A conic incident with the Br-points

We rewrite the equation (2.1.2) of the curve cp in the form

(1) {(as - 3)x

+

baY}(x2

+

y2

+

y) - y(asx

+

baY)

=

O.

From this equation and the equation (1.4) of br we deduce that the

Br-points are on the conic c represented by

or

(a3x

+

bsy){(a4

+

4bs)x - (4as - b4 - 5)y + 3}

+

- {(as - 3)x

+

baY} {a4x

+

(b4 - I)y} = 0,

(2) (a3x

+

baY){4b3x - (4as - 6)y + 3}

+

3x{a4x

+

(b4 - I)y}

=

O.