Remarks and calculations concerning a paper of R. Müller on kneecurves

Remarks and calculations concerning a paper of R. Müller on

kneecurves

Citation for published version (APA):

Meiden, van der, W. (1981). Remarks and calculations concerning a paper of R. Müller on kneecurves.

(Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8103). Technische Hogeschool

Eindhoven.

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Published: 01/01/1981

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EINDHOVEN UNIVERS

Department of Mathematics

Memorandum 81-03

February

1981

Remarks and Calculations concerning a paper of R. Muller

on Kneecurves

AMS Subject Classification

(1979)

53 A17, 70 B15

by

W. van der Meiden

Eindhoven University of Technology

Department of Mathematics

PO Box 513, Eindhoven

The Netherlands

REMARKS AND CALCULATIONS CONCERNING A PAPER OF R. ~6LLER ON KNEECURVES

a

o.

Introduction

Let OO'R'R be a plane four-bar mechanism 'vith fixed link 00'. The links OR and O'R' are thought of as edges of triangles DRS and O'R'S' respectively. Sand S' are connected Movably by two further links SK and KS', this contraption being called a knee. All motions are supposed to be in the plane of OO'R'R.

The motion of SKS', generated by that of the four-bar, forces K to,describe a curve k, the kneecurve. Muller's investigation of 1895 [ 1 ] is about the properties of this curve, 'l'he data are, according to Muller, as shown in fig. 1. It is assumed that s,s' > 0 and 0 ~ a,a' < 2n. A coordinate-frame is chosen with origin 0, x-axis along DO', and y-axis per-pendicular to 00', In view of the complexity of the calculations we once and for all define the following abbreviations: (0.1) (0.2) (0,3) (0,4) (0.5) 2 2 X :~ x + y Y : ~ (2s) -1 (X _ Q2 + s 2)

~ :~ X + iy, a + bi :~ e ia SI a' + b'i e ia' (I; - m), X : ~ S -1 l' e ia Xl ::;:;; S I -1 r I e ia '

fo, :~ X - X' ~ :~ e i (a+a') y + ia :""" e i (a-a')

c2 ::;;:::; r2 + r,2 + m2 _ n2, U:= aa' + bb', V:= alb _ ab', R2

W ::;; C 2 U

-

2m2bb' _AS := raX'Y

-

rlaXY'

2 ~

A3 ra'Y

-

r'aY' D := X'y + XY'4.

B3 := rbY' + r'b'Y E := r 2 X'y2

-

21'r'UYY' + r,2 Xy ,2

B4 := r2b2x' + r,2b,2x F ::::;: W

-

_2mA3

-

2rr'YY' x _ y2, R,2 c4XX' 2 2 T4 ::;;;

-

4m B4 + 41' TS := -8mr1"B₃V 2

-

4c mAS To := 4m 2 E

-

4r 21',2D C := _T4 + TS + T6 1',2u2 B := 21'1"F(XX' - D) + YY'C .

Complex conjugation is denoted, as usual, by a bar; hence, for example,

s

x - iy and X s -1 re -ia It is not difficult to verify that

(0,6) a 2 + b2 = X aU + bV a'X X ~~

a,2 ₊ b,2 _{x' alU}

_-

_{b'V aX' X' (t;;}

_-

_m)

₍₂

_-

_m)

2 v2 XX' i (a-a') - 1" £ a' - Re('I'li) U + U + iV e ~ (~

-

m) cos a

-

cos s' S 1'1" _~XX, 'l'XX' ss'

Th..: equation of is derived by Muller without details of the caleulat,ion. Our first aim is to give this calculation in full, OUI' notations are slightly different from chose of Muller; our method is, apart f:com the use of complex numbersl essentially his~

vie interpret, as already suggest,ed, the configuration of fig. 1 as lying in the plane of complex num-bers; the point

are related by

is represented by the complex number ~ ~ " + iy. 'l'he parameters e,6' of the motion

(1.1) n •

The locus of the point

K has to obey the equations

(1. 2) £ ,

(1. 3) I~ g'ei (e'-a')

_ml

£' .

The equation of k is obtained by elimination of 8,8' from (1.1), (1.2) and (1.3). writing for the moment i8

L, : = e , t ' , we transform (1.1) by elementary calculation into

(1.4) c2 2rr' Re(tt') + 2mr Re(t) - 2mr' Re(t')·

(1.2) is equivalent to

la

+ bi

6tl

leading to

Y a Re(t) + b Im(L)

"hieh, together with It! I, is solved by

(1.5) t X l{ay - bR + i{bY + aR)} .

Similarly

(1.6) t ' x,-l{a'Y' - b'R' + i(b'Y' + a'R')}

whence

(1. ) Re(tt' ) (XX') I{U(YY' + RR') + V(YR' - Y'R)} .

By substitution of (1.5), (1.6) and (1.7) in (1.4), and multiplying by XX' we get

2rr'UYY' - 2rr'URR' 2rr'V(YR' Y'R) + 2mrX' (aY - bR) 2mr'X(a'Y' - b'R')

and, using (0.5),

2

'i'tlis, by squaring and regroupinlj, develops into

(1. fl) 4rr'RR' (U t _ 2rr'UYY. _{2(r'VY' + rubX') (rYV} + _rub'X)}

- 2rr'UYY' - 2

'fne braced part of the left-hand side of (1.8) can be reduced to

(1. 9) or XX'P, by (0.5) ,

and the riglit-hand side of (1.8) can be expanded and regrouped into

( 1.10) 2 + XX'(- 8mrr'VB 3 - 4m B4 + + 8mrr'YY'A 3 + 4m 2 or, again by (0.5)

xx'

- 4c2roA - 4rr'YY' 5 2 2 2 ) + T4 + TS - 4rr'YY' (c U + 2m aa' - 2mA

3) +

Obviously (1.8) can be divided by XX' and then with (0.5) be made to read

( 1. 11) 4rr'RR'P 4rr'Yy'P .

We square again, enhancing

or I rearranging I

(1. 12) 8rr'F{2rr'F(XX' - DJ + YY' (T

4 + TS +

which, according to (O.S) can also be read

Brr'v'B

+ T )2 6

Wlth (0.5) i t can easily be checked that the degrees of P, C, B in x and y together are 4, 6, 10 res-pectively.

Hence the first conclusion; the kneecurve k is, In general, of degree 14. This, as well as the equation (1. 12), is in agreerJ1r:::nt with Muller.

..

- 4

The behavi.ollt' or ~: .:j,t infini ty

(cdlCU"'-~Li(~~1-Mill LhE~ focal of k; sir.ce his findings are based on arguments that

are not. ver-y C'vnvi Hel ng \-le continue i n a di fferent way ~ Iro this purpose we fi.rst of all define curves

(2.1) B 0, C 0, F ()

respectively~

Next: we ;-,omogenize t.he coordinates x,y of t.he plane with a third coordinate z (the infinite line ll"" being rep>eesented by the equation z 0) and introduce new coordinates in the plane by

(2.2) x + iy x - iy .

Our aim is to calculate equations of ~ld

KF

in (s,~,z)-coordinates. In passing we remark that the fundamental triangle for these coordinates has as its Vertices 0(0,:),1), I(O,l,O), J(1,O,O) or,

with respect to (x,y,z) coordinates, O{O,O,l),

I{i,i,O), J(1 ,-i,O). Fig. 2

/

O~---+----Behaviour of a curve at infinity includes its inter-sections with the infinite line and its tangents at these points.

We expand the expressions (0.1)-(0.5) in terms of ,s,z. We agree upon the following formalism: An expression, say E of (0.5), is explicated in separate terms according to their degrees in s,~ together, every term having its degree as a super-script; thus E ; E(6) + E(5) + E(4) + ... + E(O), and, abbreviating again, E(6,5,4),=E(6) +E(5) +E(4). Factors z are omitted again.

Straightfor'.vard but occasionally tedious calculations give

(2.3) (2.4)

x

2 aat L Y ~ +

2"

1;1; + 4 'P Y bb'~-

'4

+ 2 U v yT,l; m,

_2'Y

+ GZ:~ + mi(y + 2 ia)s :t 4 '£. 4 y /25 , ~2 lll.(<j) m -~ + y + iall;

-

'4(qJ

+ y ~ ia)r,

,

4 -2 _lll.(~ ia)2 s + y

-

ia) t; + y + 4 1012 ,

(2.5) Remark: (2.6) Remark:

w

yy' 2 ;; + + (pt. 4 2 -m hr;l, j-+ - 2 X I (s I 2-- mr,

1;;-.

2 ( ,2 + m2j}

~

+ . 2 + (5 + Sf 1 ~XX'1; 2-2 -~) l; 2-F

_{- "2}

1; +

_xx

l; + 2 2 2 _.Q,,2) +

tC

y + s' £ - + m 2 'PX(X' +

"2

{(5

-

1) + (8,2 _~' 2 _{) 'IlX'} + !l! { ₂ - l 2 -) lj>X (X' 1) + (s,2 i ' 2 -l'l'X' +

-

i 2) {rr? cos {J.'

-

,2

-

.~' 2 s 2 2 rn ) (y + ia) }r; + (c 2 m2) (y io) }~ r

-

,

-2 ~t; + t; +

~{X

(8 2 _;e2) X' (s' 2 R,.2

-

-1 - X')1;2 + + 2 Re ('!lEi) 2y)

}z;2

+ 2 - 2 m2) (y + m lj>(X'

-

1 l (e 2- _{1 )} (e 2 m2) (y + m q>(X'

-

-

-+ m2)}

If s := ~, then 0 ( kp: if s':= i', then 0' E kp .

XX' + m 2 -r;~ + m2)}

~

--2 - X') 1; + + io)}1; +

-

icrl } ~ + D

1.(2-

+ 4 2 +

1;2~3

+ 1; 3

₂

₊

_l;C

3 ) + s 5' 4s' + (1 + :::---,==-,2 2 2 =--;;':::''-It; ~2 _ m(1 + m 2s,2 -xx' -D + 2 4 ~---~--~.---==-.~---+-m=-)l;l; 2 2 2 (s -.~ l a n d 0 I· C:=m + !l!(-.!_ + 4 2 5 ,2 2-<r, C + (1; + 1;) 4s' 2 2 2 s - £ ) • 2 "In - Z ( 45' + -2 + 1;1; l + _ f,2) , + +

(L. 7) Remark: (2.8) + r' I ) I; 3-~ -2ia -2ia 2 -2io.* + rl e ) -2 2 2 2 2 ~=-.~--- ~ + ~2(r + r' ) - 2r' cos 2a'}~~ , + + 2 2 2 2 m 2 - m2 2 2 +

~

(y - (1 +

2iya)~

+

2'~'

+

~(y

- a -2 2iyo)~ 2 m 3 2 2ia 3 - m r e I ; , 4 - m\c +

1I;:~o

B3 ;

~(~

_{4i s'} + + ( B V' (5) ₃ J 2 2ia' + rl e ) (e- 2ia _ 2) + + 5

o .

1 sx 4i (;0 + 3-2 1; 1; + (B V) (4) 3 ~ {y(sX + s' 8 S i S + io s 2 -2ia 2 -2in' (r e + r' e ) msx ~2_ 4is' + _s + stxJ) S 2-3 r r ~ ~

+~[y(~+

+ (c4 _ s io s

m (y(--" ... cos a +

E.:.

cos (l') + 30

4 Sl 5 r ' . 2-2 sin a + ";;- Slon (1')};;' I; • (B. V) (3) 2 -3 m sX(y + iO") 3 I;

I;

+ j 85'

[!fl;(y

+ 2 raie ia ,2 t,2) -ia m + ia) e S s 2s'

ia -i(> + Oi{SX(S,2 i,2)

+

-

ia) (--" .. e + +

-s s 4 s' -2 1:; + s

'X'

-3 + -s-)}I;I; + ~ 2

~2)} ]1;2~

₊ + (5 s ~ 2 + s (s

-

12)} ] +

Rem!n'k: (2.9) (2.10) (B V) (0) 3 s m r 4 cos a + 0 sin a)

o .

0; and also

B3VI

~:=!O 7 -s

o .

a

2-3 +

4"

~ 1; !O + 2s,{r' cos a' - s'Re(Al} cos a') + X

cos a' + 30 s in a'){ s 2 -

~2)}

I;; C; ,

+

mii

-3 41;1,; + 2 _~{ 1 - 2 2-2 mx (s +

4"

mx(s - t ) 1; + 4 2 s 1 + -4

!r;:=o

(5) (4) T (3) 5 +

-

22) r; + 4

o .

S l y ' ~) S rr' y (."'X + S i S £2)2 irr'(J(3sX + s' +

SIX'

+ . )} s

{ rr' y (-E- cos Ct +

!..'..

cos CL') + 3 rr ' cJ( -E-s in a +

!..'..

s in a' 1 _ c 2

s· s S i ;; sX

s'x'

35X rr'y(~ + ----) + irr'(J(--- + s' s S i S + cos a' - Re(All} m3 { n ' (y + 10) (.E... e-iCl 5' 5 4r2r'oi ei(l 2 2r' s' + c (A -

SO

cos a')} + l2rr' <Ji (s~ (s' 2 ~' 2) s' X I 2 ~2) ) _c2_(x(s 2 ry X' (5,2 £.2)))]

('~+

+ m

-

+ - - - ( 5

-

~-)

-5 S

r

3 ia -ia 2 -+ - m {rr' (y la) (~ e + + + c (A cos (l' ) } ₊ 5 S 5 + m {

sx

2 9.,2) s I

X

I 2 (:2

(x

£2) - 2 £,2» ) ] -2 2rr'ai ( - ( 5 ' _s· + ---IS

-

X' (5'

-

Z;C; s +

Remark: (2.11 ) (2. 12) + T_ (1 ) :0 (0) T I

51~:~o

X, + UYY' + E(6) E (5) E(4) 8 -2 2 (5 -.~) rr' (y+ ta) [ s 2 _{[ 2} III C X - rr' (y (s 2 _ Q2) 0 0; _{and also}

T51

1;;=m

o .

2 + S m 2

2"

[y(m + 4 2 ~ {r 4 -3 1;1; + 2 - 2 - .t ) 1;1;; - m(s 2-3 - 2ml;; 1; + + (3y +

[2 2

m r + - ( - + 4 2 - s 2 2 + s' - £. ) 4yrr') + sst cos ,,] 1;1; + ia) ~ } r r r'

-;( -; -rs,l

+ "'-_-;..-c:'_ + , 2-3 J I; I; + 2-2 (; 1; +

Remark: +

EI<::~o

T6 (6) (5 ) (4 ) T (3) 6 , 2 y + i O ) ( 5 -55 2 2 2 ~(s 2 and

EI<;:~m

482 2 r,2(~ + ) ~l23 , r 2 s 8' r2 2 2r,2 2

m

2)

m

{ 2 ( r ' + - - ( r 8 s,2 r; 2 m2) + m {

-

m~) ₊ + + + 2_' 2 X' 2

--

m X (ilID + - m X' (lim 4r,2 rrl + - - - 4Y-8S') + s,2 r r ' . 2 2 2y 55' (5 + s' 2m[m4 9.2 (2

-

₂

-s

r

4 2mLm (2 rr' S5' _,2

2]

-~-) + 5,2 r,2 2 -2) +m { 5 ' ,2 2

,21

r r

J

s' + ~ 2 (3y + (3y + x'r2) E.) s 5 E.) + S + s 2

-

l2) ₊ s ,2

_-

Q,,2)2 rr' 2 iO);;<

m }

+ 2-3 1; <;

,

+ - 2 ,2 2 2 r' ) + E...-(25' - 2R,' 5,2 9,2) (3\-io r _{r' ]}

SO)

+ S r

,

]

S s ,2 2 , 2 ~) + r ) + st2 Sl

3. Remark: T (1) 6 T (J) 6 Obviously :cr'( + 10)(s2 SSI Y 2 4 C - m 10 -2 2 s - Q ) 2 2 r,2 + 5 - £ ) + s'

rm

2 s + 10) (m 2 _{+ 5,2} - io) + s' 2 +

-

£,2) ₊ s rm 2 _{_ 10) 1m2 + s,2 _ 1,2)} + s 2 -2 - ~)I; + , 2 2 2 2 2 2 2 2 2 ,2t2 y(s -£ )(2m +s' - £ ' ) -4r r' + r r 2 s (r,2 _ s

~2)

] ~ + (r,2 ? r m-)~(s s 2 _{i 2)}

₁

J

1;

,

11;:=0

- r,2); by calculation it appears that

From the results in §2 follow the following Newton diagrams for the polynomials indicated below the pictures.

Fig. 3

yy'

Z6

xx/-o

- 11

l+J+k=2n di of degree 211 is called lozenge-shap",."! or, for short.,

cients a

i ik have t.he property

( J. 1 ) 1\ [J

;)!Jserve that the property of being a lozenge depends on the degree by which the polynomial is considered; example, a polynomial of degree n will be a lozenge when considered as a polynomial of degree 2n. Therefore we 3peak of an m-lozenge if the P?lynomial is to be considered as of degree m.

It, is obvious from the Newt.on diagrams that F and YY' are 4-10zenges; '1'4 is not a 4-10zenge; T4 and TS

are 6-1ozenges , however, and So are XX, - D and

3.2. Lemma. The sum of two 2n-lozenges is a 2n-lozenge; the product of a 2n- and a 2m-lozenge is a 2(n + m)-lozenge.

P roo. f Th e s t atement a out b th e sum 1S 0 Vl0US; ' b ' i f aijk~ ~ i - j z k(i + J' + k = 2 ) n an d are typical terms of the polynomials under consideration, then their product will be

(3.2) _d b i+r-j+s k+t

ijk rst~ ~ Z

If i + r > n + m' then either i 0 or r > m and b

rst

=

0, whence 0; and if

j '~ s > n + m then, in the same way,

=

O. Since the coefficient of

r,i+r~j+szk+t

in the product is obtained by addition of terms like (3.2), it must be also zero.

0

2m)

:3.3. J->,:"oposl tion. C is a 6-1ozenge, is a 12-1ozenge, B is a 10-lozenge and FB and Srr'PB-C£ are 14-1ozenges. Proof. By repeated application of lemma 3.2.

3.4. Proposition. (Srr' (14)

Proof. The leading term of 8rr

= - 4rr I _{[ -rr'}

2 -- m b.b.

is computed with (0.5), (2.5), (2.6) and (2.13) to be

8rr'p(4) {2rr'F(4) (XX'-D) (6) + (YY') (4)c(6)} 1 3-3] + -2)}r;;; s·

o

o

3.S. Theor.."'_IIl' The kneecurve k is of degree 14 unless ,~ = 0; the condition b.

o

is equivalent to the condition tha t triangles and O'R'S' are directly similar.

if ,~ -F 0 then k is a 'I-circular curve.

Proof. The first statement is obvious from proposition 3.4, the second one from the definition of

to in (0.3).

The intersection of 7< and ~oo is determined by the equation

(3. J) (2

o

1\ [,7 _{0) v (z} 01\

e

0)

- 12

It '''ill b" clear by nuw thaT: " and J are sin,]ularities of k. Our next aim therefore must be to eluci-datee Ule nat.ure tehese singulari ties by determining the tangents there. A general remark 1s in order "ere; SUlce crle equation of

:c

is real when considered in x-y-coordinates, the tangents of k at J must

be the complex conjugates of the tangents at [; it is sufficient therefore to consider only J. Since passes tllrouqh J the equation of the whole set of tangents at J consists of those terms in 8rr'F'B-c2 '..,hieh are nomogeneous and of the lowest degree in r" that is, of degree 7 in ~; provided that these terms are present we have, moreover, that c2 does not have any bearing upon these terms, which is apparent. from the Newt:on-diagrams. Hence (under the stated restriction) :

.6, Proposition, The systems of tangents of k at I and J coincide with those of kFB

'I'he terms of degree 7 in

~

in FB constitute a polynomial which can be writ.ten as

~7h(~,Z),

where h is a homogeneous polynomium in ~,z of degree 7.

Since F is of degree 2 in ~ and B is of degree 5 in ~ (both being lozenges) we see immediately that tt(1;,z) can be written

(3.4) h(I;,z) tjJU;,z) fl(~,z) ,

where

~21j;(1;,Z)

and (~,z) represent the corresponding parts of F and B respectively. We thus have

3.7. Proposition. The system of tangents of kFB is the union of the systems of kF and k_B, as well at

r

as at J.

From (2.5) we infer that (supplying factors z again)

13.5) tjJ(t;,z)

- 2

1 -~xx't; 2 + ~

(xx' -

~)r,z + m2- 2

2 Q

(1 - X')z

%

(Xr, - mz) (X'r, - m(x' - l)z) .

In the first place We see that this elCpression can never vanish, since 1'1'1 parameters of the setup. Secondly. the tangents have equations

(3.6)

respectively. -1

fiX Z m(l X' -1 )z

1 for all values of the

o

The intersections of these tangents at with their conjugates at I are the only real points contained by them; they are the special foci of F (by definition, according to Pliicker), and read in inhomogeneous coordinates

(3.7) mx -1 - e IDS -.iu r

IDS' -ia'

Lq. 4

s'

o

13

This means (fig. 4) that the triangles

ORS

and

OO'P!

are directly similar and that the triangles O'R'S'

and are directly similar.

In an analogous but more elaborate way we find the special foci of

k

B• Since, according to (0.5), B 2rr'F(XX'-D) + YY'C, we, for the moment, write F' etc. for the coefficient of the highest power of ~,

present in F etc.

'I'hen, i.)y a repetition of the argument preceding proposition 3.7, we have

(3.8) flU;,:;:) 2rr'P'(XX'-D)' + (YY')'C' .

Obviously FO; lj;(;;,Z), and from (2.6), (2.5) and (2.7), (2,10) and (2.13) we derive

(3.9) (XX'-D)' (3.10) (YY')' whence (J. 11) B(r,,:;:) ~~ {2..(r _ mz) + s' } 4ss' s' ~ 5 t; , 4ss' l.: (7; - mz) , t;(t; - mz) 459' [ - 2rr'lj;O;;,z) _~ {2..(1; _ S· mz) + 5' ;;} 5 +

c·

l

J

'rherefore, two speci al foci of kB are

(3.12)

o

o

0' m •

By the' same reasoning, collecting the terms for C' from (2.7), (2.10) and (2.13), and omitting factors z of the terms beforehand, we get for the remaining three foci the equation

(3.13)

a ,

where (3.14) '10 - 2 - s i(u'-u) (2X' - Xl - m8c + mr,rr'{ s' e + s (n -jal r's' e ) I

This rEsult is again in agreement with Muller's.

We see at one.:; that S

o

identically iff

Z

= 0 or, equivalently, r:, O. We can now strengthen Propo-sition 3.6 into

J.8. Proposition. I f

f1"

0 then the special foci of k are 0.0', F

- 14

-If r.. F 0 tnen (3.1j) "all be written

(J.15) +

o ,

wltn (3.16) _hO m£.

,

2 2 _{~ ;l(a'

-

a) s' (a

-

Q:.') } hI m (2X'

-

Xl c + rr' _s' e +

,

s

h2 m(c 2 _X ) + m(rs e -ia 2rs e -ia X·

-

r's' e -ia' -ia

e lX' - 1) •

We conclude this section by observing another property of the formulae in §2. All the expressions the.re are self-conjugate in the variables 1; and ~ , in other words, they obey the rule

(3. 17)

Therefore, the polynomials F, B, C and Srr'FB-c2 have the same property. Inspection of the formulae makes it clear that the imaginary parts of the coefficients will become zero if sin a

=

sin (l'

=

0, implying, among others, that also cr ~ O. But in this case

0.1S) f(~,l;)

whence, from (3.17) ,

(3.19) f 1;) •

This, however, means that the curves under consideration, and in particular the kneecurve k , are

symmetric with respect to the line 00'.

"l.~. Proposition. If sin C! sin fl' 0, then the line 00' is an axis of symmetry for the kneecurve k.

(MulIers condition, a 0, differs in a non essential way from ours.)

If sin a ; sin at

°

then the foci FI and 1:'2 are on 00'; the roots of (3.15) are real or conjugate complex; hence ~O' is an axis of symmetry for the configuration of the focal points, as could be expected.

On th" other hand, for: this configuration to be symmetric with respect to 00' there are four

possi-bilities:

i c" 1 and F2 are real, and (3.15) is self-conjugate;

i i p and p _{are oonjugate, and (3.15) is self-conjugate;}

" 1 '2

i i i _P1 and _{F2 are} each conjugate to a root of (3.15) , and the third root of (3.15) is reall lV p is real, and PI conjugate to a root of (3.15) •

)

The first of these possibilities leads to sin 0. sin a'

o.

The second one gives at first

(3.20) 1

-

X' -1

-

X -I or X· -t +

-

X -1 1

,

equivalent to

(3.21 ) s cos a. t cos a' A s sin 0. sin 0.'

a

frDm t:tH.: lat t(]r ur th(:"}st;! wo .see that. either sin ex SLn ((.1

o

or sin a

t-

0 and sin a' ,. O. We assume tIld!: Sin

c'

i (), ~ln " I' 0 and that (3.21) holds. Then

2 r 2 s 2.£ cos a s

Ur,der our present assllmptions the denominator of the fraction cannot be zero; the numerator will be zero if cos (1;

,

~ ~s' ; the speciality of the case lies in

; this, by (3.21), implies that. cos a'

the fact that the triangles airS and G'R'S' ar equilateral (and similar, as we saw earlier). Supposing this also not being the case we proceed to calculate Im(h

3/h

o

)'

Apart from some factors we get (3.22) Im

ICE

cos a 1

-

~-.I sin a) (~ 1" _cos (a'

s s s'

r'

a) - cos a + i;o sin (a' a) + i sin (ll} =

(E. r' _a}

cos a 1) { - sin (al

-u) + sin

s s'

!. _{s sin} r'

(l

{S"'-

cos (a' - a) cos a} rr'

{sin (a' (X) } 2E. sin (a' a) sin

57

-

a) cos a

-

sin a cos (a'

-

+ sin a cos a

-

-

-

a

s s

NOw, by (3.21) ,

(3.23) sin (a' ex) sin a (~ cos a

-

1 ) and cos (a'

-

(X) -r' cos a + (sin 2 a

-

cos 2 a)

s'

s r rs

By inserting these expressions in (3.22), division through sin (l and equating to zero we get

(3.24) 4 r,2 2rr· 2 2sr·2

~

CL +

-s 2 5S' rs' or 2 (3.25) 4 + 2 r ~) cos u sr' s r

We apply the same method to hl/h

O' to get (3.26) 4 rs (3.27) 8 rs cos (i 0 2 0 r' ,2 r'

,

~} r

o ,

rs,2 2 2 - - i s - c ) sr,2

From each pair of these three equations the terms with cos2 a can be eliminated to give

(3.28) 2 (3.29) 2 .2 rss,2 + (3.30) 2 (2r' ,2 r s' 2 2 + 3m r' ) cos a 2 2 2

rs

12~2 2 2 rss'" + ~ - _{2r ' s'}

£

+ s 2 - m s 2 rs,2c2 2 2 - rss' - + r' s' s s s 2 3 .2 3sr' 2 +~) + m (~+ s r s r

These relations appear to be dependent when multiplied by 2, -1 and merely added and divided by s,2 they give

respectively and added. !>fuen

(3.31) 2

and (3.29), subtracted from (3.28), leads to

o .

.2

or (3. J2)

,

:.:; t s I ( - ) cos {I 2 cos a E.+~ s r ,2 sr' 2 m

---

_{2 2} m + S1 16

-Since the setup is completely symmetric in the triples (r,s,a) and (r' ,s' ,a'), we might also have derived

(3.33) 2 cos at - + rt s' Sf r'

Insertin7 now (3.32) and (3.33) into (3.21) we obtain easily

(3.34)

n .

This is clearly impossible and settles the second possibility as unsuit~ble.

Reference

[1 R. Muller, Ueber eine gewisse Klasse von Ubergeschlossenen Mechanismen. Z. Math. Phys. 40 (1895), 257-278.