ON THE BICURSAL SEXTIC. 
585 
[624 
624] ON THE BICURSAL SEXTIC. 585 
and the three 
r independent 
i three points 
The three-bar curve may be represented by means of a system of equations of 
the last-mentioned form, viz. x : y : z = Xy (aX + by,) : v 2 (cA + dy) : Xyv, where A, y, v are 
connected as above; or, taking X, Y as ordinary rectangular coordinates, x, y, and z 
are here the circular coordinates - = X+ iY, ~ = X — iY, and z— 1 ; and the parameters 
z z 
G). 
-, - denote like functions cos 6 + i sin 6, cos § + i sin </> of angles which are the 
) meets 0 = 0 
0 = 0 in the 
3, G, the six 
»ints of inter- 
•responding to 
inclinations of two bars to a fixed line. Using, for convenience, Figure 2 of my paper 
on Three-bar Motion, (p. 553 of this volume), the curve is considered as the locus of 
the vertex 0 of the triangle 0C 1 B 1 , connected by the bars G/J and B 1 B with the fixed 
points B and C respectively; and we have CG 1 = a 2 , 0G 1 = b 1 , 0B 1 =c 1 , C l B 1 = a 1 , B 1 B = a 3 . 
Also, to avoid confusion with the foregoing notation of the present paper, instead of call 
ing it a, I take BG=a 0 : the angle 0C 1 B l is — G lt and cos C 2 + i sin C 2 is taken = 7. 
n 
Hence, taking the origin at G, the axis of X coinciding with GB and that of Y 
re 0 is to be 
0 shall pass 
being at right angles to it: taking also 0, </>, \Jr for the inclinations of CG U C 1 B 1 , and 
B l B to GB, we have 
a. 2 cos 0 + a, cos <£ — a = — ci 3 cos ijr, 
0, 1), 
a 2 sin 0 + «1 sin cf) = a 3 sin \Jr ; 
9> “/)> 
viz. writing cos 0 + i sin 0 = A, cos cf> +i sin (f> = y, these give 
© 
1 
a 2 A + a x y — a 0 = — a 3 (cos yfr — i sin yjr), 
6 is to be 
= 0 shall pass 
e thus have 
1 1 / , • • 
a 2 ^ + «1 a„ = — a-g (cos y +1 sin i/r), 
that is, 
(n 2 A + a 2 y - a) (a 2 ^ + a, - - — a 3 2 = 0; 
-g> o), 
v ' \ X y j ’ 
viz. 
0, 0), 
(«o'- + Oi 2 + + «1«2 “ «0^2 (a + 1)-a 0 a x [y + l-'J = 0, 
L. 0). 
for the relation between the parameters A, y. And then 
! equations of 
, JiX -t- j~v = 0, 
iay be added 
X = a 2 cos 0 + b x cos ((f) + G0), 
Y = a 2 sin 0 +b 2 sin (</> + G0) ; 
that the cubic 
e first-mentioned 
of these points, 
of these points, 
ng from a node 
, there the cubic 
:omponent conic 
itions for 6 are, 
viz. if x, y = X + iY, X — iY, then 
(x = a 2 X + b 2 y l y, 
\ 1,1 
\y = a 2 - + b 2 , 
( A 7 x y 
which equations determine the coordinates (x, y) in terms of the parameters A, y con 
nected by the foregoing relation. 
c. ix. 74 
74