The collected mathematical papers of Arthur Cayley.

623]

ON THREE-BAR MOTION.

573

or, what is the same thing, to
1k2 ( — _) (y —) + Ack3 (a-1)(+ a/)+ Akl(/-)( +?)+ l (y-)(a + );
with the like expressions as to the values of y and z. Introducing for homogeneity
a quantity o, viz. writing, in place of %, 1, I, we have the parameters
(?, V, ', co) connected by the homogeneous equations
+ o3 = 0,
(-k2 1 + k22 + k) W2 + k ( ) - + al ( )+ k k2 (- ) = O,
and the ratios of the coordinates are
+ k31 (l- ) (G + -^) + k,1C2 (- ) (a + t)
C 2 (' - ) (a - 2) + &fi.3 (a- ( )
+ 13k3 (- 1) (w77 + + klj (7-) (k + 7
Suppose, for shortness, these are x: y zP: Q: R. Observe that the form of
the equations is tq+w = O, 1=0, and x: y z=P: Q: R, where fl and P, Q, R
are each of them a quadric function of the form (W2, W^, n)?J, w)l, q7 7,, b7), the terms
in 2,?2, '2 being wanting.
Treating (:, 7,, co) as the coordinates of a point in space, the equation + o^3=0
is a cubic surface having a binode at each of the points ( =0, o =0), (X =0, -0),
(~=0, = 0), and the second equation is that of a quadric surface passing through
these th int n three points; hence the two equations together represent a sextic in space, or
say a skew sextic, having a node at each of these three points. The equations
x: y: z =P: Q: R establish a (1, 1) correspondence between the locus of 0 and
this skew sextic. To find the degree of the locus we intersect it by the arbitrary line
ax + by + cz = 0; viz. we intersect the skew sextic by the quadric surface aP + bQ + cR = 0.
This is a surface passing through the three nodes of the skew sextic, and it therefore besides intersects the skew sextic in 12-2. 3, =6 points. Hence the locus is
(as it should be) a sextic.