PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 3 JULY 15, 1917 Number 7 
THE CAYLEYAN CURVE OF THE QUARTIC 
By Teresa Cohen 
JOHNS HOPKINS UNIVERSITY 
Communicated by E. H. Moore, May 5, 1917 
The quartic curve (axY = 0 determines a correspondence 
{axy{ay)ai = 0, i = 0, 1, 2, 
in which x is a point on the Hessian curve and y a point of the Steinerian. 
The locus of lines ^ joining corresponding points x, y is the Cayleyan, 
known to be of degree 18 in ^ and 12 in the coefficients of the quartic. 
The Cayleyan can be expressed in terms of the two contravariants of 
the quartic, (s^Y [= i\a^^\'] and {t^Y [ = i\^y^Hy a^\''\cxl3^\^], and of terms 
produced by operating with the polars of these on (axY. The working 
out of this depends on a special reference triangle which is always valid 
for the general quartic. Suppose 
(axY = axo^ + 4:aiXo^Xi + 4^2X0^X2 + 6hxoV + 12/xo^XiX2 + 6^X0^X2^ 
+ bxi^ + ^b2Xi^X2 + 6fxi'^X2^ + 4^1X1X2^ + cx2^. 
Since x and y as given above can never coincide for the general quartic, 
because 
(axY^i = 0, i = 0,1,2, 
is the condition that (axY have a double point, let x, the point of the 
Hessian, be the reference point (0, 1, 0) and let y, the point of the Stein- 
erian, be (0, 0, 1), so that Xo = 0 is a line of the Cayleyan. Then 
ai^a2ai = 0, i = 0,1,2, 
or m = &2 = / = 0. 
This reference scheme is maintained throughout, though more highly 
specialized as occasion demands. Under it the Hessian becomes 
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