MATHEMATICS: T. COHEN 
449 
There are certain other Hnes known to be lines of the Cayleyan. 
There are 21 points whose polar cubics as to the quartic break up into 
a conic and a line, which is a fourfold Kne of the Cayleyan. Let one of 
the 21 points be (1, 0, 0); then for the polar cubic to contain Xo as a 
factor requires that = Cq = n ^. Now, using this condition, re- 
quire that the highest power of in the expression for the Cayleyan 
vanish. The result will be certain conditions on the undetermined 
coefficients, but still not enough to solve. 
Instead of putting the Cayleyan again on these lines it is easier to 
proceed at once to the general reference scheme which has been the 
basis of all the work, when only m = b2 = f = 0, and require that the 
highest coefficient of vanish. This at once completes the work and 
furnishes proof of its correctness. The Cayleyan is obtained as 
+ (s^s'^y [i5{s"iy{s"a){s"'ms"'cc)imtiy 
- 57is'Wc^)is"'iy{s"'c^yimta) ] 
+ (smsms''iy[Tis'''ms'''c^y{mtcd' - iois"Wa)imtay] 
+ M {siyisms'mo' ■ (s^y 
- 1 isi)Wis"iy{s"'iy . imtay. 
Since, however, this expression has been obtained by causing a coeffi- 
cient to vanish, there is the possibility that it gives merely a syzygy 
and vanishes identically. Therefore it was tested on the special quartic 
+ -\- X2^, where the Cayleyan is known to be ^o^^i^^2^, and found 
not to vanish. 
The stationary lines of the quartic are known to be lines of the Steiner- 
ian. From the above form of the Cayleyan it can be shown that the 
contacts of these lines are the same for the Cayleyan as for the Steiner- 
ian, so that the two curves touch. We had also certain common lines 
of the Cayleyan and Hessian, which were likewise lines of {t^y. These 
lines can be shown to have the same contact as to the three curves. 
The Cayleyan and {t^Y have 108 common lines, 24 of which are absorbed 
by the flexes, leaving 84 to be accounted for here. Because of the con- 
tact of the curves each line counts for two common lines; therefore the 
Cayleyan, Hessian, and {t^^ touch in 42 points. 
Certain interesting facts come up under the reference scheme here 
employed. The polar conic of (0, 1,0) is 
hx^'^ + 2h^^i hxi^ = 0, 
