448 
MATHEMATICS: T. COHEN 
2n{bh — bo^)xi^Xo + 2ci{bh — &o^)^i% + lower powers of Xi; 
(s^y ^ bc^o' - ^cbo^o'^i + 4(- bco + boCi)^o'^, + ; 
(t^y ^ - bci'i^o' + 2(- ben + bcoci + 2boCi^)^,% + 6bcin^o% + 
In the first place, the Cayleyan is known to be on the stationary lines 
of the quartic, which are the common lines of (s^Y and (t^y. There- 
fore the Cayleyan must be made up of terms containing either {s^Y 
or {t^y at least once. 
Now let us see what are the common lines of the Cayleyan and {s^y. 
To make Xq a line of the latter requires that be = 0. If b = 0, then Xo 
is a stationary line of the quartic. If c = 0, then not only is a:o a Hne 
of {s^y, but its contact with it is (0, 0, 1), the point of the Steinerian, 
which has also become a point of the quartic. Therefore quartic, Stein- 
erian, and (s^y all meet in a point. The Steinerian, a curve of order 
12, meets the quartic in 48 points; the 48 corresponding Hnes together 
with the 24 stationary lines make up the 72 common lines of the Cay- 
leyan and (s^y. The condition that the polar point of (s^y lie on (axy 
is the vanishing of (s^Yisa) {s'^y{s'a){s''^y{s''a){s'''^y{s'''a)] this, when 
multiplied by (t^Y, is of proper degree for a term of the Cayleyan. It 
is, then, the only term of the Cayleyan not containing (s^y. 
It is now in order to ask for the common Hnes of the Cayleyan and 
(t^y. For Xo to be a line of the latter requires that bei^ = 0. Again 
setting aside the stationary Hnes, we have Ci = 0. Then it is seen that 
Xo has as its contact with {t^Y the point (0, 1, 0); furthermore, it is tan- 
gent to the Hessian at the same point. Therefore there are a certain 
number of lines of the Cayleyan which are also lines of both the Hessian 
and {t^Y, these two curves having contact on these lines. For the terms 
of the Cayleyan not containing (t^Y it is sufficient to use 
X(sj)^.(/0'(/a)(/'?)3(/'a)(/"^)3(/"«)(ij)H/a) 
+ {sms'i)Ks''m'Pis'''iXs'''amy{tc^y + ^{s"'&Ks"'cd{miay] 
where X, a, r, (p, At, v are undetermined coefficients. By requiring the 
highest power of to vanish when Ci = 0 certain relations on these 
coefficients are obtained, not enough to solve, however. To the terms 
given above it is necessary to add terms containing {t^Y; 
p{t%y . {siy(sa)(s'iy{s'a){s"iy(s"a)(s"'iy{,"'a) 
+ e{siy{s'iyis"iyitiy . is"'ay 
will be found sufficient. 
