WHICH SATISFY GIVEN CONDITIONS. 
129 
and hence 
AD— BC = — 2£| 
[kk l P+ 2c ' / k ^\] 
2^ 
1 
II 
W 
1 
O 
<1 
BD-C 2 =- | 
values which satisfy identically the equation of the surface written under the form 
(AD - BC) 2 - 4( AC - B 2 )(BD - C 2 ) = 0. 
Moreover, proceeding to form the derived equation, and to substitute therein the fore- 
going values of (#, y, z, w), we have 
dA:BB:BC:bD = 0:£ 2 :2£:3, 
and then the derived equation is 
(AD— BC)( 3A-2&B-£ 2 C) 
— 2(AC — B 2 )( 3B— y:C+^D) 
— 2(BD — C 2 )(2kA-2k t B )=0, 
that is, 
-k( 3A— 2/rB -^k 2 C) 
+ k 2 ( 3B-4/cC+£ 2 D) 
+ (2M-2PB), 
or finally 
- k(A - 3B k + 3£ 2 C - k 3 D) =0, 
which is satisfied by the foregoing values of A, B, C, D ; hence the conic is a nodal curve 
on the sextic ; and by merely changing the sign of one of the radicals \/ «, (and 
therefore interchanging k , k x ) we obtain another conic which is also a nodal curve on 
the surface, that is, we have as nodal curves the two conics 
x:y:z:w=Q\/b: Q\/ a : 1 : — W and 
x:y:z:w=Q\/b: — a \ 1 : -kfi 1 - {-j-- 
It is to be remarked that each of the nodal conics meets the cuspidal curve in two 
points, viz. writing for shortness 0= \\/ — , 0,=^- \/~' , for the intersec- 
tions of the first conic we have 
x:y:z:w=®^/a: Q^/b: ligand =-0 y/a: -0 s/b: 1 
and for the intersections with the second conic 
x:y:z: w=0 lV / a : —Q lS /b : 1 : j and = — 0 lV / a : Q x \/b : 1 : y 
The condition of passing through any arbitrary point establishes a linear relation be- 
