246 ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. [406 
this may be considered as the intersection of the quadric surface AC — B-— 0 and the 
cubic surface AD— J5O=0; and the cuspidal curve is consequently a sextic. 
The surface has also a nodal curve made up of two conics; to prove this I write for 
shortness Jc = h — '/ab, k l = h + 's/ab‘, the values of A, B, G, D then are 
A. == Sckk-i , 
B = — kk x vj — 2 chz, 
G — — ax 2 — by 2 — cz l + 2h {xy — zw), 
D = 3z (2xy — zw); 
and it is in the first place to be shown that the surface contains the conic 
x : y : z : w — 6 V& : 6 Va : 1 : — k6- + ^, 
where 6 is a variable parameter. Substituting these values, we have 
A = — 3 ckk 1} 
B = Dk^d- — c (3h + Vab), 
G = 2kk 1 d* — | (3A — Va6), 
and hence 
AD - BG = - 2k (kk x № + 
AG -R =- ¿ 2 (kk^ + '-^J, 
BD-R = - (khP+^J, 
values which satisfy identically the equation of the surface written under the form 
(AD — BCf -4 (AG- 5 2 ) (BD - C 2 ) = 0. 
Moreover, proceeding to form the derived equation, and to substitute therein the fore 
going values of (x, y, z, w), we have 
dA : dB : dC : dD = 0 : D : 2k : 3, 
and then the derived equation is 
(AD- BG) ( 3A - 2k B- DC) 
-2 (AG-R )( 3B-4>kG +DD) 
- 2(BD-C* )(2kA-2DB ) = 0,