654] 
ON CERTAIN OCTIC SURFACES. 
83 
— 2 hf {calif — b-g 2 — 2 bgaf 
2bgch) w 4 z 2 x 2 
-2 c/( 
+ 2aA( 
+ 2ca ( 
) xty 2 w 2 
) z 4 y 2 w 2 
) y 4 x 2 2 2 
— 2/gr (a&/$r — c 2 /?, 2 — 2chaf 
2chbg) w 4 x 2 y 2 
- %ag( 
+ 2 bf( 
+ 2 ab ( 
) y 4 z 2 w 2 
) X 4 Z 2 W 2 
) z i oc 2 y 2 
+ 2 £lx 2 y 2 z 2 w 2 = 0, 
where the values of the coefficients indicated by ( „ ) are at once obtained by the 
proper interchanges of the letters, and where il is an arbitrary coefficient, is a surface 
having the four nodal conics 
x = 0, . cy 2 — bz 2 + fw 2 = 0, 
y = 0, — cx 2 . + a^ 2 + gw 2 = 0, 
z = 0, bx 2 — ay 2 . hw 2 = 0, 
w — 0, — /îc 2 — gy 2 — liz 2 . = 0. 
In fact, writing the equation under the form 
w 2 © + (fx 2 + gy 2 + hz 2 ) 2 x (6W + c 2 a 2 y 4 + a 2 b 2 z 4 — 2a 2 bcy 2 z 2 — 2b 2 caz 2 x 2 — 2c 2 abx 2 y 2 ) = 0, 
we put in evidence the nodal conic w = 0, fx 2 + gy 2 + hz 2 = 0: and similarly for the other 
nodal conics. 
It is to be observed, that the complete section by the plane w = 0 is the conic 
fx 2 + gy 2 + hz 2 = 0, twice repeated, and the quartic 
b 2 c 2 af + c 2 a 2 y 4 + aïlfz 4 — 2a?bcy 2 z 2 — 2 ab 2 cz 2 x 2 — 2 abc 2 x 2 y 2 — 0 : 
the latter being the system of four lines 
V(a) V(6) V(c) ’ V(a) V(&) V(c) 
The plane in question, w = 0, meets the other nodal conics in the six points 
{x = 0, by 2 — cz 2 = 0), (y = 0, cz 2 — ax 2 — 0), (z = 0, ax 2 — by 2 = 0), 
which six points are the angles of the quadrilateral formed by the above-mentioned 
four lines. 
The four conics are such, that every line meeting three of these conics meets 
also the fourth conic. The lines in question form a double system: each of these 
11—2