461] 
ON THE GEOMETRICAL INTERPRETATION &C. 
333 
is that of the tangent plane to the torse along the line ax 2 4- 2 bxy + cy 2 = 0,. 
bx 2 + 2cxy + dy 2 = 0 : this line meets the cuspidal curve in the point whose coordinates are 
a : b : c : d = y s : — xy 2 : x 2 y : —y 3 . The equation 
H= 0 
is that of a quadric cone having the last mentioned point for its vertex, and passing 
through the cuspidal curve: and the equation 
0 = 0 
is that of the cubic surface which is the first polar of the same point in regard to 
the torse. 
The equation O 2 — V U 2 = — 4<H 3 , writing therein U=0, gives 0 2 = —4 H 3 , a result 
which implies that U — 0, H = 0 is a certain curve repeated twice, and that £6=0, 
0 = 0 is the same curve repeated three times. The curve in question is at once 
seen to be the line of contact B X U = 0, B y U = 0; it thus appears that the tangent 
plane U = 0 meets the cubic surface 0 = 0 in this line taken three times. This can 
only be the case if the equation 0 = 0 be expressible in the form MU + (8 X U) 3 — 0, or, 
what is the same thing, 
MU+(aS x U+/38 y Uy = 0, 
a. and ¡3 constants, M a quadric function of (a, 6, c, d); that is, O must be equal to 
a function of the form 
MU + (a8 x U+/38 y U) 2 . 
Seeking for this expression of O, and writing the symbols out at length, I find that 
the required identical equation is 
( a 2 d — 3abc +2b 3 ) a? 
— (— Sabd + 6ac 2 — Sb 2 c) x 2 v 
2bxy + cy 2 ) + ¡3 (Jbx 2 + 2cxy + dy 2 )} 3 = 
^ — (— ad 2 — 3bed + 2c 3 ) y 3 
(a, b, c, d\x, y) 3 .( 2a 2 , Qab , 6b 2 
ad + Sbc )t0», y) 3 (a, Q) 3 , 
6ab , 12ac + 6b 2 , 3ad + 156c, 
66 2 , 3 ad + 156c, 12bd + 6c 2 , 
— ad + Sbc, 6c 2 , 6cd , 
6c 2 
6cd 
2d 2 
6cd 
(where the f indicates that the binomial coefficients are not to be inserted, viz. the 
function on the right hand is {2a 2 ^ 3 + (5abx 2 y + Qb 2 xy 3 + (- ad + 36c) y 3 } a 3 + &c.). As a 
verification remark that for x = a, y = /3, the equation becomes simply 2U 3 = U.2U 2 .