biKMMHMMtH 
146] A MEMOIR ON CURVES OF THE THIRD ORDER. 395 
or expanding, 
(yz — Px 1 , zx — Py\ xy — Pz\ Pyz — lx 1 , Pzx — ly 2 , Pxy — lz%l 71, £) 2 = 0 ; 
or arranging in powers of x, y, z, 
{-p?- 2% -IY-21&, -p?-2l% v , W+P& W + ^№,y,zy = 0: 
and if in this equation we replace f 2 , &c. by their values in terms of a, /3, 7, as 
given by the equations (D), we obtain the equation given as that of the pair of lines 
OE, OF. 
19. It remains to prove the theorem with respect to the connexion of the lines 
EF, IJ. 
The equations (A) show that the two lines 
gx + rjy +&= 0, 
a x + fiy + £z = 0, 
(where 77, £ and a, /3, 7 have the values before attributed to them) are conjugate 
polars with respect to the curve of the third class, 
l (£ 3 + y s + £ 3 ) - 0, 
in which equation 77, £ denote current line coordinates. The curve in question is of 
the form APJJ + BQU = 0. We have, in fact, identically, 
ST.PU-4<S.QU=(l + 8 IJ \l (f + 77 3 + ?) - 3^77^}. 
It is clear that the curve in question must have the curve PU = 0 for its Hessian; 
and in fact, in the formula of my Third Memoir, [144] 
H(6aPU + {3QU) = (-2T, 48S 2 , 18TS, T 2 +l6S s %<z, ¡3) S PU 
+ ( 88, T,-8S 2 , -TS \ct, /3) 3 QU, 
the coefficient of QU is 
(8Sa+T/3) (a 2 — S/3~); 
and therefore, putting a = \T, /3 = — 4$, we find 
H(3T.PU-4S. QU) = ~l (T 2 — 64S 3 ) 2 PU. 
Article No. 20.—Theorem relating to the curve of the third class, mentioned in the 
preceding Article. 
20. The consideration of the curve ST. PU — 4$. QU= 0, gives rise to another 
geometrical theorem. Suppose that the line (f, 77, £), that is, the line whose equation 
is Ijx -f- 7777 + £z — 0, is with respect to this curve of the third class one of the four 
polars of a point {X, Y, Z) of the Hessian, and that it is required to find the envelope 
of the line %x + yy + %z = 0. 
50—2