ME. A. CAYLEY’S MEMOIE ON CUEVES OE THE THIED OEDEE. 
427 
Hence completing the system and throwing out the constant factor, the equation of the 
pair of lines is 
(3/a^+2(l+2Z^)/3y, 3Zi3^+2(l+2^^)ya, 3^y^+2(l+2Z>i3, 
{l-U^y~%V^y, {1-U^y-U^a^Jx, y, zf=:^. 
But the equation of the line EF is l^x-\-ny-\-tz=^-) and the equation of its lineo-polar 
envelope is 
n, I =0; 
X, Iz, ly 
71, Iz, y , lx 
t, ly, lx, z 
or expanding, 
{yz—l^o^, zx—iy, xy—Pz^, l^yz—laf, Vzx—y, Pxy—lz^yji, ij, ^)^= 0 ; 
or arranging in powers of x, y, z. 
-V7i^-2lt^, -ir-mn. y, zf=^. 
And if in this equation we replace P, &c. by their values in terms of a, (3, y, 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 
^x+}jy +^2=0, , 
ax+(3y+^z=0, 
(where tj, ^ and a, (3, y have the values before attributed to them) are conjugate polars 
with respect to the curve of the third class. 
in which equation >}, I denote current line coordinates. The curve in question is of 
the form APU+BQU=0. We have, in fact, identically. 
3T . PU - 4S . QU = (1 + SlJ { 
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, 
H(6aPU+i3QU)=(-2T, 48 S^ 18TS, P+16 S^I«, (3)*PU 
+ ( 8S , T ,-8S^ -TS %oi,(3fQV. 
The coejfficient of QU is 
(8S«+T/3)(a^-S|3^); 
and therefore, putting a=|T, /3= — 4S, we find 
H(3T . PU- 4S . QU)= -KP- 64S*)TU. 
3k2 
