394 
A MEMOIR ON CURVES OE THE THIRD ORDER. 
[146 
which is the equation of the line EF, that is, the line through two consecutive positions 
of T is the line EF\ or what is the same thing, the line EF touches the Pippian 
in the point T which is the harmonic of Q with respect to the points E, F. 
18. The lineo-polar envelope of the line EF, with respect to the cubic, is the 
pair of lines OE, OF. 
The equation of the pair of lines OE, OF, considered as the tangents to the 
Hessian at the points E, F, is 
|(3№ - 1 + 2l 3 YZ)x + (3l 2 Y 2 -1 + 2l 3 ZX)y + (3PZ 2 - 1 + 2PX Y)z) ] 
x {(3 PX' 2 - 1 + 2 PY'Z') x + (3 P Y' 2 - 1 + 2 PZ'X') y + (3 PZ' 2 - T+WX'Y') zj j 
Here on the left-hand side the coefficient of x 2 is 
9PX 2 X' 2 - 31 2 (1 + 21 3 ) {X 2 Y'Z' + X'-YZ) + (1 + 2I s ) 2 YY'ZZ', 
which is equal to 
9Pa 2 - 31 2 (1 + 21 3 ) (l 2 fiy + j a 2 ) + (1 + 21 3 ) 2 ¡3y, 
that is 
j (- 1 + ft) {3fa 2 +2(1+ 2l 3 ) f3y] ; 
and the coefficient of yz is 
91 4 ( Y 2 Z' 2 + Y' 2 Z 2 ) -SP{\ + 2/ 3 ) ( YY' (X Y' + X' Y) + ZZ' (XZ' + X'Z)) 
+ (1 + 2 I s ) 2 XX' ( YZ' + Y'Z), 
which is equal to 
№ (I - 2/3 7 ) - 3P (1 + 2i») ( - I /37) + (1 + 2py a ( - j a) , 
that is 
+ {(1 — 4Z S ) a 2 — 6l 2 /3y}. 
Hence completing the system and throwing out the constant factor, the equation of 
the pair of lines is 
(3la* + 2(1 + 21 3 ) /3 7 , 3Z/3 2 + 2 (1 + 2I s ) ya, Sly 2 + 2 (1 + 21 3 ) 
(1 - U 3 ) a 2 - GPfty, (1 - 4fl s ) /3 2 - 6l 2 ya, (1 - U 3 ) y 2 - GPaffix, y, z) 2 = 0. 
But the equation of the line EF is %x + rjy + £z = 0, and the equation of its lineo-polar 
envelope is 
V , 
K 
x , 
Iz. 
l v 
V, 
Iz, 
V> 
lx 
= 0; 
£ ly, V, Z