A MEMOIR ON CURVES OE THE THIRD ORDER. 
403 
146] 
as the condition which expresses that a line gx + yy + = 0 cuts harmonically its 
lineo-polar envelopes with respect to the cubic and with respect to a syzygetic cubic. 
27. To find the locus of a point such that its second or line polar with respect 
to the cubic may be a tangent of the Pippian. Let the coordinates of the point be 
(x, y, z) ; then if %x + yy + %z = 0 be the equation of the polar, we have 
£ : y ■ £= % 2 + Zlyz : y 1 + 2Izx : z 2 + 2Ixy, 
and the line in question being a tangent to the Pippian, 
-7(| 3 + t? 3 + £ 3 ) + (- 1 +4Z»)£ ? £=0. 
But the preceding values give 
£.i _|_ ^3 £3 _ ^ + 67 (¿c 3 + y :i + 2 3 ) + 36l 2 x 2 y 2 z 2 + ( — 2 + 87 :i ) (y 3 ^ 3 + z s x? + x 2 y 3 ) 
%y£ = 41 2 (a? + y 3 + z 3 ) xyz + (1 + SI 3 ) x 2 y 2 z 2 + 27 (y 3 z 3 + z 3 x s + x A y 3 ); 
and we have therefore 
7 (x 3 + y 3 + z 3 ) 2 + (107 2 — 167 5 ) (oc 3 + y 3 + z 3 ) xyz + (1 + 401 3 — 321 6 ) x 2 y 2 z 2 = 0 ; 
or introducing 77, HU in place of x 3 + y 3 4- z 3 , xyz, the equation becomes 
-S. U 2 + (HU) 2 = 0, 
which is the equation of the locus in question. 
28. The locus of a point such that its second or line polar with respect to the 
cubic is a tangent of the Quippian, is found in like manner by substituting the last- 
mentioned values of £, y, £ in the equation 
Q 77 = (1 — 107 3 ) (f 3 + y 3 + £ 3 ) — 67 2 (5 + 41 3 ) gy%. 
We find as the equation of the locus, 
(1 — 101 3 ) (x 3 + y 3 + z 3 ) 2 + 67 (1 — 307 3 - 16/®) (x 3 + y 3 + z 3 ) xyz + 67 2 (1 — 1047 3 — 327®) x 2 y 3 z 2 
— 2(1 + 8l 3 ) 2 (y 3 z 3 + z 3 x? + x 3 y 3 ) = 0, 
where the function on the left-hand side is the octicovariant ©„ U of my Third 
Memoir, the covariant having been in fact defined so as to satisfy the condition in 
question. And I have given in the memoir the following expression for ©„77, viz. 
@„77= (1 - 167 3 - 67 6 ) TJ 2 
+ (67 ) TJ.HU 
+ (67 2 )(HU) 2 
— 2(1+ 81 3 ) 2 (y 3 z 3 + z 3 x 3 + x 3 y 3 ). 
51—2