A MEMOIR, ON CURVES OF THE THIRD ORDER. 
401 
146] 
The lineo-polar envelope with respect to the cubic, of any tangent of the Pippian, 
is a pair of lines. 
And conversely, 
The Pippian is the envelope of a line such that the lineo-polar envelope of the 
line with respect to the cubic is a pair of lines. 
It is I think worth while to give an independent proof. It has been shown that 
the equation of the lineo-polar envelope with respect to the cubic, of the line 
%x + rjy + ? = 0 (where ? y, £ are arbitrary quantities), is 
2ft??, -IW- m, -IV-UH iF + fyC, W + Wt, + y> *) 2 = 0; 
and representing this equation by 
£• (a, b, c, f g, h#x, y, z) 2 = 0, 
we find 
be - f 2 = f (- ? + 8I s ? + 81 s ? + 12 
ca — g 2 =r\ (8I s ? — rf + 81 s ? + \2l 2 %y?), 
ab -h 2 = £ (81 s ? + 81 s ? - ? + 12l 2 %y?, 
gh - af= £ (21 s (? + v s + ?) + 4Z (1 + 21 s ) ££) + (1 + 81 s ) ??, 
hf-bg = v (21 s (? + v s + ?) +U(1 + 21 s ) + (1 + 81 s ) ??, 
fg-ch=£ (21 s (? + v 3 + ?) + 4£ (1 + 21 s ) %y?) + (1 + 81 s ) ??; 
and after all reductions, 
abc — a/ 2 — bg 2 — ch 2 + 2fgh 
= [ -1 (P + V 3 + ?) + ( - 1 + 4Z 3 ) M 2 = (PU) 2 , 
or the condition in order that the conic may break up into a pair of lines is PU= 0. 
25. The following formulae are given in connexion with the foregoing investigation, 
but I have not particularly considered their geometrical signification. The lineo-polar 
envelope of an arbitrary line %x + yy + ? = 0, with respect to the cubic 
x s + y s + z s 4- Qlxyz = 0, 
has been represented by 
(a, b, c, /, g, h^x, y, z) 2 = 0; 
and if in like manner we represent the lineo-polar envelope of the same line, with 
respect to a syzygetic cubic 
a? + y 3 + z s + QVxyz = 0, 
b y 
(a', b', c', f, g\ li\x, y, z) 2 = 0,