346 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
In fact the tangents at the node or critic centre are given by the equation 
+ ..., -9-^, ...) (cc, y, = 0, 
the other two critic centres are given as the intersection of the line 
with the conic 
I py , 
9 +. A 9 + ¡x 
vz 
9 + v 
= 0, 
fJL — V V — A X — IX ~ 
+ + = 0; 
X y z 
the theorem will be true if the pair of tangents and the last-mentioned conic are cut 
harmonically by the last-mentioned line. Now in general the condition in order that 
the line %x + yy + = 0, may cut harmonically the conics (a, b, c, f g, K§x, y, zf and 
(a', b', c', /', g', h') (x, y, zf = 0 is 
(be' + b'c - 2ff, ..., gh! + g'h - af' - a'f ..y, £) 2 = 0, 
and if a' — b' = c' = 0, then the condition is 
(- 2Iff, ... gh' + g'h —af, ...$£ y, O 2 = 0. 
68. In the present case the equations of the two conics may be written 
(0 + 4X, ... - 9- 2 ^-, ...) (x, y, zf = 0, 
(0, ..., fi-v, ...$> 3 y, zf = 0, 
and we have 
gh’ + g% - af = - ($ + ?£) (X - g) - (e + ff) (v - X) - 0* - ») (6 + iX), 
— 0 (X — /x + /x — v + v — X) 
-1- -Q (— vX 1 + v\fi — \/xv — X 2 /x) 4- 4X (/x — v), 
(fi-v 
and the condition is 
2X 2 
-Q ), ... (ji-v 
69. Writing this in the form 
-4X , 
2 fjiv 
2X 2 
4x), ... 
9 + X 9 y, 
’ 0+ „) ~ 
v) (9 + 
2/jlv 
9 ) \9 + X 
4- 2 2 (/x — v) ( -Q — 2X 
X 2 
/XV 
(9 + /x) (9 + y) 
= 0,