349] 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
325 
Article Nos. 33 to 38, relating to the Tangents at a Node or Critic Centre. 
33. I proceed to investigate the equation of the tangents at the node of the curve 
xyz + k (x + y + zf (Xx + fiy + vz) = 0 ; 
it will be recollected that if x, y, z are the coordinates of the node, then we have 
cc : y : * : cc + y + z : + W + g 1 —: I :1- 
Representing for a moment the equation of the curve by U = 0, then the second 
derived functions of U are 
k . 2 (Xx + fiy + vz) + 4k(x + y + z) A, 
k . 2 (Xx + /u,y + vz) + 4k (x + y + z) ¡i, 
k. 2 (Xx + ¡iy + vz) + 4k (x + y + z) v, 
x + k . 2 (Xx + gy + vz) + 2k (x + y + z) (/a + v), 
y -\-k . 2 (Xx + fiy + vz) + 2k (x + y + z) (v + A), 
z + k. 2 (Xx + ¡zy + vz) + 2k (x -1- y z) (A + /a), 
or calling these (a, h, c, f g, h) respectively, and substituting the values x = 
we find 
07 8&A 2 k /n 
a — 2k -|—Q-, — ~0 (Q + 4A), 
1 
0+1 
&c., 
with the like values for b, c; and 
j, 1 ^. 4k . . 2k ( 0 1 a a a \ 
f=e + \ Jr * k+ T (fi+v)= T\eTx. 2A + fl + 2 '‘ +2 ")’ 
where the term in ( ) is 
6 (6 + A) (6 + /a) (6 + v) 
0 + X' -10 2 
+ 0 + 2 g + 2v, 
— 2 (6 + /a) (6 + v) 
0 
+ 0 + 2/a + 2 v, 
26-2g-2v- 2f ^+0 + 2y, + 2v, 
= — 0 — 
’¿¡JbV 
NT’ 
that is