336 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
and hence, substituting for 6, — 0, its value = l 3 
coeff. Y 2 = 
2h 
©2 
2 L 
on /¿v + v\ + A/a 2 (fiv + v\ + A/a) 6A/iid 
~ ie -- + e,— + ft 
= 0-^ -Q {— S0,0. 2 2 + (/jlv + v\ + A/a) (0, + 2 0 2 ) + Q\/j,v] ; 
but we have 6, + 6, + 0 S = 0, whence 8, + 26» = 6. 2 — 6 3 = l 2 ; \iv + v\ +A/a = — (0+6» + 0,0 3 4- 0»0 3 ), 
2A/av = 6,6£ 3 , and therefore —36 1 6 2 2 + 6\ibLv = — 36 1 9'? + 30 1 d 2 d 3 = — 30 1 O 2 (9 2 — d 3 ) =— 31^02', 
and hence 
coeff. Y‘ = {- 3ftft - ftft - ft (ft + ft)), 
= ©ftft '4 4ftft + (ft + ft) 3 ), 
that is 
0,0,0» ^ ’ 
coeff. Y 2 = 
and, by merely interchanging 6» and 6 S , 
2 1,1* 
coeff. Z 2 = 
0 3 0,6 3 
50. Hence the equation of the tangents is 
or, what is the same thing, 
2I1I3 3 y„ 21,1 2 _ 8 _ _ 
0A6s 0-A0S ’ 
or putting 
l 2 s 6 2 0* T+ ls 3 6s®s Z2 
A= T№®,’ B = T№®Y (7 = к© 3 , 
the equation of the tangents at the node corresponding to 6, is BY 2 + CZ 2 = 0. And 
hence the equations of the tangents at the three nodes respectively are 
BY 2 + CZ 2 = 0, 
ax 2 . +az 2 = 0, 
AX 2 + BY 2 . = 0; 
that is, the nodes or critic centres are conjugate poles in regard to a conic 
AX 2 + BY 2 + CZ 2 = 0, 
which is the three-centre conic; and the tangents at each node are the tangents from 
such node to the conic in question.