338 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
— 0^ — 0 2 2 + (X + + v) (#! — 0 2 ) + \yv — Q-'j , 
— (@1 — @2) (^1 + 0 2 + \+fl+V — > 
= (0i — 0 2 ) (A + /x + v + 300 ; 
©1 — ©2 = 0i 3 — 0 2 3 + + y + v) (0i 2 — 02 2 ) + (^v + v\ + Ayli) (0j — 0 2 ), 
= (0 X — 0 2 ) [0! 3 + 0J02 + 0./ 4" (A + fl + v) (0! + 0 2 ) + ¡XV + v\ + AyU/}, 
= (01 — 0 2 ) • {601 2 4- 2 (A + /x + v) 0i] : 
and thence 
A ' 4- R' 4, 
T=i- 9 ?1 ^ I + X + ^ + ,). 
Moreover 
y/ V' if) ù \ f I Py I ^ 1 
- Wl ^ {(^ + X) ( ^ f + A) + (01 + fl) (02 + /x) + (0! + *) (02 + v)\ ’ 
= {e ' ~ 0,) {(5U\)' + (0,+V)* + (5T+O s } ’ 
and hence 
£ = (30i + A + /a + v 
\X fXJJ vz 
0i -f- A 0i -j- fjb 0i v 
2©j ( \x fxy vz 
+ 77\— No + 
30J 2 ((0i + A) 2 (0i + yti) 2 (0i + v)-j 
in which we have only now to substitute (A, /x, v) = (or s , /3 -3 , y -3 ) and 0j = 
have 
n . M . M n M 
0i + ^- ^, 0i + P ^^ ^ > 
where i¥ = (/3y — a 2 ) = (/3y + ya + a/3), and then observing that 
** +x+ * + -- M (?+b + ?) - jV 
-1 
a^y‘ 
the equation 8=0 becomes 
2 (/3y + ya + a/3) ^ ^ + 3 (ouc + j3y + yz) = 0, 
or, what is the same thing, 
(2/3 7 + a») l + (2 7 a + /3°) | + (2«/3 + 7 =) i = 0, 
which agrees with a former result. 
We