349] 
ON A CASE OF THE INVOLUTION OF CUBTC CURVES. 
319 
that is 
and consequently 
x + y + z = 2a? (1 + J, 
x + y + z = 2x(l + ^ = 2y (l + ^j = 2z(l + ~y 
or, what is the same thing, 
, . 2 1 
x + y + z : X : y : z = - : - 
and we have thence 
1 1 1 
+ -7T~. f 
0 X + 0 /x + 0 v + 0 ’ 
2 
# + A 0 + /x 0 + v 0 
an equation which may also be written in the form 
= 0, 
or in the form 
and we then have 
X> | /x , jj 1 A 
o + \ + o+fx + d + v ~ = ’ 
6 j — 0 {/xv 4" vX 4" — 2X/xv = 0 5 
2 1 /2\ 3 
e (e + \)(o + /i)(o+v)* w ’ 
{6 4 X) (0 + /x){6 + v) ' 
21. We see that 0 is determined by a cubic equation, and that the ratios x : y : z 
and the parameter k are rational functions of 0. There are thus three nodes or critic 
centres, and the like number of nodal curves and of critic values of k. 
The form secondly obtained for the equation in 0 shows that we may write 
x 
y : z : x + y+z : Xx + /xy + vz = 
1 1 1 
0 +X 0 + fx ‘ 0 + v 
Article Nos. 22 to 32, relating to a Twofold and a One-with-Twofold Centre. 
22. If k has a twofold and a one-with-twofold value, then 0 will have also a 
twofold and a one-with-twofold value; and conversely. The equation in 0 will have 
a twofold and a one-with-twofold root if 
{/xv + vX + X/x f - 27 X>V = 0 ; 
or, what is the same thing, if 
/xv + vX + X/x — 3 = 0, 
or if 
’<>*')" + {vX)* + (X/xf = 0,