349] 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
33T 
Article Nos. 51 and 52. Special Case of the Three-Centre Conic. 
51. Write for a moment 
X' 
\x yy vz 
0\ -f- A 6 X + yu. 6] -j- v 
Y> = Xx + M + vz 
6 2 + \ 0o+ fl 6 2 + V ’ 
Z' = — + w + vZ 
so that 
0 3 4- \ 0 3 + fl 0 3 + v ’ 
(hV (h) (h) 
y '-'1 y/ TT" tt/ r/ v - f 3 nr/ 
X ~ Us ’ ill ’ TJi ’ 
the equation of the three-centre conic expressed in terms of (X', Y', Z') is 
0, ® a 
say 
r*+^z*=o, 
LiUi 601/2 63C/3 
A'X' S + B'Y'- + C'Z' 2 = 0. 
When 0 1 = 0 2> we have C’= 00, A/ = —.B', X'=Y'\ by writing the equation in the 
form 
(A' + £') X' 2 + B' (Y' 2 - X' 2 ) + C'Z' 2 = 0, 
and observing that in the limit Y' 2 — X' 2 = 2X'(Y'— X'), we see that the equation 
will thus assume the form 
^-ejrs + f- ~z* = 0, 
tJl L7 2 1/3 
where 
_A' + B' r-r 
^ ^1-^2 A +2£ — 0 2 ’ 
is a finite function; A'=0 is the line joining the twofold centre and the one-with- 
twofold centre, S = 0 is the other tangent at the one-with-twofold centre, Z' = 0 the 
tangent at the twofold centre or cusp; the form X'S + 00 Z' 2 = 0 shows that the three- 
centre conic reduces itself to a pair of points, viz. the twofold centre or cusp, and 
the point where the tangent at the cusp is met by the other tangent (that is the 
tangent not passing through the cusp) at the one-with-twofold centre. 
52. To verify the value of S I proceed as follows: 
A'+ B' 
®i 
a, 
Ox -0* 0,- 0 2 {0, (02 - 0 3 ) 02 (0 3 - 0 1 )[’ 
= (6,-e,)(e,-e l ) • e^I, { 01 f~ w) ~~ (@l “ 0s) } ’ 
C. V. 
43