332 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
where the value in question ior X 73—--r is most readily found from the identical 
\“i "f a*/ 
equation 
X ¡x v _ d 3 — 0 (/xv + v\ + X/x) — \/xv 
0 + \^~0 + /X~^0-\-V © 
by differentiating and then writing 6 — 0 X . 
Second term is 
2/u.z/' 
= - w t(e l+ x)(e 1 + f), 
= - ~ 2 (<?, s + Xff, + 2/lv + , 
6\fXV 
— + (X + [x + v) 0 X + 2 (/xv + v\ + \fx) + q 
Whole is = ^- multiplied into 
~0 ©1 + 3 {S0 X 2 — ([XV + v\ + XyU.)] 
— 2 jsdi 2 (X + [x + v) 0 X + 2 (/xv + v\ + \fx) 4- 
5 
= -q {d/ — ([xv + z>X + \/x) 0 X — 2X[xv), 
which is = 0. 
46. We have next 
coeff. X Y = (0 X + 4X,., ., — 6 1 — 
2 ¡xv 
1 
_ v. 
' ?, + 4X _S(> + 2/w ' 
0 X / V^I + X 0\ + /•*•’ 0i + vj \ 0-i 4- X ’ 0. 2 4- [x 0., + v 
1 1 
(0 X 4 X) (d 2 + X) 
+ 
0, ) 1(01 + /1) (0, + vy (0! + v) (0, + X)j • 
First term is 
— V 1 4. qv ^ 
w 0 2 + X^ (0 1 + X)(0 2 + \)’ 
= v L_ 4. 3 V [ ^ ^ \ 
~ d,+x^0 1 -e,\0 1 +x 0 2 +\)’ 
since