349] 
ON A CASE OE THE INVOLUTION OF CUBIC CURVES. 
321 
viz. it is 
cr 4 — 6qa 2 + 16</ 2 — 128?’cr = 0, 
which is 
(7 <r 2 + 4 q) 2 — 16a- (З0- 3 + 4 qa + 8?’) = 0, 
or, putting for a moment 1 = — |, and therefore 5(1 — 2) = — 16, 
it is 
(7a- 2 + 4q) 2 + (l — 2) 5а (З0- 3 + 4^0- + 8?') = 0. 
Now writing 
x' = a + 2x = Зх + у 4- z, 
y' = cr + 2y = x + Sy + z, 
z' = a + 2z = x + у 4- 3z, 
we find 
у z + z!x' + x'y = 7 a 2 + 4 q, 
x' + y' + z = oa, 
x'y'z' = З0- 3 + 4^0- + 8r, 
so that the equation is 
(y'z' + zx' + x'y'f + (l —2) x'y'z' (x' + y' + z) = 0, 
that is 
y' 2 z~ + z' 2 x' 2 + x' 2 y'- + lx'y'z (x' + y' + z') = 0 ; 
or, putting for l its value, the equation is 
о (y' 2 z' 2 + z'-x 2 + x'-y" 2 ) — 5xy'z (x + у + z') = 0 ; 
or, as this may also be written, 
(5, 5, 0, -3, -3, -3)(I, ) r i) 3 =0; 
a form which shows that the curve has three nodes at the angles of the triangle 
x' = 0, y' = 0, z' = 0. 
25. It is easy to see that the curve is touched by the lines x = 0, y= 0, z = 0 
at their intersections with the lines y — z = 0, z—x = 0, x — y = 0 respectively, or (what 
is the same thing) in the points (0, 1, 1), (1, 0, 1), (1, 1, 0) respectively. It may be 
added that the line y—z= 0 meets the curve in the node (—4, 1, 1), being of course 
a point of twofold intersection, in the point (0, 1, 1) on the line x = 0, and besides 
in the point (16, 1, 1): and the like for the lines z — x = 0 and x—y= 0. 
26. It may be noticed that although any line passing through one of the nodes 
is in a sense a tangent to the envelope, yet that it is not a proper tangent and 
does not give rise to a twofold centre. It is in fact shown (post, Nos. 73 and 74) 
that the critic centres for a line \x + yy + vz = 0 passing through the point (—4, 1, 1) 
are three points lying, one of them on the line у + z = 0, and the other two on the 
conic x (x + у + z) — fyz = 0. 
C. V. 
41