Vol. 6, 1920 
MATHEMATICS: J. H. MCDONALD 
703 
can be drawn. Let abed be four tangents to a twisted cubic meeting 
their transversals / and /' in ABCD, A'B'C'D' and let AA', CC touch 
the cubic at points of parameters o, oo and BB' , DD' at points of parameters 
d,ip. It is found that the equations of A A' are x = y = 0, of CC z = 
= 0, of BB' X — 2ey -\- = y — 2dz + O'^w = 0, of DD' x — 2cpy 
ipH ■= y — 2ipz + <4p-w — 0. The equations of / are x — ay = dz — 
= 0, of y X — \y = Sz — kw == 0, where k,\ are connected with 
d,<p by the equations 6 -\- (p = k -\r X/2, 6(p = ^/zk/\. Putting m = 
{ ABCD } , n = [a 'B'C'D'] the cross-ratios are connected by an equation. 
The values of m and n are 
^ 3^ — - 2X e 3d — 2k 
e 3y — 2\ <p Sep — 2k : 
if d/(p = / it is found that mn = t\ m -\- n = t"^ -\- 1 — {t — 1)\ 
Eliminating t; — = 2m^n^(l — w^n^). This is the poristic 
condition or if the ranges on the transversals of four lines satisfy this 
condition an infinity of cubics can be drawn to touch them. The modular 
equation for the cubic transformation appears here. 
To determine the involutions of cubic forms of a given discriminant, 
let a^yd be four planes of a pencil of axis /; in a^y take lines abc arbi- 
trarily (distinct from /) meeting / in distinct points ABC. The cubics 
tangent to abc and the plane 5 are of multiplicity oo2 being subject 
to 10 conditions. Let d be the line in 8 tangent to a cubic and D its 
intersection with / and I' the second transversal of abed. If m n have 
the same meaning as before n is given being the cross-ratio of the pencil 
a 13yd and m has four values. D has four positions and the lines d form 
four pencils. 
To see that to a value of n correspond only four values of m it suffices 
to notice that the equation between m and w results from the elimina- 
tion of t between mn = t^, m -\- n = t"^ -\- 1 — {t — 1)^. Considering 
n as given and eliminating m there results an equation of degree four 
for t and so four values of m are found. 
The final result may be stated in this way. Let ABCD be any four points 
on the axis of a pencil of planes a^yb and suppose [ABCD] [a^yb] con- 
nected by the modular equation; then any four lines, one through each 
point to lie in the corresponding plane, are tangent to an infinity of twisted 
cubics. 
There are four inequivalent involutions determined on these cubics 
by the pencil of planes, and n is the invariant of their discriminant. Their 
construction is as follows: Given n, determine m by the modular equation. 
From 6 -\- if = V2(/c + X), B^p = Vs/cX and 
d Sd — 2\ 
m = - . 
ip Sip — 2X 
kX are determined uniquely. For mn = t"^, m -\- n = -\- 1 — (t — 1)^ 
