PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
309 
In this case the plane transforms into the surface 
P —/(h& + + t^iy, tjh + f-.hj + Q//) 
= + 2QQ/(/>//) + . (516), 
and if we take (as we may) f{bh) =:f{hJ)^), the scalar equation of the surface takes 
the form 
4x2fw = 4xh' + y\ where iv = _p y _ OQQ, 2 = SQQ (517). 
On comjDarison with (oil) we see that two of the lines of the Steiner’s quartic have 
united; for a? = 0 we have the line x, y counted four times. 
139. By a process similar to that of Arts. 123 and 124, but much simpler, we can 
determine the order {in') of the complementary of a surface of order m, and the 
order {n') of the surface into which both transform. The formula is 
4u?. 
V 
4m' 
(518), 
where v is the number of points of octads of which the surface is wholly composed.'^'' 
And this formula is proved without trouble, remembering that a line in the }) space 
transforms into a twisted quartic—tlie intersection of two quadric surfaces. 
In like maimerf for a curve (M), its complementary (iVT) and its transformed (N), 
_ 2M' 
V 8 — V 
(519). 
Thus the complementary of a connector is a twisted cubic the complementary of a 
plane is a surface of the seventh order, which cuts the plane in tlie triangle of 
connectors and in a quartic—probably the four lines of Art. 137. 
The formulm of this article are not directly applicable to tlie Jacobian, which is a 
ciitical suiface of the transformation. The twisted quartic into which a line in the 
p space transforms, cuts the Jacobian in 16 points and does not in general touch it. 
For if it did the twisted quartic would liave a double point. Consequently, the 
Jacobian transforms into a surface of the sixteenth order. Every point on the 
* For the general tran.sformation of order ji, the relation is 
t For a transformation of order ji, 
1 For example, 
ji-ni ^ jx-ni 
V - V 
■■ 11 . 
/xM_ /xM' 
[M^-V 
= N. 
2 = ^ r ^'’^lere qr, = E p = E 
1 n '1/n I •' II I Ihi 
is the equation of the twisted cuhic through si.x: points qi, q-i. . . q^■„ and it is not difficult to veiify that 
this curve and the line + Qs transform into a common line p + ^Fts if the eight points form an octad. 
