PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 307 
and let the twenty-eight points f be denoted by 
7h3=/(^ii<Z3), p^^=f{q^q^) .(506). 
It may be remarked that these relations lead to 
±2y - =/(qi ± a/ - lq.2, qi ± \/ -’1^2) • • • (507); 
so that the points (506), although real, if the points of the octad are real, have been 
ti-ansformed from imaginary points, and consequently do not lie in the same region 
(Art. 133) as the point 
The Jacobian correspondents transform into ±^12, &c. 
136. A plane transforms into a Steiner’s quartic. 
In the notation of the last article, the plane 
2' — hS'i “t“ h*h .(508) 
transforms into the surface 
P — Po ih' + + ‘^PiAPz + '^p-iihh + ■ • ■ (509) ; 
and if we write the identity connecting the five quaternions in the form 
P = PqW + p.pc + p.p! +703^.(510), 
comparison with (509) gives 
‘Zxyziv — yV + -f- .t-//®.(511) 
on elimination of the ^larameters t. This is the scalar equation of the surface (509), 
and the existence of the three intersecting double lines (y, ^ ; z, x\ and x, y), which 
characterize a Steiner’s quartic, is manifest. 
Evidently the three connectors transform into the double lines; and the points 
y>o i 7130 7^0 ii^i3 separate (Art. 133) the lines into regions intersected by 
a pair of real and a pair of imaginary sheets of the surface.'^ 
137, The nature of the surface into which a ])lane transforms may be established 
from purely geometrical considerations. A tangent plane to the surface transforms 
back into a quadric touching the plane, that is, cutting it in a pair of lines. These 
lines transform back into conics in the tangent plane and on the surface. One point of 
intersection of these conics corresponds to the point of intersection of the lines. 
The other three points must result from the union of pairs of points of octads, and 
therefore the lines must cut the sides of the triangle in points harmonically conjugate 
to the Jacobian correspondents. The .conics consecjuently intersect the lines into 
which the three connectors transform, and these three lines must be double. In terms 
It is easy to verify this by determining the greatest and least value of E4 + for real values of 
<2 and 4 . Compare (509). 
2 R 2 
