810 PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEO^IETRY. 
Jacol^ian is the union of a pair of points of an octad (Art. 132), and therefore the 
complementary surface is composed of hexads of points of octads, and its order is 
conserpiently 24, or six times that of the Jacobian, because the quartic cuts it in a 
hexad for every point of intersection with the Jacobian. 
140. The comj^lemc'tLtary of the Jacobian is the focal surface of the congruency of 
connectors.''^ 
When two points of a set transforming into a common point approach coincidence, 
they close in on tlie Jacobian, and simultaneously the remaining points of the set 
reach the complementary surface. Through any one of these remaining points two 
consecutive connectors pass ; and therefore, by Hamilton’s beautiful theory, the 
I’emaining points are focal gyoints on the rays connecting them to the coincident 
jioints. 
Every ray touches the focal surface in two points—the two focal points on the rav; 
and for a quadratic transformation it cuts that surface in twenty other points. These 
ticenty g)oints are harmonically conjugate in jjairs to the Jacohian correspondents. 
For (Art. 132) the harmonic conjugate of any one of the points belongs to the same 
octad as that point; but tbe fecal surface is complementary and is wholly composed 
or hexads of points of octads, and therefore the harmonic conjugate is also on the 
focal surface. ■ 
141. The focal surface of the transformed connectors is the transformed Jacohian. 
On transformation the harmonic conjugates on a connector unite. In the notation 
of Art. 138, the point h and the focal point unite in a focal point of the trans¬ 
formed connector, for through pass two consecutive connectors which transform 
into consecutive connectors through ffff Similarly the points c and c^ transform 
into the second focal jroint and the transformed Jacobian is consequently the focal 
surface. The twenty points of the last article transform into ten points. The 
Jacobian correspondents a and a' transform into limiting points (Art. 133). Thus we 
have accounted for the sixteen jioints in which the transformed connector meets its 
focal surface. 
2he class of the transformed Jacohian is n' = 4. In the p space draw a plane 
through an arbitrary line to touch the surface. This plane contains a jiair of 
consecutive transformed connectors, and on passing back to the g space it becomes a 
quadric containing consecutive intersecting connectors. This quadric is therefore 
a cone. The system of planes through the arbitrary line transforms into a system of 
quadiics through a twisted quartic, and four of these quadrics are cones. To these 
four cones correspond four tangent jilanes to the focal surface through the arbitrary 
line. Flence we may write down the equation of the reciprocal of the transformed 
Jacobian. The condition that the quadric S(/’(qq) = 0 should be a conef is 
* This theorem is true for the connectors of a set of points to a coincident pair of the set for all 
transformations. 
t If .f im) = then /' {Iq) = 2 AjSfoi. 
