315 
PKOFESSOK C. J. JOLY ON 
QUATEENIONS AND PEOJECTIVE 
GEOMETRY. 
having ^ as a doublG point and containing th© three triple chords which pass 
through q. 
The ciihic homogiaph ot any plane contains the critical sextic which counts thrice 
in its intersection with the octic surface of triple chords, and the remainder of the 
intersection consists of the six line-homographs of the critical points in the plane. 
The homogiaph of the surface of chords of the^i sextic, winch meet the line 
is the cone whose vertex is q and which contains the q sextic. 
The homograph of one sextic is the surface of triple chords of the other. 
One choi d can he diawn to meet two non-intersecting" triple chords in points not on 
the sextic. Its homograph is the line joining the homographs of these chords. 
The locus of the points Fq (_p)j the Jacohian correspondents of points on the critical 
curve, is a curve of the fourteenth order. For the octic surface intersects the Jacohian 
m the second critical curve counted thrice, and in a residual curve of order 14. 
148. Connectors of points with their homographs compose the complex of the 
sixth order 
Lf(PP), fipv), f{qp), r) /(qp), f{qq),r) 
= ifipi>)> fipq). f{qq), >■) i/ii'p), f{qi>), .f{qq), >•) ■ (538), 
as appears on elimination of .r, y, z and ir from 
f{xp + yq, zy> + V)q) = r .( 539 ). 
Oi in other words, this is the assemblage of lines which meet their twisted cubic 
homographs. 
The condition that two pairs of homographs should be on the same line is 
{{f{pp\ fipq), f{qq), r)) = o .(540), 
for if two sets of values of x, y, z, tv satisfy (539), the five quaternions included in 
(540) must be co-planar. Now (540) imposes two conditions on the line qtq, and 
therefore represents a congruency of lines; and from the conditions implied in (540) 
we can select but two combinatorial functions witli respect to p and q. These are 
Ui 2 P^)’f{pq)>f{qp),f{qq)) = o, {f{pp)^,f\pq) -\-f{qp),f{qq), r) = o . (541); 
and the congruency is therefore common to two complexes of the fourth and third 
orders respectively. But these complexes contain the congruency 
[/( 2 f), fipq) +f{qp)> f{qq)] = 0 .(542), 
and this is foreign to the question, being, in fact, the congruency (496) of Art. 132 of 
connectors for tiie permutable function J{pq) + J {qp)- When this is rejected, there 
remains the congruency of connectors of two pairs of homogi'aplis, and its order and 
2 s 2 
