312 PEOFESSOE C. J. JOLY OX QUATEEXIOXS AXD EEOJECTIYE OEOMETEY. 
Consequently twenty-two connectors are generators of a quadric S//(qq) = 0 ; and 
in particular the polar quadric of a jioint with respect to a cubic surface contains 
22 generators joining corresponding points on the Hessian. 
SECTION XIX. 
IIo^iOGEAriiY OF Points in Space. 
The Third Example of the Use of the Bilinear Fauction. 
Art. Easrc 
O 
143. The relation / = r establishes a one-to-one correspondence between the points p 
and ([ when r is fixed.312 
144. The homograph of a line is a twisted cubic. A line breahs off for e'\'ery intersection 
with a critical twisted sextic/q (/•) = 0. 312 
145. The homograph of a plane is a cubic surface intersecting the Jacobian I (p) = 0 in a 
critical sextic Fp (r) = 0 and a residual curve Fp' (l) = 0 .313 
146. The lines on the cubic surface. The schemes of the double-sixes and triple tangent. 
planes.. 
147. Points on a critical sextic and their line homographs.314 
148. The complex of connectors of points with their homographs, and the congruency of 
bi-connectors. .... 31.5 
149. The congruency of Jacobian connectors for the general bilinear function.316 
143. Writiug generally 
f{pq)=^r or \f{pq),r]= 0 .(525), 
and regarding 7‘ as a constant quaternion, a one-to-one relation is established 
between the points and q, so that one may be said to be the homograph of the 
other. 
This is equivalent to three relations of the form 
Sjjfq = 0, SqpUq = 0, Syi/gq = 0.(52G); 
and accordingly the bilinear function is not utilized to its full extent, but it seems 
to be the most convenient instrument for investigading the subject. 
114. AVe have generally in the notation of Arts. 107, 108, 
ql ( p) = /J (r), pj(q) = [r) .(527), 
and thus the critical curves of tbe transformation are 
id, (;') = 0 and Ffr) — 0 .(528) 
respectively ; or (compare (437)) 
{{>'’f{p<-^)>f{F^->),f{pf,f{Fd))) = 0 and (f, f fq), f{bq), f (cq), f{dq])) = 0 . (529)- 
