:.!1G PROFESSOR C. J. JOLT OX QUATEPtXIOXS AXD PPtOJECTIVE GEOMETRY. 
class are /r = 5 ( = 4 X 3 — 7), = 9 ( = 4 X 3 — 3), for the conoTuency (542) has 
l)een shown to be of the seventh order and third cla.ss. 
Equations (541) being supposed satisfied, they are equivalent to 
+ u,f{p>i) + u',f{qp) + vj{qq) = 0, 
'^\f{PP)-\-'^'-Af{pq)+f{qp))-rVoJ{qq) = r. . . . (543); 
and multiplying tlie first by t and adding it to the second, we find that t must satisfy 
tlie quadratic 
(m + O(.)(m + 0do) = (ri + 0q)(i’3 + 0./3) .... (544), 
if tlie sum can be reduced to the form (539). The roots of this equation lead to the 
determination of the two pairs of homographs. 
The bi-connectors of homographs which pass through a point are double edges of 
the cone of connectors of homographs, and those which lie in a plane are bi-tangents 
to the curve enveloped by the connectors. This appears from tlie forms of the 
ecpiations (538) and (540). 
149. The congruency of connectors of Jacobian corre.spondents is intimatelv 
connected with the theory of the last article. 
We have already considered the case in which the function is permutahle, hut 
matters now are much more complicated. 
The congruency may be expressed liy 
f{ pp) + pfipq) + pf{qi>) + '^pf{qq) = o.(545), 
and It is obvious that it is included in the quartic complex, the first of (541), and it 
is easy to verity that it is also included in the sextic complex (538) and that no 
matter v'Jiat quaternion “ r ’ rnaij he. Replacing uv by iv in (545) and substituting in 
the equations of these two complexes we find that either w = uv, or else the lines must 
belong to tlie congruency (540). In other words, the congruency of this article is 
conqilementary to the congruency of the last as regards the two complexes. But the 
rays of the former congruency count double as edges of cones or a,s tanofents in 
])lanes. Hence tlie order and class of the congruency under discussion are 
p = 14( = 4 X G - 2 X 5), z/ = G ( = 4 X G - 2 X 9). 
Ihese numbers are exactly double the corresponding numbers for the permutahle 
function, and as regards the class there is no difficulty in seeing how this arises. In 
general there are two sextic loci of Jacobian correspondents of the points in a plane 
(528), and the connectors in the plane join the six points of one to the corresponding 
six ])olnts ol the other. For the })ermutahle function the two loci coalesce, and the 
numlier of connectors is halved. 
Again, we may say that the lines of this new congruency through a point are Jixed 
edges of the cone (538), and tlie lines in a plane fixed tangents to a sextic curve, 
