PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 311 
.(520), 
where r is the vertex, for the tangent plane S//‘((/r) ^qf [Ir) = 0 must vanisli 
identically. Hence the fourth invariant of f' (Ir) must vanish, or 
(/'(H /'(^^0) = h.(521), 
and this is the equation of the reciprocal of the surface. 
Thus the transformed Jacobian is the reciprocal of a Jacobian surface, but one of 
less generality than those previously considered. We may replace (520) by four 
equations 
S//(m) 0, S//(r6) = 0, S(/'(rc) = 0, S//(rR) = 0 . . (522); 
and because / is a permutable function, on replacing r by xa + ijh + 20 + ivd and 
eliminating x, y, 2 and w, we obtain the symmetrical determinant 
S//(a«), S(/Xa5), S//(ac), ^If {ad) 
S//(od), S//(W;), S//(6c), S//(/x/) 
S//(ac-), Slf{bc), S//(cc), Slf{cd) 
S{f{ad), SIf{hd), SIf{cd), BIf{dc/) 
But (‘Three Dimensions,’ Art. 528) a surface, whose equation is a symmetrical 
determinant with constituents linear in the variables, has ten double points. This 
accounts for the class of tlie surface being 16 instead of 36 (= 4(4 — 1)^). 
In the case in which the function is self-conjugate as well as permutable, that is 
when p, q and r may be transposed in '^'pf{cp ) in any manner, we have the theory of 
the corresponding points on the Hessian of the general cubic surface 
and Mr. Hussell’s paper may be referred to for various exanqjles. 
142. The characteristics of the two congruencies are found thus. The order of the 
congruency of connectors is obviously /x = 7, as seven connectoi'S can be drawn from 
an arbitrary point to the remaining points of the octad to which the point belongs. 
The class is = 3, for three connectors lie in a plane. The order of the focal surface 
(Alt. 139) is M = 24. Its class is N = lA This follows from the relation (‘ Three 
Dimensions,’ Art. 510) 
M — N = 2 (/X — j^).(524) ; 
or independently by Mr. Hussell’s elegant methocB'' which is applicable in this more 
general case. 
For the transformed congruency, the order is y' = 28 (Art. 135), the order of the 
focal surface is M' =16, and its class is N' = 4 (Arts. 139, 141); and therefore (524) 
the class of the congruency is F = 22. 
= 0 . . . (523). 
* ‘ Proc. Roy. Iiish. Acad.,’ vol. 5, p. 473. 
