PROFESSOR C. J. -TOLY ON QUATERNIONS AND PRO-TECTIVE GEOMETRY, 291 
we shall call respectively the first and second Jacohians. Whenever a pair of 
cpiaternions satisfies the equation 
= h.(422), 
the point p must lie on the surface /(p) = 0 and q must lie on J{q) = 0 ; for f{pr), 
a linear function of r, has then one zero latent root, and f{rq) has also a zero 
latent root. 
On reference to (409), it appears that (422) is equivalent to 
S_p/j^ = = ^pfpi = Sp/py = 0 .(423); 
and m the particular case when the function is permutable, the four linear functions 
are self-conjugate, and the equations assert that the polar planes of one point (p) 
intersect in the other {q). In this case the surfaces (421) coincide witli one another 
and with the Jacobian of the four quadrics; and although it does not appear that in 
general the surfaces are the Jacohians of four quadrics, we have retained the name as 
being convenient and suggestive. 
Two points related as in (422) will be called Jacobian correspondents, or more 
particularly IJ Jacobian correspondents. 
105. When a function has a zero latent root, so has its conjugate. Consequently, 
whenever p and q are Jacobian correspondents, or whenever (422) is satisfied, it must 
be possible to find two other jioints r' and i-", so that 
f{Pq) = 0, = 
(424). 
There are thus two new types of Jacoliian correspondence; and the argument of 
Art. 102 shows that there can he no more, for the conditions (422) and (424) may he 
re-written in the form 
{fl{qP) = 0, {f"l(P' 2 >) = 0 .... (425), 
vithout alteiing the signification of the ecjuations, and we have now exhausted the 
six fundamental functions of the article cited. 
106. The points “ ” and “ 
he upon the third Jacobian K{r). 
oj the second and third Jacobian correspiondences 
A latent root of/'(I 'q) considered as a function of q (424) is zero, and therefore r' 
satisfies the equation 
(/'(»•'«)-/'(>-'0,/'(i-'c),,/-'(i'V)) = ((,/'7'(r'a), = 0 . (426), 
in tlie .second mimber of wliicli the function of q has been replaced hy its conjugate. 
But (417) the second number is equivalent to 
if" {<rr), f" {br), f" (cr), f" {dr)) = 0 .(427), 
and consequently r", whicli satisfies (427), satisfies also (426), or r'and r" lie upon the 
same quartic surface. 
2 P 2 
