PEOFESSOE C. J. JOEY OX QUATEENIONS AND PEOJECTIVE GEOMETEY. 305 
To a plane in the j) space corresponds a quadric, or 
S/i; = 0, S//’(^ry) = 0 ........ (494) 
transfoiin one into the other j and thus to one point 2 ^ corresjDond eight jDoints _ 
the intei sections of three quadrics—and to one point q corresponds in general one 
point 
We use the word octcid to denote the group of eight points corresponding to ^ 1 . 
132. The right line q = a -{■ tb transforms into the conic 
—/(««) +2(/’(a5) + (65).( 495 )^ 
and / {aa) and f{hh) are two points on the conic, while f iah) is tlie iiole of their 
chord. 
The condition for the cohinearity of these three points is 
[/('^«)> f{ah), f{hh)-] = 0 .(496); 
and tins equation may he replaced by 
/(«o^) + (x + y)f{al>) + xyf (66) = 0 , or f{a + x6, a + yh) = 0 . (497); 
and this expresses tliat the original line joins Jacobian correspondents. Tlais lines 
joining Jacobian corresjoondents transform into lines. 
In this case (Art. 104) of the permutable function, if 
/{rr') = 0 =f{rJ) .( 493 ), 
the points r and r' are conjugate to every quadric of the system (494). 
We may replace (498) by 
J {>' =b r ± tr') =f{rr) + t'ff'd) .( 499 )^ 
ov jwints harmonically conjugate to a jxdr of Jacobian corresjmndcnts transform into 
a single q)<^int. 
Thus we may speak of the rays of the assemblage of lines represented liy ( 49 G) as 
connectors, (1) of a pair of Jacobian correspondents, (2) of a pair of points of an octad, 
(3) of an infinite number of pairs of points of octads. 
It is evident that when two points of an octad coincide 
Jacobian; and that every point on the Jacobian is the union of 
octad. 
, they unite on the 
a pair of points of an 
133. The Jacobian correspondents transform into limiting jooints, sejmrating the 
points derived from reed from those derived, from imaginary points. 
The points on tlie transformed connector 
P — f sf f r). . . 
2 R 
VOL. CCI.—A. 
(500) 
