PKOFESSOE C. J. JOLY UN QUATERNIONS AND PROJECTIVE GEOMETRY. 301 
touches it ill 10 points on the critical curve (Art. 121), and the four points correspond 
to the intersection of the line Avith the plane in the p space, while to the ten points 
correspond lines of the type mentioned in Art. 109. We learn, therefore, that an 
arbitrary right line in the p space intersects ten of these lines, and that they compose 
a critical surface of the tenth order. This is otheiavise justified from the considera¬ 
tion that an arbitrary cpiartic transformation converts a plane into a surface of the 
sixteenth oidei , and the fact that a plane transforms into a sextic shows that a 
critical surface of the tenth order has been discarded. 
The equation of the complementary of the Jacobian J (p) = 0 Avill be found in 
Art. 127. 
124. In like manner, taking an arbitrary curve in the q space of order M, let its 
complementary be of order and let both transform into a curve of order N. The 
curve, being arbitrary, Avill not intersect the critical curve, and the 4M points in Avhlcli 
it cats the quaitic, tiansfoimed from an arliitrary plane m the p> space, will correspond 
point for point to the N points in Avhich the transformed curve cuts tlie iilane Tims 
N = 4M. 
Consider further tlie intersections of the curve and its complementary with an 
arbitrary surface (/a) and its complementary {in'). The ciii Am meets the complementary 
of the surface in Mni points, and the complementary of the curve meets the surface 
in M m jioints. In general, each point of one set corresjionds to one point of the 
other set, and the tAAm sets compose pairs of united points. Thus Mud = M'a?, or 
M = 3M by (479); and accordingly AA^e liaA^e the complete formula 
N = 4M = :‘“'.(481). 
The Avhole set of jioints of intersection of the curve and surface and their com- 
plementaries is arranged as folloAvs The Mm points unite Avith 3Mia of the Wm' 
points in M/n tetrads. The M/rd points and the Wrn unite Avith 2 (M/?d + Mbn) of 
the MW points to form tetrads, and thus by (481) and (479) all the M'nd points are 
exhausted , and theie am but 4M/// (=: Mm -j- MudMbu) tetrad.s. But the curve 
(N) intersects the surface (//) in Nn = 4M X 6/n points, and consequently there 
remain over 20 Mm points, Avhich are critical points on the transformed curve and 
surface. These points evidently must lie on tiie critical surface of Art. 123. 
When a curve is Avholly composed of pairs of united points, the order of the 
transformed curve is N = 2M, and from symmetry the order of the complementary is 
M' = M. 
An arbitrary surface and its complementary do not intersect in a curve Avholly 
composed of jiairs of united points, though of course the curve of intersection Avill 
contain all the pairs of united points aaTiicIi he on the surface. It does not seem to 
be easy to a.ssign any general relation connecting the order of a curve of this nature 
Avith that of its transformed curve. I'lius 7 is tlie order of the curve transformed 
from the cubic intersection of a plane Avith its complementarA' (Art. 116). 
