302 PROFESSOK C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
125. We may account for the curve of intersection of the pair of sextics derived 
from two arbitrary planes in the following manner. 
Call the two planes P and P', and their complementary cuhics C and O'. The 
complementary of the line (PP') forms part of the intersection of the cuhics 0 and O', 
and this curve is a cubic (481). There remains, therefore, a .sextic as part of the 
intersection of C and O'. The complementary of the cubic curve (PC') is a curve of 
the ninth order, part being the cubic (P'C), and the remaining part the residual sextic 
on C and O'. This sextic is wholly composed of pairs of united jJoints. The line and 
its complementary cubic transform into a common quartic. The cubic (PC'), the 
cubic (P'C) and the residual sextic transform into a common curve of order 
3X4 = 2x6 = 12 (compare the last article). Thus we can only account for a 
curve of order 16 ( = 4 + 12), and the sextics consequently intersect in a singular 
curve of order 20. 
126. The complex of lines joining pairs of united points is of the fourth order. 
It o and h are any two points on a line joining united points, 
f{p, a) = .xa + gh, f{p, h) = to, + ivh .(482), 
where p determines the function. The theory of quaternion arrays allows us to 
write the condition that these two equations should be simultaneously satisfied in 
the form"^ 
r /(<^1«) /(eo«) f[ega) f{epi) a h () 0^ 
^ 0 . . (483) 
L,/'(^i^^) f{ef) fief) 0 0 a hj 
where e^, cq, Cg, are arbitrary quaternions ; and by the rules of expansion of arrays, 
this equation is equivalent to 
^ ± ifieif, fie.pi), o, />) (fief), f{ef), o, 6) = 0 . . . (484), 
wheie the signs follow the rules of determinants. As this is of the fourth order in 
a and h, and also combinatorial with respect to both, it represents a complex of the 
fourth order. 
127. By (433) and (434) we have 
f{pcj) = qJ {q), p = Ffq) .( 485 ); 
and throughout this article we shall suppose p expressed as a quartic function of q. 
One loot of the latent quartic of j i 'pq) is thus equal to J iq), so that when we 
substitute in the equation of that quartic (Art. 103 (420)), we have identically 
d {qf - ./ {gf 1 "' ( y,) + ./(,y)y P'fp) _ ./ (^) I' q_ ^ 0 . ( 486 ). 
The equations of the various assemblages of chords of Art. 113 may also be discussed by the aid of 
arra3^s. 
