528 PROFESSOR CHRYSTAL ON 
thus obtained are very interesting, but we pass over them, simply remarking 
that in this case the equatorial surface corresponding to the given plane plays 
the part of (24). 
On the Congruency formed by the lines of Single Resultants. 
We might regard the congruency of single resultants as the rays common 
to the complexes of the second and fourth order determined by the relations 
(5) and (7). 
A simpler method will suffice to deduce Minpine’s theorem for the case 
Fs: 
Let (yz) be any point on the ray passing through (adc), we have 
4 = —- &c. where d’?=(a#—a)?+ (y—b)?+(z—c?’), 
Nae eal peioneere 
1 ayy +bp,+en,’ 
én being co-ordinates of the foot of the perpendicular, 
and 7 — p? = (ad, + bu, + 6r,)? where 7?=a7+6?+0?. 
also &e., 
Hence equations (8) and (9) give us 
SY + py? + hiv? = pt + f7(a—y r/7? — p?)? 
+ &. 
+ &e. 
ptt (SA? + Gm? + h?v,”) (7? — p*) 
+ f2a2 +9? + hc? 
—2(f7ar, + 92bp, + h?ev,)(ad, + bp, + €7;) 
II 
p'+7°p—p*+ &e. — &e., 
whence 
SPP Mt P =P ne + AE Pyne 
+2( 70d, + Php +1?er1)(@y + 0p, +O)=H=PPC + PV +N... (25) 
From (8) we get 
PV AG ey try + (Gr, + bp, +e, =7". : ; (26) 
The common Equations, (25) and (26), represent quadric cones having their common 
congruency is 
of the fourth vertex at (abc). These have in general four generators in common. Hence 
through the point (abc) there pass in general four rays of the congruency. 
