230 
ON THE ANALYTICAL THEORY OF THE CONGRUENCY. 
[928 
For the point 2, we have 
. liy — gz + aw = 0, 
— hx . + fz + bw = 0, 
gx-fy . +cw = 0, 
— ax — by — cz . =0, 
Ky — g 2 3 + a 2 w = 0, 
— h 2 x . + f 2 z + b 2 w = 0, 
goX -f 2 y . + c 2 w = 0, 
— a 2 x — b 2 y — c 2 z . = 0, 
each set of four equations being equivalent to two equations,' in virtue of the relations 
a/+ bg + ch= 0, a 2 f 2 + b 2 g 2 + c 2 h, = 0 respectively. There is no completely symmetrical 
expression for the values of x, y, z, w; according as we derive them from the first 
equations, the second equations, the third equations, or the fourth equations of each 
set, we obtain 
where 
x : y : z : w= ©j : ag 2 — ga 2 : ah 2 — licu 2 : gh 2 — hg 2 , 
— kfz f b 2 : © 2 : bh 2 — hb, : hf 2 — fh 2 , 
= c/ 2 -fc 2 : eg, - gc 2 : © 3 : fg 2 - gf 2 , 
= be 2 — cb 2 : ca, — ac 2 : ab 2 — ba 2 : © ; 
©! = —fa 2 - bg 2 - ch 2 , 
©2 = - af ~ gb 2 - cli 2 , 
©3 = - af, - bg 2 - he,, 
© = - af 2 -bg 2 -ch 2 , 
= af 2 + gb 2 + hc 2 , 
=f<h + bg 2 + hc 2 , 
=fa, + gb 2 + ch 2 , 
=fa 2 + gb 2 + hc 2 . 
For the point 3, we have, of course, the same formulae, with the suffix 3 instead 
of 2.