ANALYTICAL RESEARCHES CONNECTED WITH 
where it is necessary, for the present purpose, to give opposite signs to the radicals. 
For if the radicals had the same sign, it would follow that 
/3\ - (¡3 + ^) fi + /3v - 2p 
1 
7 L 
7A, + 7/i - ^7 + v — 2p 
= 0; 
hence the section \x + py + vz + pw = 0 would pass through the point 
A 11 2 2 
x : y : z : w = 0 : -5- : — : - -= + -; 
P 2 7 P 7 
or the section would be a tangent section of the two determinators of the same 
class with the resultor x — 0, which ought not to be the case. The proper formula is 
/3\ - (/3 + ^) fi + /3v - 2p 
+ 
7\ + 7/x — ( 7 + -) v — 2p 
= 0: 
and this equation being satisfied, the section 
\x + fiy + vz + piv = 0 
passes through a point 
o 1 1 
, * : y : i : w=2 : 
2 _ 2 
£ 7 
The bisector passes through this point and the line of intersection of the determi 
nators ; its equation is 
1/1 _ 1 N1/1 1 \ „ 
or reducing and completing the system, the equations of the bisectors are 
h) y+ ( 1+ w) * + (H) w=0 ' 
2/3* 2r) x (' + ¿7- 
1 + 2f 
2a 2 
/1 IN 
-(1 + 
2/3 2 
«+ 1 + 
2 a 
In order to verify the geometrical construction, it only remains to show that 
each bisector touches two tactors. Consider the bisector and factor 
~ (* + 2/3 2 ) X + (* + 2a 2 ) V + (2a 2 ” 2/S 2 ) z + (a " /5?) w = °’ 
— 2a (aa — V — 26c) # + (c + a) y + (6 + a) z + (2aa — V — 26c) ?c = 0; 
and represent these for a moment by 
\x + py + vz + pw = 0, \'x 4- p!y 4- vz 4- pw — 0 ;