STEINER’S EXTENSION OF MALFATTl’S PROBLEM. 
257 
and the values of a, b, c are therefore 
a'.h : c={2(3y-\-l)( — a+/3-{-y+2aj37) 
: (2ya+l)(a — ^ + yH-2a/3y) 
; (2a/3+l)(a+^ — 7+2a/3y), 
which are to be substituted for a, b, c in the equations of the separators and tactors 
respectively. 
Now proceeding to find the bisectors, let 'Kx-\-(jijy-\-vz-\-§w=0 be the equation of a 
section touching the determinators, 
^x+~y-yz-\-w=0. 
And suppose, as before, A^=K~-\-iM^-\-r—2(jbu—2vX—2X[j(j — 2f; the conditions of con- 
tact are 
+^A = px— 2^ 
-I-7A = yX +y[x— (y V — 2 §, 
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 
or the equation Xx-\-(j,y-]-uz-\-^w=0 w'ould pass through the point 
11 2 2 
x:y:z:w=0; 
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 
2f] -f-[7X-}-7/A- (yH--)!/— 2^ =0. 
And this equation being satisfied, the section 
Xx -f- /A?/ -}- t'Z + g>w; = 0 
passes through a point 
^ 1 12 2 
x:y:z:w=2:-j,:-;^:-^--- 
The bisector passes through this point and the line of intersection of the determi- 
nators ; its equation is 
or reducing and completing the system, the equations of the bisectors are 
(1+^)3/+ 
2 L 2 
