steiner’s extension of malfatti’s problem. 
59 
114 
§ 2. 
In order to state in the most simple form the geometrical construction for the 
solution of Steiner s extension of Malfatti’s problem, let the given sections be called 
for conciseness the determinators 1 ; any two of these sections lie in two different cones, 
the vertices of which determine with the line of intersection of the planes of the 
determinators, two planes which may be termed bisectors ; the six bisectors pass three 
and three through four straight lines; and it will be convenient to use the term 
bisectors to denote, not the entire system, but any three bisectors passing through the 
same line. Consider three sections, which may be termed tactors, each of them touching 
a determinator and two bisectors, and three other sections (which may be termed 
separators) each of them passing through the point of contact of a determinator and 
factor and touching the other two tactors ; the separators will intersect in a line which 
passes through the point of intersection of the determinators. The three required 
sections, or as I shall term them the resultors, are determined by the conditions that 
each resultor touches two determinators and two separators, the possibility of the 
construction being implied as a theorem. The a posteriori verification may be obtained 
as follows:— 
§ 3. 
Let x = 0, y = 0, z = 0 be the equations of the resultors, w — 0 the equation of the 
polar of the point of intersection of the resultors. Since the resultors touch two and 
two, the equation of the surface is easily seen to be of the form 
2 yz + 2 zx + 2 xy + w 2 — 0. ( 2 ) 
The determinators are sections each of them touching two resultors, but otherwise 
arbitrary; their equations are 
- m+ k y+ k e+v,=0 ’ 
20 x ~ +p J + * = 
^* + ^¡/-7* +» = 
o, 
0. 
The separators are sections each of them touching two resultors at their point of 
contact (or what is the same thing, passing through the line of intersection of two 
resultors), and all of them having a line in common. Their equations may be taken 
to be 
cy — bz — 0, az — cx — 0, bx - ay = 0, 
1 I use the words “ determinators,” &c. to denote indifferently the sections or the planes of the sections; 
the context is always sufficient to prevent ambiguity. 
2 The reciprocal form is, it should be noted, 
x 2 + y2 + z -2 _ 2yz - 2zx - 2xy - 2w 2 — 0.