STEINER’S EXTENSION OF MALFATTl’S PROBLEM. 
255 
of a determinator and tactor 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 verifica- 
tion may be obtained as follows : — 
^ 3 . 
Let j?=0, 2=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 
2yz-\-2zx-\-2xy-{-iif=0*. 
The determinators are sections each of them touching two resultors, but otherwise 
arbitrary; their equations are 
— aT + ^ = 0 
~x-(3y+j^z + iv=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 re- 
sultors), and all of them having a line in common. Their equations may be taken 
to be 
cy—hz = 0, az — cx—0, hx—ay = 0, 
the values of n, b, c remaining to be determined. Now before fixing the values of 
these quantities, we may find three sections each of them touching a determinator at 
a point of intersection with the section which corresponds to it of the sections 
cy — hz=0, az — cx=0, hx—ay—Q, and touching the other two of the last-mentioned 
sections ; and when a, c have their proper values the sections so found are the 
tactors. For, let Kx-\-^jy-\-vz-\-^w=0 be the equation of a section touching the de- 
terminator — ax-\-^y-\-~z-\-w=0, and the two sections bx—ay=.Q, az—cx=0, and 
suppose -\-v^ — 2iJjv—2v^—2'k^— 2f, 
the conditions of contact with the sections hx—ay=0, az—cx=0 are found to be 
{b a) ^=^{b a)}^— {b a)(jij — (b—a)v 
(c-}-«)A=(c-1-«)A— (c—a)pj—{c-{-a)v, 
values, however, which suppose a correspondence in the signs of the radicals. Thence 
* The reciprocal form is, it should be noted, 
x'^-\-i/^+z'^—2i/z—2zx—2a:y — 2w^=0. 
2 L 
MDCCCLII. 
