114] 
STEINER S EXTENSION OF MALFATTIS PROBLEM. 
67 
where 
+ V<Ï3 + $$ V<2D) ; 
*4 
+ 
and consequently 
i, C 4>(+)=(Va® - «5) (Vais - a?) j/fz+va (r+z>+/+ 20 vg), 
a reduction, which on account of its peculiarity, I have thought right to work out in 
full. 
The condition of contact is 
<S> ( + ) = V'V" = ~ (V1ÔD - (VII - H) vr + ÿî VïTzî 
Hence finally, the condition in order that the sections 
x Vtif + (hx + by + fz) + — 6p Vl + F 2 = 0, 
x V33 + y Vj^ + (<7# +fy + cz)Z + w V — cp Vl + £ 2 = 0, 
(the former of which is a section touching 2 = 0, & = 0, and the latter a section touching 
¿r = 0, y — 0) may touch, is 
fYZ + ^M(Y+Z) + (f+ 26 VI) - Vfo Vl + F 2 Vl +F 2 = 0. 
The preceding researches show that the solution of Steiner’s extension of Malfatti’s 
problem depends on a system of equations, such as the system mentioned at the 
commencement of the following section. 
Consider the system of equations 
a + 0 (Y + Z) + y YZ + 8 VF+T 2 VF-VF 2 =0, 
a' + 0' (Z +X) + 7' ZX + 8' Vl + Z 2 Vl + X 2 = 0, 
a" + 0" (X + F) + 7 "X F + S" Vl + X 2 Vl + F 2 = 0 ; 
these equations may, it will be seen, be solved by quadratics only, when the coefficients 
satisfy the relations 
0 _ F 
7 - a V - a 7" - a " ’ 
/3 2 + 7 2 - S 2 _ 0'* + V 2 -8'* _ 0"' + y">-8"> 
7 2 
_ a a 7 /2 - a' 2 7" 2 “ «" 2 
9—2