[114 
114] 
steiner’s extension of malfatti’s problem. 
83 
forming 
It is obvious, from the equations, that each separator passes through the point of 
contact of a factor and determinator, it consequently only remains to be shown that 
each separator touches two factors. Consider the factor which has been represented 
by ax + (3y + yz + 8w = 0, the unreduced values of the coefficients give 
+ P>/3 + ® 7 = Z 2 V1, 
»«+J9/S+JF7=AL(pj + „), 
«5«+iF/9 + ®7-J| (« + /.), 
V a “’ + • • • f 8! = Vf № + m + <%) = 
Represent for a moment the separator 
VI 
1 _ iw-4Ai - Cu - C s 4-il - 
J, 
V33 
J, 
h VOD 
J. 
z = 0 
K 
by lx + my + nz + sw = 0. Then putting %[1 2 + ... — s 2 = Q 2 , since 
P 
® a l+...+-8s = K>\W& + -^(13 + v) + ^(<& + y) 
va 
va 
= K* -I f 1 - ^ 
l 
J 
^ V-t?) ( a - 5)-< f -g> V - !r) i 1 - ?) 
^ ! |-(f-g) s +h(f+g)- ? ® 
the condition of contact becomes 
□ =Ij-(f-g)’ + h(f+g)-^}; 
or, forming the value of [J 2 and substituting, 
f ! (i - jJ+g 2 (1 - $)'+( f - g) 2 (1 - jJ + 2 ( x - j?) (f “ g) g i 1 “ ~ J. 
-2 1- 
2g2 
/ 2 
(f-g)f(i-y(i-rr) "M 1 
J, 
a* 
j s 
j, 
= L(-(f-g)=+h(f+g)- 2 ®) ! , 
which may be verified without difficulty, and thus the construction for the resultors 
may 
is shown to be true, 
11—2