114]
steiner’s extension of malfatti’s problem.
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and from the mode of formation of these equations it is obvious that the separators
have a line in common.
The equations of the determinators being x = 0, y = 0, z = 0, the equations of the
tactors are
V33^r— V©y = 0, V©# —Va.z = 0, Vay-V<ï3æ = 0 ;
and if ax + fiy + yz + 8w = 0 be the equation of the tactor touching
x = 0, V©# — Va.z = 0 and Vay —V23« = 0,
the conditions of contact are
= j(Va23 - 3^)0 Va - /3 V33) + 7 (© V33 - V©)j 2 ,
= j(Va© - (K) (« va - 7 v©) + vod - jp va>j 2 ,
whence
(Vaaa - ?§) va« - (Vaaa - 3^) V23/3 + (© vas - jp va) 7 ,
(Va© - ® ) va«+ (3^ v© - j va) /3 - (Va© - ©) 7 ,
c/3 2 4- by 2 — 2f/3<y + — 8 2 = 0,
and putting for a moment
fi = va© - © - A va© (Va© - é\
v=vas - 3^ - n/2 vaaa (Vaas - 3^);
after some reductions, and observing that the ratios only of the quantities a, /3, 7, 8
are material, we obtain
C. II.
11