63 
1 1 4] STEINERS EXTENSION OF MALFATTl’s PROBLEM, 
if A be the same as before, and A' the like function of A', y\ p \ also if 
= XV + yy' + vv ~(yv' + y'v) - (vX' + v'X)~ (Xy' + X» - 2pp, 
then 
A' 2 = (2aa 2 - 2a V - 2bc + b + cf, 
a , =M! ( 2 + -Ly_ 2a V- a()C ( 2+c I) + o( 2 + I,) ; 
and the condition of contact AA' = 6? (taking the proper sign for the radicals) be 
comes 
(2aa 2 - 2a V — 26c 4- 6 + c) = aa 2 
— 2a V — 26c 
+ c 
or reducing, 
aa — 6/3 + c 
a — /3 
2a/3 + l 
= 0, 
an equation which is evidently not altered by the interchange of a, a and 6, /3. The 
conditions, in order that each bisector may touch two tactors, reduce themselves to 
the three equations, 
aa — 6/3 + c 
a — /3 
2a/3 + 1 
= 0, 
/3-7 
2/3 7 + l 
+ 6/3 - c 7 = 0, 
— aa + 
6 
2a 7 + 1 
+ cy = 0, 
which are satisfied by the values above found for the quantities a, 6, c. The possi 
bility and truth of the geometrical construction are thus demonstrated. 
§ 
Let it be in the first instance proposed to find the equation of a section touching 
all or any of the sections x = 0, y = 0, z = 0 of the surface of the second order, 
ax 2 + 6 if + cz 2 + 2fyz + 2gzx + 2 hxy + par = 0. 
Any section whatever of this surface may be written in the form 
(aX + hy + gv)x + (/iA + by + fv) y + (gX+fy + cv) z + v -p Vw= 0, 
where 
V 2 = aX 2 + by 2 + cv 2 + 2fyv + 2gvX + 2hXy - K,