114] steiner’s extension of malfatti’s problem. 
values which give 
K 2 = - f 4 - g 4 - h 4 + 2g 2 h 3 + 2h 2 f 2 + 2f 2 g 2 - ; 
the equation of the section in question is 
VM'(~ f2 + Z 2 + h2 ) x + ¡s/lg ( f2 - S 2 + h2 ) y + 7g (f 2 + g 2 — h 8 )s + ^-^~^? w = 0. 
1 proceed to investigate a transformation of the equation for the section with an 
indeterminate parameter X, which touches the two sections y = 0, z = 0. We have 
nV- = (aX + hy + gv) 2 + (Qhy 2 + 23 v 2 — 2Jfyv) — 23(£' + jp 2 ; 
or putting for y and v their values V&J, V(2D in the second term, 
aV 3 = (aX + hy + gv) 2 + (Vlj}<2D - §) 2 ; 
and introducing instead of X an indeterminate quantity X, such that 
aX + hy + gv ~ (V33(iD — jp) X, 
we have 
aV 2 = (VÌ3© - Jp) Vf+T 2 ; 
also introducing throughout X instead of X, and completing the substitution of V23, V(2D, 
for y, v, the equation of the section touching y — 0, z — 0, becomes 
(1ax + liy + gz) X + y\/(& + z\/23 J rW'J— op Vl + J 2 = 0. 
It may be remarked here in passing, that this is a very convenient form for the 
demonstration of the theorem ; “ If two sections of a surface of the second order touch 
each other, and are also tangent sections (of the same class) to two fixed sections, 
then considering the planes through the axis of the fixed sections and the poles of 
the tangent sections, and also the tangent planes through this axis, the anharmonic 
ratio of the four planes is independent of the position of the moveable tangent sections;” 
where by the axis of the fixed sections is to be understood the line joining their poles. 
The sections touching z = 0, x = 0, and x = 0, y = 0, are of course 
x V (2D + (hx + by + fz) Y + z A '/$L + w*i/ — bp\/1 + l 72 — 0, 
x ^23 + y + (gx +fy + cz)Z + w^ — cp l J 1 + X- = 0, 
hx' + by' +fv' = (MW- <£) Y, X' = VC y! = y, v' = M, 
gX"+fy" + Cv" = (M23-^t)Z, X" = VC y" = M v" = v". 
The conditions of contact of the sections represented by the above written equations 
would be perhaps most simply obtained directly from the lemma, but it is proper to 
deduce it from the formula for contact used in the present memoii. If for shortness 
(±) = otX'X" + by'y" + cv V + f(yv" + y"v') + g (v'X" + v"X') + h (X'y" + X"y') ± K, 
C. 'll. »