The collected mathematical papers of Arthur Cayley.

622] ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTI'S PROBLEM. 547
where the conics P, Q, R each pass through the two points I, J, and meet the conic
e in two points, viz.,
the conics P, ~ meet in two points P1, P2,,, Q, Q,, Q1, Q2,,, R,,,,, R1, R2.
For this purpose, writing
V = Aa - Ff, = Bb - Gg, = Cc - Hh, = ab -fgh,
= (X + Y+ Z), = ax+ by2 + cz2 -fyz-gzx - hxy - V,
0 = acx + b/3y + cyz,
Aa Ff Ff
= x- x+ y+la,
Gg Bb Gg
Hh Hh Cc
a - x + y- -z,
then we have identically
aAf - VX = 0,
bBf - V Y = Or,
cCQD- VZ = O'.
In fact, the first of these equations, written at full length, is
aA (a2 + by2 + cz2 -fyz - gzx -hxy- V )- V (by2 cz2 - 2fy - aA)
= (aax + by + ceyz) ( x + -,fy + z
where on the left-hand side the constant term is = 0. Comparing, first, the coefficients
of X2, y2, 2, on the two sides respectively, these are Aa2, (Aa- V)b, (Aa - V)c, and
Aa2, Ffb, Ffc, which are equal. Comparing the coefficients of yz, zx, xy, the equations
which remain to be verified are
-(aA -2V)f= Ff( o + b),
-aAg = Faf - + Aac 7
-aAh = Faf + Aab3;
or, as these may be written,
- (aA - 2V ) /8 = F (cy2 + b32),
- Agya = FfL2 + Acy2,
- Aha43 = Ff2 + Ab/2;
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