550 
ON A SYSTEM OF EQUATIONS CONNECTED WITH MALFATTI S PROBLEM. 
[622 
that is, 
0 = A (EG - F'0 - F (G'H' - AF), 
= K' (Aa + F'f), = K (abc +fgh) ; 
which is right, in virtue of the relation K' — 0. Taking the signs all positive, we find for 
(a; 2 , y 1 , z 2 , yz, zx, xy) the values (A, B, G, F', G', IF), giving two points of intersection 
{^A, 
and I — VA, 
G'_ 
\lA ’ VV ’ 
V a ’ \/a j 
Taking the signs one positive and the other two negative, say 
\/BC = F', *JCA = - G', *JAB = - H', 
we find for (x 2 , y 2 , z 2 , yz, zx, xy) the values 
(°, T , —, F, 0, o), 
viz. we have thus two intersections 
(o. v't- Vf)- (°- - Vt- -'Vi 
and the other combinations of signs give the remaining two pairs of intersections 
;Vs- °- >Jt)- (- G Vis> °- V- 
b_ 
GcJ ’ 
Aa 
c 
and 
But the most convenient statement of the result is that the values of (ax 2 , by 2 , cz 2 , yz, zx, xy), 
for the four pairs of points respectively, are 
Vr- "V®- °> (Vf- - H '\Zwy °> 
(a A, 
bB, 
cG, 
F\ 
O', 
H'), 
( o , 
cG, 
bB, 
F, 
o, 
o), 
(cG, 
0 , 
a A, 
0, 
<¥, 
0), 
(bB, 
aA, 
0 , 
0, 
0, 
H'): 
substituting 
these 
values in 
the 
verifying that the equations are in each case satisfied.