LL4] STEINERS EXTENSION OF MALFATTl’s PROBLEM, 
then the equations of the required sections are 
{ax +hy + gz)X + y*J(E, + z V23 + w V - ap Vl + X* = 0, 
x'/<& + (hx + by+fe) Y + z Vl + w V- bp Vl + F 2 = 0, 
^ ^23 + y Vl + (g# +/y + cs) Z + w V - cp Vl + Z 2 = 0, 
where A, F, Z are to be determined by the following' equations, 
(/+ 29 Vl) + Vl (F + Z)+fYZ - ^bc Vf+T 5 Vl+^ = 0, 
(g + 20 V23) + V33 (Z + Z) +gZX -^ca Vl+ Z 2 Vl + Z 2 = 0, 
(A +26>V^)+V©(Z+ F) + hXY - Vo6 Vr+ZWiTF = 0; 
and the solution of which, putting 
t-№J3m=§, g =h=^®7vaj3-?% /=72 
is given by the equations 
A r X=?fe h +( -f+g+h)> —2(—f+g + h)/, 
AT=?fe- + ( f — g + h) 2 — 2 ( f-g + h)J, 
KZ = ^f + ( f+g-h) 2 -2( f + g-h)/. (') 
Instead of the direct but very tedious process by which these values of X, F, Z 
have been obtained, we may substitute the following a- posteriori verification. 
We have 
K‘ (1 + X‘) = 4 (- f + g + h) 2 J> (l +1) (l - I) (l - , 
if. VTTT ! VIT® 5 = 4 (f» - (g 2 -h)) /■ (1 - i) V 1 - S V 1 “ ’ 
JT*(1 + YZ) =4(1- jj {(J‘ - gh) (f 2 - (g - h) 2 ) - 2gh (g - h) 2 ], 
AT(F + Z)~ 2f 2 -2g 2 -2h 2 + 4/ 2 = 4p -y(/ 2 -gh). 
Putting also 
p - g 2 - h=+Söü = (f 2 - ( g - hy) - M ? h) 
J 2 
X 2 
= (f 2 -(g-h) 2 ) (g+h) 2 -f 2 --^ 
4g 2 h 2 \ 4g 2 h 2 (g — h) 2 
J 2 
1 It is perhaps worth noticing that the value of the quantity X previously made use of, 
x= _i!-№-g 2 -h2 + (j+f-g-hpl. 
Wa 1 J '