STEINER’S EXTENSION OF MALFATTl’S PROBLEM. 
269 
whence 
Y+Z= (l -2. 
«+«/ \ _ , 
K x/^,y, A 
YZ = 
2 \/2 v/ 
And by forming the analogous expressions for Z-bX and ZX, X+Y and XY, the 
values of X, Y, Z may be determined. But the equations in question simplify them- 
selves in a remarkable manner by the notation before alluded to. 
Suppose 
s=^s/VM-f, g=-yS\/VM-e, h=^/c^/^/a£-®, j=V2^m€, 
these values give 
V Jv 
K\/bc 
W 
W =2sh-\/ l-fiV 1-S 
= P_g»_h»+ j« 
PV ^-J2 
2g^h^ 
=-f"-g"-h^-l-2P 
=-P-g*-h^-l-2gV-|-2liT-t-2Pg^-^^^". 
Applying these results to the preceding formulae and forming for that purpose the 
equations 
/— / /7^ — \^2\^u + cn, J Vot, f 
2y2v/H^,y(3;/,=4gh. — ,7^=J 
J2 
we have 
K(Y+Z) + 2K(l=4(J«-gh)(l-j) 
K=YZ+K' =4{(.P-gh)(f«-(g_hY)-2gh(g-h)’}(l-j) 
the former of which, combined with the similar equations for Z-j-X and X-j-Y, gives 
for X, Y, Z the values to be presently stated, and these values will of course verify 
the second equation and the corresponding equations for ZX and XY. 
Recapitulating the preceding notation, if x = 0, y~0, z=0 are the equations of the 
given sections, iv=0 the equation of the polar plane of their point of intersection 
with respect to the surface 
ax^-\-hy^-\-cz^-\- 2fyz -f- 2gzx -|- 2 hxy -\-pw^ = 0, 
