76 
ANALYTICAL RESEARCHES CONNECTED WITH 
[ 114 
Suppose 
f=7a'/Aae-j, g =7as 7v®a-es, h=7®vVaì3-i% /=V2 7aîB®, 
these values give 
K6 = — f 2 — g 2 — h 2 + 2J 2 , 
K 2 = _ f4 _g4 _ jp + 2g 2 h 2 + 2^^ + 2f 2 g 2 . 
u 
Applying these results to the preceding formulae and forming for that purpose 
the equations 
2V , 2v / a + a V(3 / y / — 4gh, 
ghKÔ + 7M Kf= ~ gh ^ ^ f2 ~ “ 2gh ( g ~ h ) 2 > 
we have 
K (F-f Z) + 2AT0 = 4 (J 2 — gh) ( 1 — 
i( J' - gh) (P-(g- h)‘) - 2gh (g - h)>} (l - I) ; 
K-YZ+1P 
the former of which, combined with the similar equations for Z + X and X + Y, gives 
for X, F, Z the values to be presently stated, and the^e 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, w- 0 the equation of the polar plane of their point of intersection 
with respect to the surface, 
ax 2 + by 2 + cz 2 + 2fyz 4- 2gzx -f 2 hxy + pw 2 = 0 
the equation of the surface, gl, 23, C, §, Ær, K as usual, and 
1 
e =K('/M№ + +