378 
ME. A. CAYLEY ON THE CONIC OE EB'E-POINTIC 
and let (D"U), See. be the corresponding values of D^U See., then we have 
aX+5Y+cZ= 
D^U 
DU’ 
ax,+by,+cz,:= 
17 
7 /D^UX 
+ ( J)U J^ ’ 
and if from the four equations vre eliminate a, h, c, we find 
XDU, 
YDU, 
ZDU, 
D^U 
^ , 
y ^ 
2 , 
2(m-2) 
■> 
■> 
^1 7 
/D^UX 
VDUj* 
X 2 , 
y^ ■> 
^2 , 
/D^UX 
1 DU 
for the equation of the conic in question; x,y,zhe\n^ the coordinates of the point of 
contact, and X, Y, Z current coordinates. 
II. Demonstration of Identities assumed in the 'preceding section. 
Proof of the expressions for (bf + 3^1^2)11 and (Bl+bBlBg)!! : 
17. It will be remembered that Bj, Ba stand originally for 
dx B ^+ dy Bj, + dz B^, 
d^xb^-\-d^y'by-^d^z'd^, 
and that A, B, C being the first derived functions of U, dx, dy, dz are changed into 
Etz-Cf-t-, Cx—Av, 
and that the resulting value of Bj, viz. 
(Bj'-Cp)B,+(Cx-Aj/)B^4-(A(U— BX)B,, 
is also represented by D, so that B, = D- The corresponding values of d'X, dy/, d'z are 
d^x~v t^B — gjdC, 
d^y =XdQ — V dA, 
di^z —gjdA~Xd^, 
where we have 
dA—adx-{-hdy-\-gdz, See., 
in which dx, dy, dz are to be replaced by the values 
Bv — C/i), Cx — Av, Agj — Bx ; 
and Ba really denotes what 
d^x'b ^. + d‘^y'by + I^z'b, 
becomes when the above values are substituted for di^x, d^y, d^z. But in the expressions 
