398 
me. a. CAYLEY ON THE CONIC OF FIYE-POINTIC 
or substituting for and the values -^-Uxyz and (^+U.yz)- le- 
spectively, this is 
We have thus 
{x’ 4 - 6 Myz + Vll^o^yz — xy^z^W 
=(1 + 8 P ) 
+ {2M xhjz — — 2 lxfz^)S:Z 
and the equation _^^^H,_j.vS+(l+8?)(QW-3a?/s-®,)=0 
is thus verified. 
“S““5rr:.rr.^— 
ing section. 
V Extension of the last preceding method to a cur ve of any oi do . 
write Xy+^i, +/^ 5 tVip term involving X’" vanishes, and 
point of the curve, and X, Y, Z current coordinates . the teim 
dividing out the factor g> the equation becomes 
X— DU+iX->D>U +4).>-y D»u + A;X“- + &c. = 0. 
Making the like substitution in D»U-nDU=0, the assumed equation of the tive- 
pointic conic, the factor ^ divides out and the equation becomes 
2 x(m-l)DU+f<.D’U-(kP+/vn)D’U= 0 , 
or, what is the same thing, 
xDU(2(7?i- 1)~P)+M^'U— nDU)=0 ; 
T • 1 -/V Y // tVip TPsnlt throwing out the factor DXj , is 
and if from the two equations we elimmate X, g^, the lesiu , o 
(D»U-n0Ur‘ -i(2(m-l)-P)D=D(D-U-nDtJr-=+&c.=0, 
where all the terms after the second contain the factor ^ 
the equation may divide by DU, is consequently 2(,«-l)-P-^, Ldhm'bv DU the 
condition of a three-pointic contact. Substituting this value, and dividing by DU, 
equation becomes 
_n(D«U-nDU)->+fD>U(D'U-nDUr--iD-U.DU(D=U-nDU)-<+&c. = 0. 
