CONTACT AT ANT POINT OP A PLANE CUEVE. 
385 
Hence writing 
and then the equation of the five-pointic conic is 
D^U' - 1 + ADU'^ DU' = 0. 
I take as the equation of the cubic, 
'{J =:cif -\-y^ -\-z^ -^Qlxyz=^ •, 
the formulge Table No. 70 of my Third Memoir on Quantics^, putting H for HU, give 
H = -\-y^-{-z^)—{l-\-2 l^)xyz. 
U' = ^{x^ + 3 /® + 2 :® + 6 Ixyz ) , 
the first derived functions are 
l{x^+2lyz), ^{f+2lzx), K^^+2%); 
the second derived functions, or (a, b, c,f, g, A), are {x, y, z, lx, ly, Iz), whence H'=: — H ; 
the inverse coefficients (91, 23, C, (S, are 
{yz—Vx^, zx—l^y‘^, xy—l^z^, Vyz—lx^, Vzx — ly'^, l\vy — lz^)-, 
and putting U'=-|-U and H'=— H, we have 
■ 9 H 3 
+ 4(a, 23, C, f, (B, IIB.H, B,H, 
and the equation of the five-pointic conic is 
D^U- (||gDH+iADu)DU=0. 
31. We have ^ 
(3, 23, C, f, 0, B,)^H 
= (yz—l^x^).QPx 
-f- (zx—l'^y^).QPy 
-f (xy—Pz^ ).<dl^z 
+ 2{Pyz-lx^).~{lJr2l^)x 
■\-Wzx-lf).-{\^2l^)y 
which IS 
= \Wiyz—^ l\x^ -\-9/+ 2 ®) 
-QP(l+2l^)xyz+2l(l-^2l^){x^+2/+z^) 
= (12l^-12l^)xyz-\-(2l-2l%x^J^ifi-z^) 
= ^l-l^X^^+f+z^ + Qlxyz), 
or we have 
(a B, C, f, 0, b,)®H=-2S.U, 
where S is the quartinvariant (see the Table No. 70). For the present purpose U=:0, 
and consequently 
(a, 23, C, 0, IXb,, B,)®H=0. 
* Philosophical Transactions, vol. cxlvi. (1856), pp. 627-647. 
