402 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[146 
then we have 
a' (be — f) + P (ca — g 2 ) + c' (ab — h 2 ) + 2f (gh — af) + 2g' (hf— bg) + 2h! (fg — ch) 
(l' 2 + 2 lf(^ + v 3 + ^) 2 
+ (2V + 41- 32W + 8Z 4 ) (f + ?? 3 + 0 
+ (24ZZ' 2 + 48Z 4 Z' 2 - 72Z 2 Z' + 24Z 3 + 3) |V£ 2 , 
which may be verified by writing V = Z, in which case the right-hand side becomes as 
1 _|_ 2l 3 
it should do, 3 (P/7) 2 . If l' — , that is, if the syzygetic cubic be the Hessian, 
then the formula becomes 
, n ~ ? 1 f (l + 4Z 3 + 76Z«)(f + f + Ç 3 ) 2 ) 
a (be - f 2 ) + &c. = I 
1 + 12Z 2 (-1 + 26Z 3 + 56Z 6 )(f 3 + T? 3 +> 
which is equal to 
[+121 (2 + 57Z 3 + 168Z« + 16Z 10 ) fV£ 2 
1 
36Z 4 
r 4 QP -24S.PP . 
26. The equation 
(6c' + Pc - 2ff, ... gh! + g'h - af - a/, .. 77, £) 2 = 0 
is the equation in line coordinates of a conic, the envelope of the line which cuts 
harmonically the conics 
(a, b, c, f g, h ~$x, y, z) 2 = 0, 
(a', P, c', f, g', h'\x, y, zf = 0 ; 
and if a, b, &c., a', &c. have the values before given to them, then the coefficients 
of the equation are 
be' + b'c -2ff «£{-£» + 4«' <4 + Z') ( v 3 + £ 3 ) + (16«' - 2Z 2 - 21' 2 ) Ç V Ç, 
ca' + c'a -2gg' =v{~ V 3 + 4IV (Z + V) (£ 3 + f ) + (16ZZ' - 21 2 - 21' 2 ) ij n Ç, 
ab' + a'b - 2 hh' =£{-?+ 4,11' (l + V) (f 3 + f) + (1611' - 2l 2 - 21' 2 ) Ç V Ç, 
gh' + g'h — af - af= f {(Z 2 + l' 2 ) (f + f + £ 3 ) + (21 + 21' + 8№) Ç V Ç] + (1 + 4ZZ' (l +l')) 
hf + hf- bg' -b'g=y {(l 2 +1' 2 ) (£» + v * + £ 3 ) + (21 + 21' + 81H' 2 ) &Ç} + (1 + 411' (l +1')) Ç 2 ?, 
ff +f9 ~ ch' -c'h = Ç {(l 2 + l' 2 ) (f 3 + f + £ 3 ) + (21 + 21' + 81H' 2 )+ (1 + 4ZZ' (l +1')) £y ; 
and we thence obtain 
(be' + Pc - 2ff,.., gh' + g'h - af - a'f .. ££ v , Ç) 2 = 
~(£ 3 + 7? 3 +£ 3 ) 2 
+ ( Z 2 + Z' 2 + 16ZZ') (£ 3 + 7? 3 + £ 3 ) ÇgÇ 
+ (6Z + 6Z' + 24Z 2 Z' 2 ) ^fÇ 2 
+ (4 +16(Z 2 Z , + ZZ ,2 ))(t 7 3 ^+^ 3 + |Y), =0