ME. A. CATEET’S MEMOIE ON CEETES OP THE THIRD ORDER. 
433 
5rA-ff/=?(2f'(f+,*+r) + 4i(l+2/»)?,2)+(l+8Z>T, 
7j/'-ij = ,(2P(P+,»+2’(+4Z(l+2P)|,2)+(l + 8P)rP, 
/y-a=j(27’(P+,*+i:>)+47(l+27*)?,j)+(l+8P)p,». 
and after all reductions, 
ohc — af^ —hg^— elf + '^fgli 
= [-/(^^-^^+^)-{-(-l+4Z%rT=(PU)^ 
or the condition in order that the conic may break up into a pair of lines is PU = 0. 
25. The following formulae are given in connexion with the foregoing investigation, 
but I have not particularly considered them geometrical signification. The lineo-polar 
envelope of an arbitrary line ^A’+??y+^ 2 := 0 , with respect to the cubic, 
of -\-y'^ + 2:^+6 Ixyz = 0 , 
has been represented by 
(«, h, c,f, g, hjx, y, zf=0-, 
and if in like manner we represent the lineo-polar envelope of the same line, with respect 
to a syzygetic cubic, 
of -^y^-\-z^-\-% Uxyz = 0 , 
by 
{a!, b', c\f\ g\ h'Jx, y, 2 )^= 0 , 
then we have 
(f{bc-f)-\-b'{ca-g^)-\-d{ab-]f)+2f{gli-af)+2g'{1if-hg)-\-2h!{fg-cJi) 
=(P+2/^)(f4-^*+r7 
which may be verified by writing V — l^ in which case the right-hand side becomes as it 
1 + 2 ? 
should do, 3(PU)^ If ?= i®? if syzygetic cubic be the Hessian, then the 
formula becomes 
which is equal to 
+12Z(2-{-57Z^+168/«+16^'«)?Vr 
^4|qu'-24S.PU'|. 
26. The equation 
i^c’+b'c—2ff',..gh'+g’h—af — af,..'X^, lY=0 
is the equation in line coordinates of a conic, the envelope of the line which cuts harmo- 
nically the conics 
(«, h c, /, g, h Xx, y, 2 )"= 0 , 
(«', b', d,f, g', A'X.r, y, zf^O; 
MDCCCLVII. 3 L 
