'MR. A. CAYLEY’S MEMOIE ON CUEVES OE THE THIED OEDEE. 
435 
We find as the equation of the locus, 
(1 — 1 + 6 /( 1 — 30^® — 1 6 -{-y^ -\-z^)xyz 
+6/^(1 — 104/^ — 32 f)x^y‘^z‘‘ 
-2(1+ w)\fz ^ + s V +^y ) = 0, 
where the function on the left-hand side is the octicovariant of my Third Memou, 
the covariant having been in fact defined so as to satisfy the condition in question. 
And I have given in the memoir- the following expression for ©;;U, viz. 
0^^U=(l-16^^-6/«)U^ 
+(6/ )U.HU 
+(6P XHU)^ 
—2(1+8 l^yiyfz ^ + ^ V + x^y^). 
Article Nos, 29 to 31 . — Formulae for the intersection of a cubic curve and a line. 
29. If the line %x-\-riy-\-'Cz=() meet the cubic 
xi^ -\-y^z ^ + 6 Ixyz = 0 
in the points 
(^n y\i ^1)5 (^2? y2? ^2)? ('^3? 3/3? ^3)? 
then we have 
^,.372^3 : y,y.,y^ : z,z^z^=n'—V : T — f : f — 
It will be convenient to represent the equation of the cubic by the abbreviated notation 
(1, 1, 1, IX.X, y^ zX=0; we have the two equations 
(1, 1, 1, Vfx, y, zy=o, 
^x+rjy-\-lz =0 ; 
and if to these we join a linear equation with arbitrary coefficients, 
ax-]-i3y+yz=0, 
then the second and thii’d equations give 
x\y:z=^l — yn'. ar,—^^; 
and substituting these values in the first equation, we obtain the resultant of the system. 
But this resultant will also be obtained by substituting, in the third equation, a system 
of simultaneous roots of the fii’st and second equations, and equating to zero the product 
of the functions so obtained*. We must have therefore 
( 1 , 1 , 1 , IX^l—yyi, ci'/i—ft'iy={aXi-^(5y^-{-yz,)(ax^-{-^y.,+yz^){aX3~\-(iys+'yZ3); 
and equating the coefficients of a^, /3®, y*, we obtain the above-mentioned relations. 
30. If a tangent to the cubic 
F-\-y^+z^-{-Qlxyz=:0 
* This is in fact the general process of elimination given in Schlapli’s Memoir, “ Ueber die Eesultante 
einer Systemes mehrerer algebraischer Glleichimgen,” Vienna Trans. 1852. 
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