413] 
A MEMOIR ON ABSTRACT GEOMETRY. 
465 
48. Instead of the general notation 
we may write 
(a, y, ...Y,..., 
where the coefficients are now indicated by (aand the degrees are g, y, ... 
49. In the cases where the particular values of the coefficients have to be attended 
to, we write down the entire series of coefficients, or at least refer thereto by the 
notation (a,...); and it is to be understood that the coefficients expressed or referred 
to are each to be multiplied by the appropriate numerical coefficient, viz. for the term 
x a y p ... x' a 'y'^... this numerical coefficient is 
№ o y... 
[a]» 
50. It is sometimes convenient not to introduce these numerical 
we then use the notation 
y, ...y (x, y', ...y..., 
or 
(a, ...\x, y,...>(V, y',. 
multipliers, and 
In particular (a, b, c§x, y) 2 , (a, b, c, d\x, y) s &c. denote respectively 
ax 2 4- 2 bxy + cy 2 , 
ax 3 + 3 bx 2 y + Sexy 2 + dy 3 , 
&c. ; 
but (a, b, c\x, y) 2 , (a, b, c, d\x, yf, &c. denote 
ax 2 + bxy + cy 2 , 
ax 3 + bx 2 y + cxy 2 + dy 3 , 
&c., 
and so (a, b, c, f, g, li$x, y, z) 2 and (a, b, c, f, g, h^x, y, z) 2 denote respectively 
ax 2 -f by 2 + cz 2 + 2fyz + 2gzx + 2bxy, 
and 
ax 2 + by 2 + cz 2 + fyz + gzx + hxy. 
51. To show which are the coefficients that belong to the several terms respectively, 
it is obviously proper that the quantic should be once written out at full length ; 
thus, in speaking of a ternary cubic function, we say let U= (a, ...][x, y, z) 3 
= (a, b, c, f, g, h, i, j, k, l\x, y, z) 3 
= a« 3 + by 3 + cz 3 
4- 3 (fy-z + gz 2 x + hx 2 y 4- lyz 2 +jzx 2 4- kxy 2 ) 
+ 6lxyz, 
and the like in other cases. 
C. VI. 
59