PROFESSOR CAYLEY ON ABSTRACT GEOMETRY. 
bo 
but (a, b, cjx, y)~, (a, b, c, (l^x, ?/)'\ &c. denote 
ax 2 +bxy-\-cy 2 , 
ax 3 + bx-y + cxy~ + (hf, 
&c., 
and so (a, b, c,f,g , hfx, y, zf and ( a , b, c,f, g, h'Jx, y, z ) 2 denote respectively 
and 
ax- 4 by- 4 cz~ 4 2 fyz 4 2 gzx 4 2 hxy 
ax 2 -\-by- + cz 3 4 fyz 4 gzx 4 bxy. 
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 — . . .fx, y, z) 3 
= («, b, c, /, g, Ji, i,j, k, IJoc, y, f 3 
= ax 3 4 by 3 4 cz 3 
+ 3 (ff~ + 9 Z ' X + Wy 4- lyz 1 +jzx 2 4- kxy-) 
and the like in other cases. 
52. A onefold relation between the coordinates is expressible by means of an equa- 
tion of the form 
(*}>, y, .. .) (O =0. 
53. The expression “ an equation ” used without explanation may be taken to mean an 
equation of the form in question, viz. the equation obtained by putting a quantic equal 
to zero ; the quantic is said to be the nilf actum of the equation. We may consequently 
say simply that a onefold relation between the coordinates is always expressible by an 
equation. 
54. It is frequently convenient to denote the quantic or nilfactum by a single letter, 
and to use a locution such as “ the equation U==(43 Ca V • -f ] = 0^ ,, which really means 
that the single letter U stands for the quantic Vi • • *) ( ^ so we are afterwards 
at liberty to write U = 0 as an abbreviated expression for (*fx, y, . . .j^O. We may 
also speak of the equation or function U = 0, meaning thereby the equation U = 0, or 
the function U. 
55. A /r-fold relation between the coordinates is (as has been shown) equivalent to a 
system of k or more onefold relations ; each of these is expressible by an equation U=0, 
and the /r-fold relation is thus expressible by a system of k or more such equations. 
Representing by ((U)) the system of functions which are the nilfacta of these equations 
respectively, the /r-fold relations may be represented thus, ((U)) = 0 ; or more completely, 
the relation being /1-fold, and the number of equations being =s , by the notation 
((U>)(Mold) = 0. 
