PROFESSOR CAYLEY ON ABSTRACT GEOMETRY. 
59 
Qualities, Notation, &c. Article Nos. 45 to 55. 
45. A homogeneous function of the coordinates [x, >/,-■•) is represented by a notation 
such as 
(*Jx, y,. ..) (,) 
(where (*) indicates the coefficients and ( • ) the degree), and it is said to be a quantic ; 
and in reference to the quantic the quantities or coordinates ( x , y, . . .) are also termed 
facients. More generally a quantic involving two or more sets of coordinates, or facients, 
is represented by the similar notation 
y>- • y,. • 
46. The quantic is unipartite, bipartite, tripartite, &c., according as the number of 
sets is one, two, three, &c. ; and with respect to any set of coordinates, it is binary, ter- 
nary, quaternary, . . . (m + l)ary, according as the number of the coordinates is two, three, 
four, or to+ 1 ; and it is linear, quadric, cubic, quartic, . . ., according as the degree in 
regard to the coordinates in question is 1, 2, 3, 4 . . . 
47. A quantic involving two or more sets of coordinates, and linear in regard to each 
of them, is said to be tantipartite ; or, in particular, when there are only two sets, it is 
said to be lineo-linear ; we may even extend the epithet lineo-linear to the case of any 
number of sets. 
48. Instead of the general notation 
(*)(x,y,.. y\ ...) (:) . .. 
we may write ' 
(a, . . .)(,t, y, . . .y j '(x , y , . . .y , . . . , 
where the coefficients are now indicated by (a, . . .), and the degrees are y., yJ , . . . 
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 ft . . . x' a 'y '^'. . . this 
numerical coefficient is 
M lx W ^ • • • 
= M a ' [PY--' 
50. It is sometimes convenient not to introduce these numerical multipliers, and we 
then use the notation 
0, • • y , . . .fix', ij, . . .y . . . , 
or 
(a, . . .yx, y, ... r ) yx', ij, . . yy . . 
In particular {a, b, c^x, y)' 2 , (a, b, c, dyx, y ) 3 &c. denote respectively 
airN-bxyNcy 1 , 
ax 3 + 3 bx 2 y + 3 cxy 1 -}- dy 3 , 
&c. ; 
] 2 
