CURVE. 
785] 
467 
in a point. And we thus see how the theorem extends to curves, their points and 
tangents : if there is in the first figure a curve of the order to, any line meets it 
in to points ; and hence from the corresponding point in the second figure there must 
be to the corresponding curve on tangents; that is, the corresponding curve must be 
of the class on. 
Trilinear coordinates (to be again referred to) were first used by Bobillier in the 
memoir, “ Essai sur un nouveau mode de recherche des propriétés de l’étendue ” 
(Gerg., t. XVIII., 1827—28). It is convenient to use these rather than Cartesian coordi 
nates. We represent a curve of the order to by an equation (*]£«, y, z) m = 0, the 
function on the left-hand being a homogeneous rational and integral function of the 
order on of the three coordinates (x, y, z) ; clearly the number of constants is the 
same as for the equation (*]£#, y, l) m = 0 in Cartesian coordinates. 
The theory of duality is considered and developed, but chiefly in regard to its 
metrical applications, by Chasles in the “Mémoire de géométrie sur deux principes 
généraux de la science, la dualité et l’homographie,” which forms a sequel to the 
“ Aperçu historique sur l’origine et le développement des méthodes en géométrie ” 
(Mem. de Brux., t. xi., 1837). 
We now come to Plücker ; his “ six equations ” were given in a short memoir in 
Or elle (1842) preceding his great work, the Theorie dev algebraischen Gurven (1844). 
Plücker first gave a scientific dual definition of a curve, viz. “ A curve is a locus 
generated by a point, and enveloped by a line,—the point .moving continuously along 
the line, while the line rotates continuously about the point ” ; the point is a point 
(ineunt) of the curve, the line is a tangent of the curve. 
And, assuming the above theory of geometrical imaginaries, a curve such that on 
of its points are situate in an arbitrary line is said to be of the order m; a curve 
such that oi of its tangents pass through an arbitrary point is said to be of the 
class n ; as already appearing, this notion of the order and the class of a curve is, how 
ever, due to Gergonne. Thus the line is a curve of the order 1 and the class 0 ; 
and corresponding dually thereto, we have the point as a curve of the order 0 and the 
class 1. 
Plücker moreover imagined a system of line-coordinates (tangential coordinates). 
The Cartesian coordinates (x, y) and trilinear coordinates (x, y, z) are point-coordinates 
for determining the position of a point ; the new coordinates, say (£, y, Ç), are line- 
coordinates for determining the position of a line. It is possible, and (not so much 
for any application thereof as in order to more fully establish the analogy between 
the two kinds of coordinates) important, to give independent quantitative definitions 
of the two kinds of coordinates; but we may also derive the notion of line-coordinates 
from that of point-coordinates ; viz. taking %x + yy + Çz — 0 to be the equation of a 
line, we say that (£, y, Ç) are the line-coordinates of this line. A linear relation 
+ by + c£= 0 between these coordinates determines a point, viz. the point whose 
point-coordinates are (a, b, c) ; in fact, the equation in question ci% + by + c£ = 0 expresses 
that the equation %x + yy+Çz= 0, where (x, y, z) are current point-coordinates, is 
satisfied on writing therein x, y, z — a, b, c; or that the line in question passes through 
59—2