258 GENERAL THEORY OF LINEAR SERIES Book III 
Chap. I 
order N in a space of r dimensions that groups of the series corre 
spond to hyperplane sections of the curve. If it contain a series 
of dimension r and order N composed of an involution of grade y, 
there is a y-to-one correspondence between the plane curve and a 
curve of order ¡i in a space of r dimensions, the groups of the 
given series corresponding to the hyperplane sections of the space 
curve. 
Suppose that we are in a space of three dimensions, and on 
a given curve we choose such a point that a line through it 
meeting the curve once will not necessarily meet the curve 
twice. There must be such points, for there is a three-parameter 
system of lines meeting the curve once, and only a two-para- 
meter meeting it twice. This point will be the vertex of a cone 
whence the curve is projected simply on a plane. 
More generally, take a curve in a space of r dimensions. It 
is not possible that every space of r—2 dimensions that meets 
the curve once should meet it twice. We see, in fact, that the 
number of parameters giving the spaces of r— 2 dimensions 
through a point of the space of r dimensions is the number of 
sets of r— 2 linearly independent straight lines through the 
point, less the number through the point in a space of r—2 
dimensions, and is 2(r—2). The number of parameters giving 
the spaces of r—2 dimensions through a line in r dimensions 
will similarly be 2(r—3), The lines through a point of a curve 
which meet it again form a one-parameter system. The spaces 
of r—2 dimensions through a point on the curve which meet it 
again depend on 2r— 5 parameters, which is less than the para 
meter number of the system of all spaces of r—2 dimensions 
through that point. Hence we may find plenty of spaces of r—2 
dimensions which meet the curve once, but no more. 
Let P be a point of the curve; V v V 2 ,..., V r _ 2 independent points 
of a space of r—2 dimensions through P which does not meet 
the curve again. Then a general space of r—2 dimensions 
through the space of r—3 dimensions determined by V x , V 2 ,..., V r _ 2 
which meets the curve once, will not, automatically, meet it 
again; so that the curve may be simply projected from the space 
of r—3 dimensions upon an arbitrary plane, for in r dimensions, 
a plane and a space of r—2 dimensions meet once. This same 
thing can be done even when some of the points Y i he on the 
given curve, 
plane curve i 
of r—3 dime] 
and the plai 
coordinates J 
and space cu 
X 0 — x 0 
where the p< 
one greater 1 
More gene 
another of ai 
relation is bi 
Definition 
be obtained 
contained oi 
example of 
lies complet 
a cubic in i 
space which 
in fact, thal 
points upon 
curve with 
of the fourt 
less would 1 
a conic, wh; 
Suppose 1 
which has t 
planes is u 
dimension ( 
contained i 
0 be a pom 
(0,0,0,...,!