MATHEMATICS: E. J. WILCZYNSKI 
249 
of which ji, yi, yz form a fundamental system of solutions, and of 
which Cy shall be said to be an integral curve. All other integral curves 
of (2), associated with different fundamental systems of solutions, are 
projective transforms of Cy. 
Since the coordinates yi, y^, y^ are homogeneous, only those combina- 
tions of the coefficients pi, p2, pz can be of interest for the geometry of 
the curve Cy which depend only upon the ratios yi '.yi '.y^. These 
combinations, the so-called seminvariants of (2), are all expressible as 
functions of 
F2= p2 - pi' - pi, Pz = p3 - 3pip2 + 2pi^ - pi" (3) 
and of their derivatives. The seminvariants of (2) are not altered if 
(2) is transformed by putting y = X(x)y where \(x) is an arbitrary func- 
tion oi x.^ 
Although the seminvariants depend only upon the ratios yi \yi \yz, 
they are still not adequate to represent the purely geometric properties 
of the curve Cy. The values of and Pz depend also upon the special 
parametric representation which has been chosen for Cy. We may 
change this parametric representation in the most general way by put- 
ting % = ^{x)^ where ^(x) is an arbitrary function of x. Those combina- 
tions of the seminvariants, called absolute projective differential invariants, 
which are left unaltered by all possible transformations of this sort, 
express intrinsic properties of the curve Cy. Moreover these properties 
are projective properties, since any projective transform of Cy may be 
regarded as an integral curve of (2). 
Every absolute projective differential invariant can be expressed as 
a quotient of two relative invariants. The simplest of these relative 
invariants is^ 
Q,=.F,-.lpi (3) 
The property of ^3 which justifies us in speaking of it as a relative 
invariant, is the following. Let us transform (2) by putting 
~x = ^{x), y = \(ix)y, (4) 
where ^(x) and \{x) are arbitrary functions of x. From the coefficients 
of the resulting differential equation between x and y let us form the 
quantity dz{x) according to the same rule which was used in forming 
6z from the coefficients of (2). We shall find^ 
^3®=^- (4) 
