COVARIANTS OF PLANE CURVES. 
1197 
in terms of (Xg, &c., the coefficient of the highest power of y in a contra variant is 
a solution of = 0. 
The proof just given is connected with the theory of eduction, but it is perhaps 
more simple to notice that the relation 
— 77 — y — 0 
must hold not only for the original coordinates, but also for the coordinates in the 
general homographic transformations, and hence to deduce the dual relationship from 
the essential principles of this paper. 
That is, if we make infinitesimal homographic transformation for both tt, and 
a, y, indicated by 
^ IT 1 
f + B^tt' ~ tt' “ A|' + Btt' + 1 ’ 
« 7 1 
«' + Di 7 ' Gu + Dy' + 1 ’ 
as in Art. 2, 
then, in consequence of 
— TT — y = 0, 
Bj^Tr'ci' + Di^'y' = tt' (Ca + Dy') fi- y' (A^' + Btt'). 
and 
= C, A = Di, D + B = 0. 
Or A is replaced by (C or by) B^, and B^ by (D^ or by) A, while B is replaced by (D or 
by) - B. 
Hence the conditions derived from A and B^ are interchanged, while those derived 
from B remain unaltered (Art. G). That is and £1^ are interchanged, and fig is 
unaltered. 
20 . Relations of the differential invariants, and the solutions of GL-^f = 0 and 
Gl^f=Qto the ordinary invariants of a curve. 
Differential invariants appear in the processes of eduction, that is ultimately as the 
result of differentiation, in three ways. Firstly, in a series of differential invariants 
only; secondly, from a series of solutions of Gl-^f = 0, i.e., a series of differential 
invariants for change of coordinates or semi-invariants ; and thirdly, from the series 
which we have called matrices, which are semi-invariants in the correlated system of 
coordinates. 
The differential equation to a curve involves differential invariants only. If we 
find the series of first integrals (in the next section I illustrate, on the cubic, the 
general method of finding this series), we may form from among them the functions 
of differential invariants, of semi-invariants, and of matrices which are constant along 
the curve. These functions may be properly called (each in its proper sphere, as 
homographic or Cartesian) differential expressions for the invariants of the curve. 
It is perhaps worthy of remark that in showing the mode of integrating the 
