COVARIANTS OF PLANE CURVES. 
1173 
include tlie integrals of the differential-invariantal equations, provided these integrals 
are rational, integral, algebraic functions. 
Having found a covariant curve, its polars, as well as its ordinary covariants, will 
be included among the differential-covariant curves. 
To find the degree of contact with the standard curve, we substitute for the 
covariant ^ — x = h, rj — y = y-Ji + ! + &c., if the equation is identically 
satisfied by yi = y^, —Vo, • • • , as far as the coefficient of the contact is of the 
order. Equating to zero the coefficient of we obtain the condition for contact 
of the n order. Since this condition is generally capable of being satisfied, and 
since the relation is unaltered by homographic transformation, the coefficient of /i” ^ is 
a differential invariant. 
The first section of this paper investigates the conditions which must be satisfied in 
order that (except as concerns a factor), the form of a function may be unaltered in 
consequence of any homographic transformation whatever, and finds the form of the 
factor. 
The results are given in the equations (15). 
In the second section, the relations of certain partial differential operators fitg, 
and fig with one another, and djdx are considered. A method of eduction of 
covariants is established, and a method of development of a covariant from a source 
or matrix, as well as a process of eduction of matrices. The matrices are shown to be 
dependent upon differential invariants and two fundamental matrices and Lg, and 
from the mode in which these enter, the order of the covariant can be predicted. 
This section prepares the way for the application of the theory in the two last 
sections. 
The third section deals with the dual reciprocal relationship) between the operators, 
and the deduction of reciprocal and contra variant functions. 
The fourth section contains applications to the cubic of the methods which have 
been considered. The equation to the osculating cubic is determined when the cubic 
is non-singular, nodal, and cuspidal, and the differential equation in each case found. 
The fifth section takes a new view of the subject. It is shown that the ])osition of 
any point which is “ homographically persistent ” with regard to the standard curve 
(which, for example, is a point of intersection of two curves which are each 
covariant) is determined by the ratios of three differential-invariant functions. 
These can therefore be considered as invariantal coordinates of the point. It is also 
shown that there are similarly invariantal coordinates of a line, and the relations of 
these line and point coordinates are investigated, showing that the general methods of 
higher geometry are applicable to the novel system of coordinates. 
The condition that the p)oint should lie on a covariant curve is now given in terms 
of the invariantal coordinates by an equation in which the coefficients are the 
differential invariants which determine the character of the curve. For this reason I 
call it the intrinsic invariantal equation to the curve. 
