COVARIANTS OF PLANE CURVES. 
1177 
coefficients (independent of their order), and that the difference of the weight of the 
coefficient of a term and the index of the power of ^ — x which it contains is uniform 
throughout. 
These determine the character of the general equation of a curve of the cP degree, 
osculating a given curve, the order of the highest differential coefficient being 
d{d-f- 3)/2 — 1. 
In its differential equation the degree is greater than this value by d, the weight 
by ^d (c? + 3), and the order by 1. 
If the curve has any imposed condition, such as the possession of a double point, 
these criteria are altered, 
2. Determination of Differential Covariants of Plane Curves. 
Abandoning the condition, as yet imposed, of considering the equation to an oscu¬ 
lating curve, the question proposed is to examine the conditions, both simple and 
in the form of linear partial differential equations, which fff — x, rj — y, y^, y^,. . .) 
must satisfy in order that it may become 
^-cl _x,7]-Y, Y„ Y„...) . . . (36) 
under the general linear homographic transformation stated above. 
The condition that dj. and lu are necessarily connected with d, by the relations 
just found, will not be required, but they wiU serve as criteria for recognising 
general cases. 
The simple conditions are exactly analogous to those found by Halphen for 
Differential Invariants (‘These,’ p. 21), and result from the same simple transfor¬ 
mations. 
1. From putting x = Y, ^ y = BgY, rj = we conclude, as before, that the 
function is homogeneous in y — y and the several differential coefficients of y. 
2. From putting x = A^X, ^ = A;^^, y = Y, y = y, we conclude that the weight 
of the coefficient of each power of ^ — x is uniform, and that this weight diminished 
by the index of the power of ^ — a? is uniform throughout the function. 
3. We have chosen the form of the expression, so that we draw no further 
conclusion from a mere change of origin of coordinates. The form of the multiplying 
factor as well as that of the function might have been found from the method which 
follows, and its correctness is established from it. " 
To obtain the linear partial differential equations which the functions must satisfy, 
we consider infinitesimal transformations. Being infinitesimal we may consider 
them separately. The transformations are four in number. The first two would 
apply to a general infinitesimal Cartesian transformation, and so the results are 
the partial differential equations suitable to that case. The other two only concern 
our more general transformation, and, as we shall see, the results have a different 
character from those of the first two. 
7 L 
MDCCCXCIII.—A. 
