COVARIANTS OF PLANE CURVES. 
1195 
valent function of 2 / 3 , y^, kc., and Halphen has shown the relationshij) between 
invariant functions. 
Our concern now is with covariants and especially with matrices. 
Obviously, the conditions to which a function is subject, in order that it may be a 
covariant function, are of the same form, whether the coordinates are point- or line- 
coordinates, and therefore the matrix of the reciprocal of a covariant, rega,rded as a 
function of Yg, Yg, &c., is a solution of XlgF = 0 . It is proposed, in the first place, to 
deduce the equation to the reciprocal from the equation to the original covariant, 
and, found in this way, the coefficient which we have called the matrix will be a 
function of y^, &c., and as a function of these quantities it will not be a solution of 
£lj= 0 . 
To find the equation which it does satisfy, we will simplify the writing by taking 
the point x, y as origin, so that we may write tt for tt — p and ^ for ^ — x. The 
equation to the covariant is a rational integral function in tt and 
Let y and a be the corresponding coordinates on the reciprocal curve with the 
relation 
a.^ — TT — y=0. 
where the third letters which are usually inserted to make the equation homogeneous 
are omitted as unnecessary and, for the purpose, undesirable. 
Following the usual course (Salmon, ‘Higher Plane Curves,’ 2 nd ed., p. 73), the 
equation to the covariaut is now made homogeneous by using this equation, and the 
discriminant of the resulting equation, considered as a binary quantic in tt and f, 
equated to zero, is the required equation to the reciprocal. 
It is not necessary that the whole of this process should be performed, for all that 
is necessary is to obtain the coefficient of the highest power of y in the resulting 
equation. We obtain it by putting a = 0 , and hence the coefficient of the highest 
power of y is the discriminant of the highest grovp of homogeneous term in the 
original ecpiation in tt and treated as a binary quantic. 
This coefficient is, of course, found as a function of y^, y^, , and when replaced 
by the corresponding value in terms of Yg, Yg, it will he the matrix of the reciprocal 
and a solution of XlgF = 0 . 
From the mode of formation of the coefficient of the highest power of y, it appears 
that, as a function of y^, ^ 3 , • • • > it is an invariant for changes of Cartesian coordi¬ 
nates, and therefore, as has been remarked in finding the condition resulting from 
the coefficient in the original homographic transformation ( 12 ), it will satisfy the 
equation giffi= 0 . 
Before continuing the more general consideration of this relation we will illustrate 
this on the general osculating conic and its reciprocal. 
The terms of the highest order in the covariant conic are 
— ag^) tt® fi- aia^n^ + 
7 N 2 
