COVARIANTS OF PLANE CURVES. 
1189 
where 
^ denotes deduction by the operator 
\ 
?5 
) J 
J ) 
^3 j 
in which we can easily trace the relations (16) 
— X2'g£^.')j 
According to this mode of procedure is the matrix from which the other 
coefficients in the covariant are develo23ed. It is necessary that should be 
a homogeneous isobaric function of the differential coefficients satisfying the differential 
equation Ilj/’ = 0. It is also essential that it shall not be an invariant as then no 
development ensues. Taking the solution of = 0, including only differential 
coefficients up to the fourth order, we obtain ~ from it the 
general covariant of conics may be developed. 
13. General solution of — 0, and theory of eduction of matrices. 
Confining ourselves to the coefficients in the covariant functions, the equations (18) 
may be written 
- A - 2/3 + d-) 
dx dx 
_d _f 
dx dx 
d d 
do: dx 
d d 
X, ^ ^3 ~ 2/2^3 
ff 
. . . . (18 a). 
Hence, if is a homogeneous isobaric function of the differential coefficients such 
that = 0, then 
^2 - cG) </>, 
and if the weight and degree of (f) are such that 2w = cC, then dcfiflx also satisfies 
the same differential equation, and it is homogeneous and isobaric. 
The solutions of fig /= 0 up to the fourth order are 
2/2 and By^y^ - Ayf ■ 
from these we form, so as to satisfy 2w — d^ — 0, 
^ = {32/j2/r - ^y-if 2/2 
