33. General method of finding the form of the intrinsic invariant equation to a curve of any order.
It will not be necessary to pass through all the steps which I have taken in developing this theory of intrinsic invariant equations.
If we have an equation to a covariant curve, say , and if ,
or , is the corresponding intrinsic invariant equation where and stand
for and respectively, then the relations between and are essentially of the character of a homographic transformation. Hence, if for points near the
origin in the covariant curve is written and is put for , for
while in the intrinsic invariant equation is written where is put for , and if , , , &c., are written for the invariantive portions of , , , &c., then , , differ from the values of , , , &c., by factors of
the character which we have considered in the earlier part of this paper.
Thus , where and stand for the expressions corresponding to those written and in (2).
In general , and at the origin
,.
Therefore,
, or ,
therefore
, where ,
and, comparing with (33), generally
.
Hence the value which, near the origin, is to replace is
(59).
Taking the general equation of the intrinsic invariant of conics as
,
and substituting for and , the values shown above, we find all the invariant coefficients vanish except , and thus