Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/1336

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MR. R. F. GWYTHER ON THE DIFFERENTIAL

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