1212 
MR, R. P. 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 ecj^uation to a covariant curve, say f {rr, = 0, and if F (X:p,:v) = 0, 
or F (X, Y) = 0, is the corresponding intrinsic invariant equation where X and Y stand 
for gjv and \jv respectively, then the relations between tt, ^ and X, Y are essentially 
of the character of a homographic transformation. Hence, if for points near the 
origin in the covariant curve tt is written afd + + • • • j and h is put 
for f, while in the intrinsic invariant equation Y is written A 3 /F + + A^/d + . . ., 
where h is put for X, and if aj, ag, a^, &c., are written for the invariantive portions of 
a. 2 , ttg, ofy, &c., then A 3 , Ag, A^ differ from the values of a 3 , ag, a,~, &c., by factors of 
the character which we have considered in the earlier part of this paper. 
Thus a,j = where and q stand for the expressions corresponding to 
those written g and X in (2). 
In general D = — ufufi and at the origin 
Therefore, 
therefore 
a« = — W3%5® k,„ or A,, = 
A3 = — 1 
A5 = - yfi 
A 7 = - 
As = - where Wg = Vg + 2^^5^ 
and, comparing with (33), generally 
K,^ = — ufiWn. 
Hence the value which, near the origin, is to replace Y is 
_ 43 _ uf {Id + JJfd + Wg/^s + . . . ).(59). 
Taking the general equation of the intrinsic invariant of conics as 
Ofd -f- dfiii + (^3 + ^ 3 ) "k + 0]^[xv Ov" =0, 
and substituting for X/v and /x/v, the values shown above, we find all the Invariant 
coefficients vanish except 6^, and thus 
