1210 
MR. R. F. GWYTHER ON THE DIFFERENTIAL 
where X and Y have been written for [xjv and \lv respectively. Here X and Y are the 
variables as we pass along a curve, and cPYjclX.^, &c., are to be found from the intrinsic 
invariant equation to the curve which we may write f (X, Y) = 0 . 
The relations between tt and X, Y are of the form of a homographic transformation, 
and therefore any of the functions of &c., which we know to be differential 
invariants, will only differ from the similar functions of cV'Yjd'X}, &c., by a factor, of 
Avhich we know the form, which will contain u^, Lq, and ctg. 
It will not be possible to express the value of a differential invariant at (X, Y) in 
terms of X, Y and differential invariants at the origin only, unless the said differential 
invariant is absolute, but the invariantal coordinates of a point of homographic 
singularity will be found by applying the condition immediately to f {Y, Y) = 0 . 
Obviously, if the intrinsic invariantal equation is regarded as the equation to a 
curve in the ordinary sense, that curve will have the same homographic singularities 
as the covariant from which it is derived, and the coefficients in the equation are 
differential invariants. Hence we are led to the simpler method of finding the 
conditions for the existence of homographic singularities in terms of differential 
invariants, namely, the immediate application of the ordinary theory of forms to the 
intrinsic invariantal equation treated as a ternary quantic. For this reason, the equation 
in this form may be regarded as the canonical form of the equation to the curve, 
since the coefiScients are the differential invariants which characterise the curve. 
I proceed to illustrate this upon the conic and cubic. 
To obtain the intrinsic invariant equation, we may omit all terms which are not 
purely invariant from the coefficients in the covariant form of the equation and 
substitute — u^u-^Xjv or — u^u~^Y for tt, and u^u-ixlv or for f. I shall retain 
X, fl, V. 
Thus, the intrinsic invariant equation to the osculating conic is 
Xv [jd' = 0 
(57). 
32. Illustrations in the case of the cubic. 
The intrinsic invariantal equation to the general non-singular cubic is (omitting a 
factor containing 
u-^X^ (VgX + U^p.) + (Uy^X — Vg/x + Uot') (Xr- -f- f) = 0 . . . (58), 
and that of its Hessian is consequently 
Q>ufY f-\-2uf\] 2ufUqX-\-2\Jq^ix —VgV, 2Ur-^X — Vg/x-|-2Uyr' 
2uf\Jq^X-\-2\] —Ygv, 2Uy^X — 6YgjU- + 2X1^^’, — VgX -|- 2Uy/x 
2 U/X — Yg/x -f 2'\Jqv, — VgX + 2 U 7 /X, 2 U 7 X 
= 0. 
