1172 
MR. R. F. GWYTHER OX THE DlFPEREiTTIAL 
A, 1890, 2 ?. 19), by Professor L. J. Rogers, “ Conjugate Annihilators ” Mess, of 
Maths.,’ vol. 18, p. 153-158, 1889), and by Major MacMahox, “ On Multilinear 
Partial Differential Operators ” (‘ Lond. Math. Soc. Proc.,’ vols. 18 and 19, 1887 
and 1888). 
Starting with the mode of formation of Differential Co variants, I attempt to give 
greater j^rominence to the geometrical character of the j^roblem of invariancy, which 
seems to have been j^rominent in Halphen’s view of the subject, and hope that while 
tlie results are jaartly coextensive with the work of Professor Sylvester and others, 
they may still be complementary, and may be found applicable in investigating the 
characteristics of higher jdane curves. 
§ 1. General Statement of the Frohlem Considered, as far as relates to 
Plane Curves. 
If a plane curve is transformed by the general homographic transformation, the 
transformed curve remains of the same order and possesses the same perspective 
singularities. If we introduce no other complications we may say that Hxlphex’s 
Differential Invariants indicate projoerties which, if they exist at any 2 )oint of the 
original curve, exist at the same point of the transformed curve. Any function of 
the differential coefficients at any jioint of the curve which, a definite factor 
retains its form under the general homograjihic transformation is a differential 
invariant. I shall write these differential coefficients y-^, y^, 1/3 ... . In this |)aper I 
introduce a curve, the current coordinates of which I write rj, and of which the 
equation consists of an algebraic relation between rj and y ,y\, y^, y^, • • • • {x, y 
being the coordinates of a point on the curve which I shall call the standard curve). 
Then, if this equation is such that, a definite factor 'pres, its form is unaltered under 
the general homograjihic transformation, I call it a differential covariant of the 
standard curve. 
Besides the sense in which a differential invariant has here been spoken of as 
indicating a jiersjiective singularity at a point in a curve, it may, equated to zero, be 
the differential equation to a curve, each point of which has that singularity. (I here 
consider a sextactic point and other points of similar characteristics, as singular 
jioints for homographic projection.) 
A conic is one of the curves which, by a general homographic transformation, can 
be transformed into itself, so that any one arbitrary point can be replaced by any other. 
Hence, if V = 0 is the differential condition for a sextactic jjoint, it is the differential 
invariant equation of conics, and if f[^, y, y^, y^, y^, yj = 0 is a covariant equa¬ 
tion of the second degree, such that when ^ = x and 7 ^ — y we have = yg, 
yg = yg, y^ = y^, it is the equation to the conic osculating the standard curve at 
(a;, y). Also regarding y as the variables and x, y, yi . . . as arbitrary constants, 
it is the complete integral of V = 0 . Generally the differential covariants will 
