L 1171 ] 
XXIL On the Differential Covariants of Plane Curves, and the Operators employed 
in their Develojjrnent. 
By R. F. Gwyther, M.A., Fielden Lecturer in Mathematics, Oivens College, 
Manchester. 
Communicated hy Professor Horace Lamb, F.R.S. 
Received Apidl 14.—Read May 4, 1893. 
The subject of the earlier parts of this paper is immediately allied with the late 
M. Halphen’s investigations in his Thesis, “ Snr les Invariants Dilferentiels,” 
(Gauthier-Villars, Paris, 1878), and in his Memoir, “Snr les Invariants Difteren- 
tiels des courbes gauches” (‘Journal de I’Ecole Polytechnique,’ vol. 28, cahier 47, 1880), 
and in the latter paper the word covariant is applied to a function having the 
character of the functions which are generally treated of in this paper. 
The connection is perhaps greater with a previous, and, I think, less familiar 
paper of the same author, “ Sur la recherche des points d’une courbe algebrique 
plane, qui satisfont a une condition exprlmee par une equation differentielle 
algebrique, et sur les questions analogues dans I’espace ” (‘ Journal de Math.,’ 3rd 
series, vol. 2, 1876). I am not able to point out any paper which touches upon 
the actual investigation attempted in this paper, and the mode of investigation can 
not claim any resemblance to any of Halphen’s work, and does not offer in itself any 
element of novelty. 
The subject of Differential Invariants has been further treated by Sophus Lie, 
“ Uber Differentialinvarianten ” (‘Mathem. Annalen,’ vol. 24, p. 564, 1884), and by 
Sylvester, in his lectures “On the Theory of Reciprocants ” (‘Amer. Journ. Math.,’ 
vols. 8, 9, and 10, 1886 and 1887). 
The literature on the cognate subject of Reciprocants has become extensive, several 
papers by Elliott, Forsyth, Hammond, Leudesdorf, and Rogers have appeared in 
the ‘Proceedings of the Mathematical Society’ (vols. 17, 18, and 19), and in the 
‘Messenger of Mathematics’ (vols. 17, 18, and 19). Of these papers, those of 
which the analytical method is most similar to mine are by Mr. Forsyth, “Homo¬ 
graphic Invariants and Quotient Derivatives” (‘Mess, of Maths.,’ vol. 17, No. 10, 
p. 154, 1888); “On a class of Functional Invariants” (‘Phil. Trans.,’ A, 1889, 
p. 71). I may also mention, as cognate matter, papers by Mr. Elliott “ On the 
Interchange of Variables in Certain Linear Differential Operators” (‘Phil. Trans.,’ 
7 K 2 10.1.94 
