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The subject of the earlier parts of this paper is immediately allied with the late M. Halphen’s investigations in his Thesis, “ Sur les Invariants Différentiels,”(Gauthier-Villars, Paris, 1878), and in his Memoir, “ Sur les Invariants Différen-tiels des courbes gauches” (‘ Journal de l'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 algébrique plane, qui satisfont à une condition exprimée par une Equation différentielle algébrique, et sur les questions analogues dans l’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.

XXII. On the differential covariants of plane curves, and the operators employed in their development