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The principal object of the present Memoir is the establishment of the partial differential equation of the third order satisfied by the parameter of a family of surfaces belonging to a triple orthogonal system. It was first remarked by Bouquet that a given family of surfaces does not in general belong to an orthogonal system, but that (in order to its doing so) a condition must be satisfied; it was afterwards shown by Serret that the condition is that the parameter, considered as a function of the coordinates, must satisfy a partial differential equation of the third order: this equation was not obtained by him or the other French geometers engaged on the subject, although methods of obtaining it, essentially equivalent but differing in form, were given by Darboux and Levy; the last-named writer even found a particular form of the equation, viz. what the general equation becomes on writing therein X = 0, Y = 0 (X, Y, Z the first derived functions, or quantities proportional to the cosine-inclinations of the normal). Using Levy’s method, I obtained the general equation, and communicated it to the French Academy. My result was, however, of a very complicated form, owing, as I afterwards discovered, to its being encumbered with the extraneous factor X2 + Y2 + Z2 ; I succeeded, by some difficult reductions, in getting rid of this factor, and so obtaining the equation in the form given in the present memoir, viz. ((A), (B), (C), (F), (G), (H))(δa, δb, δc, 2δf, 2δg, 2δh) —2((A), (B), (C), (F), (G), (H))(a̅, b̅, c̅, 2f̅, 2g̅, 2h̅) = 0: but the method was an inconvenient one, and I was led to reconsider the question. The present investigation, although the analytical transformations are very long, is in theory extremely simple: I consider a given surface, and at each point thereof take along the normal an infinitesimal length ζ (not a constant, but an arbitrary function of the coordinates), the extremities of these distances forming a new surface, say the vicinal surface; and the points on the same normal being considered as corresponding points, say this is the conormal correspondence of vicinal surfaces. In order that the two surfaces may belong to an orthogonal system, it is necessary and sufficient that at each point of the given surface the principal tangents (tangents to the curves of curvature) shall correspond to the principal tangents at the corresponding point of the vicinal surface; and the condition for this is that ζ shall satisfy a partial differential equation of the second order, ((A), (B), (C), (F), (G), (H))(dx , dy , dz )2 ζ = 0, where the coefficients depend on the first and second differential coefficients of U, if U = 0 is the equation of the given surface. Now, considering the given surface as belonging to a family, or writing its equation in the formr -r (x, y, z ) = 0 ( the lastr a functional symbol), the condition in order that the vicinal surface shall belong to this family, or say that it shall coincide with the surfacer + δr -r (x, y, z ) = 0, is ζ = δr /V, where V = √X2 + Y2 + Z2 , if X, Y, Z are the first differential coefficients ofr (x, y, z ), that is, of the parameterr considered as a function of the coordinates; we have thus the equation ((A), (B), (C), (F), (G), (H))(dx , dy , dz )2 1/V = 0, viz. the coefficients being functions of the first and second differential coefficients ofr , and V being a function of the first differential coefficients ofr , this is in fact a relation involving the first, second, and third differential coefficients ofr , or it is the partial differential equation to be satisfied by the parameterr considered as a function of the coordinates. After all reductions, this equation assumes the form previously mentioned.

IV. On curvature and orthogonal surfaces