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IV. On Curvature and Orthogonal Surfaces. By A. Cayley, LL.D., F.B.S., Sadlerian 
Professor of Mathematics in the University of Cambridge. 
Received December 27, 1872, — Read February 13, 1873. 
The principal object of the present Memoir is the establishment of the partial differ- 
ential 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 
rhe 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 X 2 +Y 2 -j-Z 2 ; 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), U, Ic, 2 If, 2 ig, 2 ih) 
-2((A), (B), (C), (F), (G), (H)X«, b, c, % 2y, 2A)=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 (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 g shall satisfy a partial differential equation of the 
second order, 
((A), (B), (C), (F), (G), (H )Jd„ d,, d x )^= 0, 
9 r 
iU I 
MDCCCLXXIII. 
