1873.] 



and Orthogonal Surfaces. 



167 



nations of the normal). Using Levy's method, I obtained the general 

 equation, and communicated it to the Trench 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 + T 2 + Z 2 : I suc- 

 ceeded, 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), (GO, (H)X^, S6, to, W, %, 237*) 



-2((A), (B), (C), (F), (G), (H)X«, h, c, 2f, 2g, 2h)=0 : 



but the method was an inconvenient one, and I was led to reconsider the 

 question. The present investigation, although the analytical transforma- 

 tions 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 p (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 are considered as corre- 

 sponding points, say this is the conormal correspondence of vicinal sur- 

 faces. In order that the two surfaces may belong to an orthogonal 

 system, it is necessary and sufficient that at each point of the given sur- 

 face 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 jo shall satisfy a partial 

 differential equation of the second order, 



((A), (B), (C), (P), (G-), (H)X4, d y , d x y P =0, 



where the coefficients depend on the first and second differential coeffi- 

 cients of TJ, if U = is the equation of the given surface. Now, consi- 

 dering the given surface as belonging to a family, or writing its equation 

 in the form r — r(cc, y, z) = (the last r a functional symbol), the condi- 

 tion in order that the vicinal surface shall belong to this family, or say 



that it shall coincide with the surface r + 8r— r(cc, y, z) = 0, is p=y> 



where V= x/X* + Y* + Z 2 , and X, T, Z are the first differential coeffi- 

 cients of r(x, y, z), that is, of the parameter r considered as a function of 

 the coordinates ; we have thus the equation 



((A), (B), (C), (P), (Gr), (H)J4, d s , 4)4 =°> 



viz. the coefficients being functions of the first and second differential 

 coefficients of r, and Y being a function of the first differential coefficients 

 of r, this is in fact a relation involving the first, second, and third dif- 

 ferential coefficients of r, or it is the partial differential equation to be 

 satisfied by the parameter r considered as a function of the coordinates. 

 After all reductions, this equation assumes the form previously mentioned. 



