303 



the Comment ationes Recentiores, but of which af'Latin reprint has been 

 since very judiciously given, near the beginning of the Second Part 

 (Deuxieme Partie, Paris, 1850) of Ltouville's Edition* of Monge, it 

 still appears that there is room for some not useless Additions to the 

 Theory of Lines and Radii of Curvature, for any given Curved Surface, 

 when treated by what Gauss calls the Second Method of discussing 

 the General Properties of Surfaces. In fact, the Method here alluded to, 

 and which consists chiefly in treating the three co-ordinates of the sur- 

 face as being so many functions of two independent variables, does not 

 seem to have been used at all by Gauss, for the determination of the 

 Directions of the Lines of Curvature ; and as regards the Radii of Cur- 

 vature of the Normal Sections which touch those Lines of Curvature, he 

 appears to have employed the Method, only for the Product, and not also 

 for the Sum, of the Reciprocals, of those Two Radii. 



2. As regards the notations, let x, y, z be the rectangular co-ordi- 

 nates of a point p upon a surface (S), considered as three functions of 

 two independent variables, t and u; and let the 15 partial derivatives, 

 or 15 partial differential coefficients, of x, y, z taken with respect to t 

 and u, be given by the nine differential expressions. 



( dx = x'dt + x,du ; dx ! = x"dt + x/du ; dx t = x/dt + x„du ; 

 (a) . . \dy = y'dt + y,du ; dy' = y"dt + y/du ; dy / - y/dt + y u du ; 

 \dz- z'dt + z t du ; dz r = z' f dt + %/du ; dz y = %/dt + % u du. 



3. Writing also, for abridgment, 



(b) . . e = x' 2 + y' 2 + % 2 ; e' = x r x t + y'y / + %'z, ; e" = x 2 + yf + zf 

 we shall have (c) . . ee"-e! 2 = K\ if (d) . . K 2 = L 2 +M> + 1ST 2 , 

 and (e) . . L = y% - z'y / ; M= z'x, - x\ ; W- xy, - y'x / ; 



so that (f ) . . Lx + My' + Nz f = 0, Lx i + My, +Nz, - 0. 



Hence K~ y L, K~ X M, K~ X N are the direction-cosines of the normal to the 

 surface (S) at p; and if x, y, z be the co-ordinates of any other point q 

 of the same normal, we shall have the equations, 



(g) . . K (X-x) = LR ; K ( Y- y) = MR ; K(Z-z) = NR; 

 with (h) . . R 2 = (X- xf + ( Y- y) 2 + {Z- z) 2 ; 



where R denotes the normal line pq, considered as changing sign in 

 passing through zero. 



4. The following, however, is for some purposes a more convenient 

 form (comp. (f)) of the Equations of the Normal; 



(i) . . (X-x)x' + ( Y- y)y> + (Z- z)z' = 0 ; 

 (j) . . (X-x)x / + (Y- y)y t + (Z-z)z / = 0. 



* The foregoing dates, or references, are taken from a note to page 505 of that 

 Edition. 



