304 



Differentiating these, as if X, Y, Z were constant, that is, treating 

 the point q, as an intersection of two consecutive normals, we obtain 

 these two other equations, 



>. f (X - x)dx' + ( Y - y)(hf \ (Z - x)dz' = x'dx + y'dy + z'dz ; 

 ^ ' ' ' \ (X - x)dxj + ( Y - y)dy } + (Z- x)dz / = x t dx + y,dy + z<dz. 



If, then, we write, for abridgment, 



W * * \ E' - Lx/ + My f ' + Nz/ ; E" = Lx u + My it + Nz,, ; 



we shall have, by (a) (b) )g), the two important formulas : 



(m) ..R(E+ E'v) = K{e + e'v) ; R [E' + E"v) = K (e f + e"v) ; 

 which we propose to call the two general Equations of Curvature. 



5. In fact, by elimination of R, these equations (m) conduct to a 

 quadratic in v, of which the roots may be denoted by v x 'and v 2> which 

 first presents itself under the form, 



(of which we shall shortly see the geometrical signification), between 

 the coefficients, A, B, C, of the joint differential equation of the system of 

 the two Lines of Curvatare on the surface. 



6. The root v x of the quadratic (o) determines the direction of what 

 may be called the First Line of Curvature, through the point p of that 

 surface ; and the First Radius of Curvature, for the same point p, or the 

 radius R x of curvature of the normal section of the surface which touches 

 that first line, may be obtained from either of the two equations (m), as 

 the value of R which corresponds in that equation to the value v x of v. 

 And in like manner, the Second Radius of Curvature of the same surface 

 at the same point has the value R 2 , which answers to the value v 2 of v, 

 in each of the same two Equations of Curvature (m). We see, then, 

 that this name for those two equations is justified by observing that when 

 the two independent variables t and u are given or known ; and there- 

 fore also the seven functions of them, above denoted by e, e', e" , E, E', E", 

 and K. The equations (m) are satisfied by two (but only two) systems 



of values, v x , R x , and v 2 , R 2 , of (I.) the differential quotient v, or — , 



which determines the direction of a line of curvature on the surface ; 

 and (II.) the symbol R, which determines (comp. No. 4) at once the 

 length and the direction, of the radius of curvature corresponding to that 

 line. 



E = Lx" + My" + Nz" ; 



(n) . . (e + e'v) {E f + E"v) = (e f + e"v) (E + E'v), 



but may easily be thus transformed, 



( s f Av 2 - Bv + C = 0, or Adu* - Bdtdu + Cdt* = 0, 

 w ' ' 1 with A = e'E" - e"E\ B = e''E ~ eE" , C = eE" - e' 



so that we have the following general relation, 



(p) . .eA+e'B + e"C=0, 



