PROBLEMS OF ALGEBRA AND ANALYSIS 521 



A from Lame, who, in his Lecons sur les codrdonnees curvilignes, 

 defined in 1859 his differential parameters 



d 2 ? 6 2 <f> d 2 ? 



/J 9 /O ^= 



* f C\ o ' <N 9 *\ 9 



dx 2 oy 2 02- 

 for the three-dimensional case where the arc-element is of the form 



ds 2 =dx*+dy 2 +dz 2 . 



Lame recognized the fundamental importance of these quantities 

 and made a systematical use of them on account of their in variance 

 with respect to any point-transformation preserving the form ds 2 . 



The general theory of invariants defines the differential parameters 

 J, and J, for the case of n variables. From these general expressions 

 Beltrami's differential parameters are directly obtained for n=2, 

 Lame's quantities (Ji) 2 an< i -^ for the special form of ds 2 in the case 

 n=3. 



The number of differential parameters is of course infinite, but 

 Darboux in his Lecons sur la theorie generate des surfaces has proved 

 that all of them are expressible by means of J,, A.,, p and the evident 

 differential parameter 



d<p d(f> dtp dfi 

 du 3v dv 3u 



VEG-F 2 



(by forming, for instance, Ji(J 2 ^>) etc.) - - an important theorem 

 which has later been extended by Staeckel to an analogous theorem 

 for the case of n variables. 



The expression J^> occurs already in Gauss's Disquisitiones. 

 By taking as parameter curves a singly infinite system of geodesies 

 and its orthogonal trajectories he transforms the arc-element into 

 the form 



ds 2 =dr 2 +m 2 d<p 2 



and shows that r satisfies the differential equation 



J,r = l. 



An important differential parameter is the geodesic curvature. 

 Its expression was thrown by Bonnet into a form which is easily 

 recognized as a differential parameter (of the second order). Its 

 numerator =0 represents the differential equation of geodesic lines 

 in an invariant form. 



Since a transformation of the two independent variables u, v which 

 preserves the same value of ds 2 can also be considered as a transfor- 

 mation of two surfaces which are applicable to each other, it follows 

 that all invariants of ds 2 are also invariants of a surface with respect 

 to the process of bending. From this reason these invariants have 



