PROBLEMS OF ALGEBRA AND ANALYSIS 525 



on geodesic polar coordinates for n=2 admit a perfect analogon 

 in hyperspace. Also in hyperspace then the determination of systems 

 of geodesies amounts to the integration of the partial differential 

 equation 



J^ = l. 



This leads now to the application of differential quadratics to 

 analytic mechanics. If we write down the expression of th'e vis 

 viva of a (holonomous) material system in terms of generalized 

 coordinates q v q 2 , -q n - 



T = \Za ik d ^ *** 

 dt dt 



we have at once in 



2Tdt*=ds 2 

 a differential quadratic before us. 



If no external forces act on the system, then a geodesic line of ds 2 

 represents at once, as also Beltrami has shown, a path of the sys- 

 tem. Thus the mechanical problem is practically reduced to the 

 integration of the equation Ajp = \. 



In the case of the existence of external forces having a potential 

 U, the above differential quantic has to be replaced by 



I '(U +h)a ik dq i dqk 



and the mechanical problem is equivalent to the integration of the 

 equation 



Ai<p = U+h 



where A\<p is the differential parameter of the quadratic form de- 

 noted before by ds 2 . 



A detailed exposition of the above-mentioned researches of Bel- 

 trami, as well as this application to mechanics, is given in the second 

 volume of Darboux's Lecons sur la theorie des surfaces. 



Passing now to the second part of my address, the purely ana- 

 lytic theory of invariants of differential quadratics, I have first 

 to discuss that paper which forms the foundation of almost all 

 later literature on the subject: Christoffel's article in Crelle's Jour- 

 nal, vol. LXX (1870), "Ueber die Transformation der homogenen 

 Differentialausdriicke des zweiten Grades." 



Christoffel puts his problem in this form: Given two differential 

 quadratics 



A la^dxidxk and A' =Ia' ikdyidy^, 



what are the necessary and sufficient conditions for the equivalence 

 of the two quadratics, that is, for the existence of a transformation 

 of one quantic into the other; and if these conditions are established 

 how can the required transformation be determined? (I should men- 

 tion that Lame" in his already quoted work, Lecons sur les coor- 



