PROBLEMS OF ALGEBRA AND ANALYSIS 519 



ject: geometry has here as well as in many other branches of mathe- 

 matics indicated the problems which in their later development 

 turned out to be of paramount interest in pure analysis. 



A few preliminary remarks concerning the nomenclature of the 

 different types of invariant expressions will be necessary. 



To a given differential quadratic form 



n 



A = % a ik dxidx k ,(a ki =a ik ) 



where the a^'s are functions of the n independent variables Xi,x 

 x n , we apply a general point transformation of the variables x, 



z , 



We observe that the differentials dx are then transformed into 

 linear expressions of the differentials dy with the Jacobian of the 

 x's with respect 'to the y's as the substitution-determinant which 

 we shall call r. 



By this transformation A goes into 



A' =Ia' ik dijidy k . 



Let now be a function 



(a) of the coefficients a ik and their first, second, . . . derivatives, 



(b) of U, V, . . . and their derivatives, where U, V, . . . are any 

 arbitrary functions of x l} x 2 , . . . x n . 



If then $ remains the same whether formed for the new or for 

 the old quantities, that is, if 



(D(a f z 8a ' ik U' U> V }-0)(a-i aik II U , 



w\u, lk , -- , . . . , u , ,..., v ,,..) w\a lk , - _,..., L/, _ 



oijX dyX dxk dxX 



...7,...) 

 we say that is an invariant (in the wider sense) of A. 



If (J> contains only the a^-'s and their derivatives, w r e call it an 

 invariant proper, and its order the order of the highest derivative 

 occurring in it. If $ contains also one or more arbitrary functions 

 U,V, . . . we call it a differential parameter, the definition of order 

 being the same as before. 



If more than one differential quadratic is given it is easily under- 

 stood what is meant by simultaneous invariants and simultaneous 

 differential parameters. 



In strict analogy with the algebraic theory of invariants we call 

 covariants expressions of the above invariantive nature, provided 

 that we also allow the differentials dx to enter into 0. 



The first and the most important example of a differential quad- 

 ratic quantic is the square of the arc-element on a surface 



ds 2 = Edu 2 + 2Fdudv+Gdv*. 



It was Gauss who made (1827), in his Disquisitiones generales 

 circa superficies curvas, this expression the fundamental object of 



