526 ALGEBRA AND ANALYSIS 



donnees curvilignes, treats and solves the analogous problem for the 

 case A =dx 2 +dy 2 + dz 2 ). 



Since the differentials dx are substituted linearly in terms of the 

 dy there exists one and only one algebraic condition for the trans- 

 formation, namely, 



\a'ik\=r 2 \a ik \. 



This condition would be sufficient if the coefficients a t -fc and the 

 elements of the determinant r were constants. In our case, how- 

 ever, other conditions must be satisfied, namely, the conditions of 

 integrability in order that the expressions for the dx's are com- 

 plete differentials. This is the way in which Christoffel introduces 

 his problem to the reader. 



The difficulty lies in the fact that the integrability conditions 

 lead at once to a great number of partial differential equations of 

 an apparently highly complex character. But Christoffel succeeds 

 in substituting for all these partial differential equations a purely 

 algebraic problem: The equivalence of two finite systems of alge- 

 braic forms in the sense of the algebraic theory of invariants. If 

 this equivalence is satisfied, --which is merely a question of algebra, 



- no further discussion of the integrability conditions is required; 

 they are all taken care of by the equivalence of the two systems. 



For the following it will be necessary to sketch briefly the char- 

 acter of these forms. 



The first is the quadratic form A itself. The next form is a quad- 

 rilinear covariant (r 4 in four sets of differentials dx 1 , dx 2 , dx 3 , dx 4 , 

 the coefficients of which are precisely the quantities (i k r s ) - - the 

 "Christoffel quadruple index symbols" or the "Riemann symbols" 



- which occur in the expression for the Riemann curvature: 



It is highly interesting to observe how the quantities ( i k r s ) 

 have entered into the theory from two so apparently different stand- 

 points. Christoffel found these expressions quite independently. 

 Though Riemann's paper was written in 1861, that is, before Chris- 

 toff el's article which appeared in 1870, it was only published in 1876, 

 ten years after Riemann's death, by Weber-Dedekind. 



For the deduction of the following forms G 5 , (? 6 , . . . these 

 forms are covariants linear in resp. 5 , e, . . . sets of differentials - 

 Christoffel uses a certain reduction process. The coefficients (Xikr s) 

 for instance of 0$ are obtained from (i k r s) first by differentiating 

 (i k r s} with respect to o; A and then by the addition of a sum of 5n 

 terms which are linear in the different symbols ( i k r s ) with co- 

 efficients depending on the so-called Christoffel triple index sym- 

 bols of the second kind - - expressions involving the quantities 

 and their first derivatives. 



