PROBLEMS OF ALGEBRA AND ANALYSIS 527 



Continuing in this way Christoffel obtains a well-defined set of 

 covariants G 5 , (?, . . . , and this is his final result: the necessary 

 and sufficient condition for the equivalence of the two differential 

 quadratics is the algebraic equivalence - - in the sense of the alge- 

 braic theory of invariants of the forms A, G t , G^, . . . G^and Pi? , 

 G' 4 , G' 5 . . . G'^, where , is a certain finite number. 



In several papers covering the period from about 1884 up to the 

 present time Ricci has worked out in a systematical way the funda- 

 mental principles of ChristoffePs investigation, and has applied his 

 theory to many problems in analysis, geometry, mechanics, and 

 mathematical physics. He recognized in particular the importance 

 of ChristoffePs deduction of the covariants G A+1 from <7 A . He found 

 that this process of deduction can be applied with a proper modifi- 

 cation to any functions of the x's and the a^-'s and that whenever 

 invariantive relations with respect to the fundamental differential 

 quadratic A come into question, this process is always of vital im- 

 portance. He calls this process covariantive differentiation with 

 respect to the fundamental quadratic A. On the systematical use 

 of this covariantive differentiation Ricci based a calculus which he 

 called Calcolo differ enziale assoluto. 



A collection of all his various investigations is given in two places: 



(1) In a paper published, together with Levi-Civitta in the Math. 

 Annalen, vol. LIV. 



(2) In his Lezioni sulla teoria delle superficie, Verona, Padua, 1898. 

 In the introduction of these autographed lectures he presents a 



complete exposition of his absolute differential calculus. Charac- 

 teristic is the way in which he treats in his Lezioni the differential 

 geometry. He divides it into two parts: 



(1) Properties of surfaces depending on the one differential quad- 

 ratic ds 2 . 



(2) Properties of surfaces depending on the two quadratics 



ds 2 

 ds 2 and . 



P 

 We are here chiefly interested in his applications to the theory of 



differential invariants. This is the result in his language: In order 

 to obtain all invariants proper and differential parameters of order /*. 

 it is sufficient to determine the algebraic invariants of the system 

 of the following forms: 



(1) The fundamental differential quantic A. 



(2) The covariantive derivatives of the arbitrary functions 



U, V, . . . up to the order /*. 



(3) (for ,>!) the quadrilinear covariant G^ and its covariantive 

 derivatives up to the order 2. 



Another treatment of the invariant theory of differential quan- 



