PROBLEMS OF ALGEBRA AND ANALYSIS 529 



variant expression provided that the symbols f, <p, . . . , F, 4>, . . . 

 occur in such a connection as to permit actual meaning. 



The symbolic representation of invariant expressions suggested 

 by the case n = 2 can without essential difficulty be extended to the 

 general case of n variables. In this treatment of the subject all the 

 essential quantities entering into the theory present themselves quite 

 naturally; they lie, so to say, on the surface; so, for instance, all the 

 Christoffel symbols of the different kinds including the Riemann 

 symbols and in particular also the process of covariantive differ- 

 entiation. 



The results of my investigation are chiefly laid down in the paper 

 "A symbolic treatment of the theory of invariants of quadratic 

 differential qualities of n variables," Transactions of the American 

 Mathematical Society, vol. iv. 



A third method of investigation of our theory of invariants is 

 based on Lie's theory of continuous groups. The general point 

 transformation by which A is transformed into A' defines a so- 

 called "infinite" continuous group. In order to obtain the invari- 

 ants of A, this group must first be "extended" in Lie's sense to 

 include the coefficients a^ of A and also the arbitrary functions 

 involved in the differential parameters. 



Lie himself developed a short outline of the determination of 

 invariants in the second volume of the Mathematische Annalen for 

 the case n=2, and indicated in particular how the Gaussian curv- 

 ature and the parameter Ai<p could be found. The general plan of 

 investigation was taken up in the sixteenth volume of the Acta 

 Mathematica by Zorowski, who studied the case n = 2 in detail, adding 

 the complete computation of the Gaussian curvature and the most 

 important differential parameters. 



An extension of Lie's methods to the general case of n variables 

 as far as the actual determination of invariants is concerned has, 

 so far as I know, not yet been made; only the problem of deter- 

 mining the number of functionally independent invariants of a given 

 order has been taken up. It seems that Lie's method is especially 

 well adapted to this particular problem. In a paper in the Atti del 

 Reale Institute Veneto (1897), Levi-Civitta found a lower limit for the 

 number of invariants of a given order. The actual number was 

 determined by Haskins in the Transactions of the American Mathe- 

 matical Society, vol. in, for the case of invariants proper (including 

 also simultaneous invariants) and in vol. v, of differential parameters. 



I am at the end of my paper. I have attempted to show, in a 

 compendious way, what has been done in this attractive field of 

 research which is so closely connected with various interesting parts 



