4 SCIENCE PROGRESS 



the instrument most adapted for the study of the homographies 

 of a Riemann matrix, i.e. the set of periods of an Abehan 

 function. In the first part of a long memoir {Rend. Circ. Mat. 

 Palermo, 45, 1921, 1-204), he develops the theory of such 

 algebras, basing his account on Dickson's tract and on papers 

 by Wedderburn and Frobenius, but adding a few novelties of 

 his own. Unfortunately-, " ragioni linguistiche " have made 

 him invent several new names ; thus a nilpotent algebra is 

 called by him pseudonulla, an idempoieni element an antojuodiilo, 

 and the identical equation the eqiiazione minima. In the second 

 part he applies this instrument to obtain new results in the 

 general theory of Riemann matrices. He introduces a new 

 number characteristic of such a matrix, the rank, which turns 

 out to be, in the case of a matrix connected with a curve, the 

 maximum degree of the minimum equation of a correspondence 

 on the curve. He also makes use of the theory of Hermitian 

 forms, and shows that non-singular Abelian functions with 

 positive indices of multiplicabilit}' fall into two classes, one 

 containing only functions v.'ith an even number of variables, 

 the other containing functions with any number of variables 

 and being the most natural generalisation of elliptic functions 

 with complex multiplication, 



A divergent series of positive terms which continually 

 decrease and tend to zero becomes convergent if alternate 

 terms are given a minus sign. Generalising this, H. Rade- 

 macher {Math. Zs., 11, 192 1, 276-88) investigates sequences 

 e-i, e^ . . . e^ . . . which, multiplied in order into the terms of 

 such a divergent series, make it convergent, obtaining there- 

 from a theorem of Hardy and Littlewood on the as}-mptotic 

 value of the number of digits equal to X in the first n of a 

 decimal :v (0 <( a; <( i ). 



G. Doetsch {^Iath. Zs., 11, 1921, 161-79) proves a number 

 of theorems on the Cesaro summability of series and extends 

 the idea of the Cesaro mean to integrable functions. 



G. Sannia {Atti Torino, 56, 1921, 34-40) examines series 

 which are absolutely summable in Borel's sense. 



Since Lebesgue's extension, published in 1904, of the notion 

 of integration, it has been the object of several investigations 

 to determine methods of arriving at the Lebesgue integral by 

 means of the sums used in Riemann integration. Lebesgue 

 himself gave one such method ; others are due to de Geocze 

 and Hahn. Denjoy, with whose name is associated a more 

 general definition of integration than Lebesgue's, also contri- 

 buted ; one of his pupils, T. J. Boks of Hilversum, investigates 

 in detail one of his methods {Rend. Circ. Mat. Palermo, 45 

 (1921), 211-64). This he calls integration (B) ; it is more 

 general than integration (L), but a function which is " totalis- 



