tiler Beihen cmf der Convergenzgrenze. 



'SS7 



number results which enumerates the combinations with which the 

 problem is concerned. The problems to which the method is applicable 

 are those which are, directly or indirectly, associated with lattices, and 

 it is remarkable that, for the most part, they are such as have been 

 hitherto regarded as unassailable by the processes of pure mathe- 

 matics. 



If a problem be presented for solution, it may be a difficult matter 

 to design the function and the operation, the combination of which 

 furnishes the solution. The plan here adopted is to consider certain 

 functions and operations, and then to inquire into the nature of the 

 associated problems. The author has attempted to place the method 

 on a sure foundation, to give illustrative examples of gradually 

 increasing complexity, and to indicate some promising lines of future 

 investigation. The only published work connected with the subject 

 is the author's paper, entitled " A New Method in Combinatory 

 Analysis, with application to Latin Squares and Associated Questions," 

 which will be found in the ' Trans. Camb. Phil. Soc.,' vol. xvi. Part 

 IV, p. 262. 



" tiber Eeihen auf der Convergenzgrenze." Von EMA^ifUEL Lasker, 

 Dr. Philos. Communicated by Major MacMahon, F.E.S. 

 Pteceived March 15, — Eead April 5, 1900. 



(Abstract.) 

 The essay is divided into three chapters. 



A limit operation of any kind, for instance, simple or multiple 

 integration ; the formation of infinite sums or products ; or any com- 

 bination of these operations gives rise to certain typical considerations 

 which form the subject of the first chapter. To fix the ideas, let 



Y{x) = III + u'2 -f -f- Un + , 



where the Ui are analytical functions of x ; and let a; = ^ be a point on 

 the curve which forms the boundary of the region of convergence of 

 the series. Let C be a curve within the region of convergence ter- 

 minating in X = In that case F(^) is generally diff'erent from 



lim. Y{x), 



(lim. X = 



varying on the curve C. It is a matter of the greatest importance 

 to examine the relations between these two mathematical conceptions. 

 liui + ti.2-{- +Un+ converges where x = ^, r(^) has a definite 



