BOUNDARY VALUE OF INFINITE SERIES 199 



6. Differences of opinion in mathematical controversies are often re- 

 solved into a difference of interpretation of the conventional definitions 

 upon which repose subsequent reasoning. No generalization of a defini- 

 tion can be accepted which changes any case to which a previously accepted 

 and conventionally established definition applies. But a definition which 

 expressed in more general form and which includes the older definition 

 in all its applications and at the same time embraces and gives definite 

 interpretation to conditions which the older definition is inadequate to 

 reach or properly interpret, is a legitimate definition and permits a true 

 and real extension of analysis. 



Thus the definition that (1) is uniformly convergent for all values of 

 X for which (2) has a unique limit, includes the older definition of uniform 

 convergence and at the same time brings out clearly Cayley's remark that 

 non-uniform convergence is always associated with discontinuity, and 

 furthermore characterizes such a point as an essential singularity in gen- 

 eral. In particular at points of finite discontinuity of a real function it 

 introduces a shear or vertical segment which unites by a continuous path 

 branches which the older definition leaves separated by a vacuum. 



If we refer to the function / {z) of a Taylor's, Teixeira's or Fourier's 

 series as the generating function of that series within its region of conver- 

 gent equality, the definition shows that the series severs its connexion with 

 the function by passing through an essential singularity on the boundary. 



The definition of the infinite series given in § 1, 



-i/=o 7 = K 



*§ = L 2 Uy{x + tj), 

 «=« 7=1 



as a function of two variables x and y one of which y has the limit zero 

 when n = CO , is"in a sense analagous to Weierstrass's generalization in his 

 idea of the continuation of the series 



fiy) + {x-y)r (y) + ^^^/"(2/) + . . . 



over the connected region of the two variables x and y for which the series 

 is uniformly convergent as representing and defining one and the same 

 function / (x) . 



East Lawn, May, 1911. 



