48 



GANESH PRASAD, 



I>art III. 



Improperly Contiimous Tbeory^). 



Notation of Improperly Continuous Analysis. 



44, By an improperly continuous analysis I mean an analysis in which, at 

 least, one independent variable has for its domain an everywhere dense but enu- 

 merable aggregate; it is thus distinguished from continuous analysis in which 

 every independent variable has for its domain a continuum. I proceed to specify 

 a System of notation which it will be convenient to adopt. 



I will use the Grreek letter | for a variable of which the domain is an 

 everywhere dense but enumerable aggregate. The notation relating to limit, 

 infinital relations, derivation, or Integration is precisely the same in improperly 

 continuous as in continuous analysis. 



Thus, cp{^) being a function of |, and l^, a particular value of |, lim 



means the limit to which q){^) approaches as | approaches from the right side; 

 and lim (p (|) has a similar significance. Also , if + 1' be any value of ^ in 



the right neighbourhood of 1^, 



1) In working out this theory I liave received great lielp from 



Brodän's „FunctionentheoretiscLe Bemerkungen und Sätze," Acta üniv. Lund, Bd. 8; and 



Karl Pearson's „Grammar of Science, 2nd edition," specially Chapter VII. 

 I am also considerably indebted to the following publications : 



Larmor's „Address to the Mathematical and Pbysical Section of the British Association, 

 1900" (Nature, Vol. 62), pp. 451—55. 



Poincarö's „Relations entre la Physique Experimentale et la Physique Math^matique" 

 (Rapports pr^sentös au Congres International de Physique, Paris 1900). 



Weierstrass's „Ueber die analytische Darstellbarkeit sogenannter willkürlicher Functionen 

 einer reellen Veränderlichen" (Sitzungsberichte der kgl. preussischen Akademie der Wissenschaften 

 zu Berlin, 1885, pp. 633—39 and 789—805 ; Math. Werke, Bd. III, pp. 1—37). 



9p'(|„ + 0) = lim 

 l' = +o 



y(|,+r)-y(lo) 



r 



