54 GANESH PßASAD, 



Therefore 



lim (3(1, /) = 0, lim Qilt) = 0. 



| = — jr+O | = 7r — 0 



51. Thus the group of conditions ivhich is necessary and sufficient, in Order 

 that % t) he the required cJiaracteristic fimction, is the following : 



i. For every value of | , there exists a finite constant P such that , for 

 anj value of t however small, 



\\V' (x, t)^^_ ^\ is less than P, — « < ^ < jt. 



ii. Lim F(|, f) = cp (|) if the limit exists, or, the limit does not exist, and 

 t=+o 



then cp (S) is contained in the aggregate of values assumed by F(|, t) as t ap- 

 proaches zero. 



Necessary and sufficient conditions for 93 (|). 



52. The group of conditions, given in the last article, leads, with the help 

 of the results given in Arts. 19 and 21 , to certain necessary and sufficient 

 conditions , for cp (|) , of an applicability sufficiently extensive for the purposes 

 of this essay. I give the simplest and most important of these conditions below, 



i. If cpi^) is a continuous function of | — and, consequently , its Single 

 associate, which is continuous in the interval (— tc, tc), is the limital ') function 

 corresponding to it — then, in Order that x{i„t) be the characteristic function 

 of the problem , it is sufficient that for every value of § within its domain, 



either | D (|, x') | «xj 1 or ± D x') rsj x' ^'^^ cos | ?/> (a?') } where ip {x')>~--^ 



1; +v>l. 



ii. If any associate f{x) of (p (|) is discontinuous in the interval (— 71, Jt) 

 then the condition relating to D x') , given in i , together with one of the 

 following conditions is sufficient: 



(a) If lim 3f (I, x') exists, it is equal to cp (|). 



x' = +0 



(b) If x') = M,(^, x') + 3I,{^, x') such that lim M, exists and 



x' = +0 



M ^_ cos \ il^ {x'}]wh.eve ip{x')>~ II'-y] j fhen the condition is that lim M^= 93 (|). 



\"x I = + 0 



(c) M (I, x') rsj cos ^ (x') where ip {x') l {^r^ ] further, cp (|) = 0. 



iii. % (§, t) ceases to be the characteristic function , if at any point |, 

 lim .r') exists and is difierent from cp {^). 



iv. i) ceases to be the characteristic function, if 9) (|) possesses an 

 associate f{x) which is such that, for eveiy value of § within its domain, 



± B (I, x') either >^ 1 ov x' ^'^^^ cos (x'') | where ip {x') l 

 1) This is Brodön's Limitäre Function. 



