00. 





198 DR. J. MEBCKR: 8TUKM-LIOUVILLE SERIES OF NORMAL FUNCTIONS, ETC. 

 converges to lira $[(* y t ) + (x+y 3 )], as n increases indefinitely, at each point of 



jr*-0 



the open interval (a, b) where this limit exists as a finite number ; moreover, at a 

 point where this limit has one of the improper values oo, it diverges to this value 

 and is non-oscillatory. If the set of points at which F (x) is continuous includes a 

 closed interval lyinf/ within (a, b), then the' arithmetic mean converges to F (x) 

 uniformly in this interval. 



If the functions V, (JT) do not satisfy the boundary condition v = at X ' f , then, 



X ^^ t) 



as n inci-eaaes indefinitely, the arithmetic mean of the first n partial sums of the 

 St,,,-m-Liouville series (39) converges at * = % to pfc^QJ , whenever this limit exists 



as a finite number ; moreover, when this limit has one of the improper values 

 the arithmetic mean diverges to this value and is non-oscillatory. 



