435 



satisfy the special condition mentioned here, are not gi\en by 

 Nielsen, but may easily be imagined. 



5. Finally we want to make a remark on another manner to 

 derive the development (6) from the series (2); a way which has 

 been followed both by Pincheht.e and by Nielsen. We may write 



ip{t)(\-tY-^ =a,{\~tY-^ + a,t{\^tY-^ + . . . + ant'iX—ty-^ + . .. , 



and thus 



1 1 



Ccp{t) (1 - ty -' dt =C\c,,{\—tY-^ + . • . + a„«"(i -0— 1 4-. . . ] dt. 







If this series is integrated term by term between the limits and 1, 

 we obtain the series of factorials required. In case the latter con- 

 verges, Nielsen (p. 239, Handbuch) derives from it that this series 

 is equal to tiie integral in the left-hand member of the preceding 

 equation; his reasoning is based upon a certain proposition of Dini 

 (Grundlagen, p. 523). If this reasoning were right, then, besides the 

 special case treated of above, we should have a more general one, 

 in which the integral (i) may be developed into a conditionally con- 

 verging series of factorials: viz. always when the series of factorials 

 to be derived from that integral, whether by means of integration 

 by parts or in the manner described just now, is a converging one. 



But, in our opinion, the proposition of Dini in question has been 

 applied in a wrong manner by Nielsen. The fact is that, in applying 

 that proposition, we should first integrate the function in question 

 over the interval 0<.t<u, where u <^ \ . Now it is at once to be seen 

 that in such an interval we may integrate the abo\e series term by 

 term (on account of its uniform convergence in that interval); in 

 other words we shall have 



u 



i(p{t){\—tY-^dt = 





 u u u 



= a, Ul—tY-^dt + a, it{\ — tY^^dt-\- . . . -f aj <"(1— 0^-^(i<4- .. ,(20) 







this equation being valid, however little a differs from unity. The 

 left-hand member of this equation is by definition equal to the 

 integral (1). But the right-hand member is only then equal to the 

 series of limit-integrals, if this member, considered as a function of 

 u, is continuous on the left in the point u=i\. This condition is 

 explicitly added by Dini, but seems to have been forgotten by 



