Mathematics. — "Observations on the expansion of a function 

 in a series of factorials:' H. By Dr. H. B. A. Bookwinkkt,. 

 (CorTimunicated by Prof. H. A. I.orkntz). 



(Communicated in the meeting of September 29, 1918). 



5. We now consider another example of Niei.skn's theorem, not 

 belonging to the cases mentioned nnder N". 4 of the remarks made 

 in tlie preceding paragraph. We choose 



where 6 is a nnmber between and 2.t, not eqnal to one of these 

 nnmbers. For this function we have 



;i=z: — 00 , A' = 0, 



the first of these equations resulting from the fact that t=V is an 

 ordinary point of the function. It is furtlier easily found that the 

 ?i^'' derivative of (f> (t) satisfies tlie equation 



y(>0(0(l-t)^> »-» ^ jir{n)_ f}:zl\' (1-0°-' ,01) 



l~t 

 The modulus of the expression -— is given by the relation 



(?'' — t 



}:Z1\^1 W-co^^) ^ (22) 



ei^—t\ 1— <-f-l/l-2t cos <9 i «' 



and it is not very difficult to see that it increases monotonously 

 from the value to 1, if / decreases from 1 to 0. 



We divide the interval (0,1) of t into two parts, (0, v] and (r, 1), 

 where r is a number given by 



r = n«i-i (0<d, <1) (23) 



so that V depends on n and approaches to zero as a limit when n 

 becomes, indefinitely large. The positive number (f^ is at our disposal 

 and will be fixed immediately. The maximum value of the modulus 

 (22) then differs from unity by a quantity greater than 



if t lies in the second interval, k being a certain positive number, 

 which is independent of ?i and /; thus we have in this interval 



