451 



the functions (21) form part of the F. F., of T, in other words, 

 that any function of the series 



^0' §!'•••' S/« ? (^2) 



into wliich (21) is transformed by T, is regular in a neighbour- 

 hood of ^Q. 



This tacit supposition, however, is essential. To see this we 

 observe that, if we start from the supposition that a series of the 

 form (1) represents the given transmutation, the existence of the ^'s 

 follows from this. For, that series is finite for any function of the 

 series (21), consequently its convergence is beyond doubt; we find, 

 if ??ii, 7»2, . , . are the binomial coefficients of m, 



^,„=a;^"a, + m,a;>>^--^a, + . . . + a,n = {.v -\- a)>» . . . (23) 



in which the last member is a symbolic form, indicating that in 

 the expansion of the binomium a^ must be replaced by a^, and in 

 the term without a, that is the one with ^"', the factor a" must 

 be added. 



Thus, if we should suppose that ^'"' has not a transmuted for 

 every integer m, the existence of the series (1) would at once be 

 excluded. 



On the other hand, however, the a's are completely determined 

 by the §'s ; for from (23), by respectively writing m = 0,1,2 .. ., 

 we can solve the functions a^, a^, a^, . . ., giving 



a,„ = (§-.r)'« ........ (24) 



where the second member is again a symbolic binomium, in the 

 expansion of which §^ must be replaced by |^ whereas in the 

 terra without §, that is in ( — a?)"', the factor '§ must be added. 



The second supposition we have in view, not less essential for 

 the the^orem, is since follows: Thei^e is a circle [a) to tuhich all |'s 

 belong. If this were not the case, the lower limit of the radii of 

 convergence of the functions (22) would be equal to zero (without 

 the limit being reached, since this would not agree with the supposition 

 already raade, that all ^'s are regular in .fp). The existence of the 

 series (1), in a neighbourhood of .i^, for other than rational integral 

 functions, is then apparently excluded as well, however small this 

 neighbourhood may be taken. Only in the point d\ itself a result 

 would not necessarily be impossible, but in every case I'o could 

 not be considered as a centi'e of a domain. 



We may therefore assume that it must have been Bourlet's 

 intention to make this second supposition as well. 



That there is a circle (o) to which all |'s belong, also agrees 



