936 



and /(!)<; J. We then have a {0) = c, a {!) = i and within the 

 interval (0,1), a{a) =zf{«), so that, if 



f{0) = C-^p, /il) = l-q, 

 where p and q are positive numbers satisfying the inequality 



there is at « = on the right a salt us of a (/t) of the amount p 

 and at « = 1 one of the amount q. 



Examples of 4:b. Take 



a,p (x) = c'« -{- xy'" + m ! .i"«^ 

 where y^f{x) is a function as considered in the preceding example, 

 without the restriction as to the value of /(J) being necessary. 



In all these cases we have imagined A to be essentially different 

 from R, but matters do not alter substantially if we suppose ^ = /^. 

 We may construct examples of this case by taking in the preceding 

 ones the function y = f{x) such that its radius of convergence is 

 unity, and then we may either choose f{x) so as to be finite for 

 a; = 1 or infinite. , 



Returning to the functional operations we remark that, in conse- 

 quence of the theorem of N". 4, such an operation is complete only 

 in domains («) such that cc <;^ A, or perhaps a =z A. We therefore 

 have only to deal with domains of the latter kind and for those 

 a{a) has now appeared to be a continuous function of «, except 

 perhaps at « = and at a ^ A. The case which we supposed to 

 have been realized, in order to simplify our statements relating to the 

 resultant of two complete transmutations, thus appears to be the 

 only one possible. Moreover, as regards possible discontinuity at 

 a z= 0, this is of no interest for the complete transmutation. If the 

 function a(«) at «^0 has a saltus from the value b^ to the value 

 h (the limit of a («) at « = on the rigiit), then though the series 

 P produces for all functions belonging to the circle (/;<,) and not to 

 {b) a transmuted at .v = 0, it does not for all functions belonging 

 to a circle ib') greater than {b^) but smaller than {b) produce a 

 transmuted in a certain domain of x^, however small. We therefore 

 shall assume as the domain ((i) corresponding to a = the circle of 

 radius b instead of the circle of radius b^ and then the disconti- 

 nuity of a («) at « :=: has been removed. 



A possible discontinuity on the left of a = A can, however, not 

 be removed. 



From formula (7) of N°. 4 



^ = a^a{a) (7) 



it now follows that the number ji coi-responding to a is also conti- 



