942 



correspoMding to an arbitrarily chosen number e there exists a point 



1 



in the domain («), where the quantity |a„,|"' for an injinite number 



of ?7i-values is greater than A{a) — g, and also in that case the upper 



limit a{(t) of ax is equal to A{(t). The condition A is satisfied then 



and it will be quite an ordinary thin^ if the condition B is also realized, 



though we do no longer see the necessity of it. If, on the contrary, 



there can be assigned an amount y. such that in no point of the 



1 



domain [a) the quantity |a,«|'" is greater than A{a) — x for an itifimte 

 number of ?/i-values, then a{a) is certainly not greater than A{a) — a. 

 The condition A is not satisfied now ; for if this were the case, 

 then, since A„i{ci) is the modulus of <:i,„(.iO for at least one point at 

 the circumference of {a), there would, corresponding to any arbitrarily 

 small quantity e, be an integer Ns such that, from and after m= iV^. 

 we should have A,n{c() <^ [«(«) + s]'", or 

 1 



^m («) < a («) -f 8 < yl («) — X + f , 



and this is impossible, if s be chosen less than x, since AUt) is the 



1 



upper limit for m z= qo, of A"^ («). As to the condition B it might 



or might not be satisfied. 



We have made the preceding observations in order to elucidate 

 a few more cases as considered here. Meanwhile such cases, if thej' 

 are possible at all, may undoubtedly be regarded as pathological 

 ones rarely occurring in practice : the observations made may be 

 able to make this even clearer. 



With this our considerations on complete transmutation have come 

 to an end. 



