Mathematics. — "Some Co7isiderations on Complete Transmutation.'' 

 (Third Communication). Bj Dr. H. B. A. Bockwinkel. (Com- 

 municated bj Prof. L. E. J. Bkouwer.) 



(Communicated in the meeting of November 25, J 91 6.) 



14. In connection with the preceding considerations the con- 

 tinuity of a transmutation we shall refer to an inaccuracy that occurs 

 in the proof of theorem X of Bourlet, which will give us an op- 

 portunity to observe that the theorem itself stands in need of a 

 clearer statement. 



Theorem X is as follows Toute transmutation additive, uniforme, 

 continue et reguliere est donnee par la fomule 



Tu = a^u -j- —u' -\- —u" -^ .. ., (1) 



oil a^, «,, . . . designent des fonctions régulieres et u' , u" , ... les déri- 

 vées successives de la fonction reguliere u. 



If this theorem, and the proof Bourlet gives of it, is considered 

 more accurately, its meaning appears to be not over-clear, and this 

 is to be attributed to the fact that Bourlet has omitted to fix a 

 functional and a numerical field in which the transmutation is to 

 be thought as defined. Are we to suppose that the transmutation, 

 for any function that is regular in the neighbourhood of a certain 

 point, produces a transmuted that is regular in the neighbourhood 

 of the same point? Are there points that form an exception to 

 this? Or functions for which this is the case? Are only analytical 

 functions to be thought of as objects of the transmutation? Or also 

 such as coincide in a certain N.F. with the analytical function /i(j,"), 

 and in another N.F., lying outside it, with the analytical function 

 ƒ, (ct)? Is the transmuted an analytical function? 



The matter does not become quite clear, for the definitions given 

 by Bourlet, are all highly incomplete. By a regular transmutation 

 he understands, according to his own words, a transmutation that 

 makes a regular function pass into an equally regidar function. But 

 what is a regular function ? Bourlet answers this in a footnote : 

 "Je prends pour definition de la fonction reguliere celle de M. M. 

 MÉRAY, Weierstrass, Fuchs, etc.: "Une fonction de la variable x 

 est dite reguliere, dans le domaine de rayon q, an tour du point 



