1100 
may be used as a valuable instrument (not yet put into practice) 
to determine if we have to do with alkylated-nitro-anilines (toluidines) 
or with their nitrosamines. Just as in the case of the 3.4-dinitro- 
phenyl-methyl-nitrosamine — a compound of whose characteristic 
behaviour we hope soon to give a description — the refractometric 
determination furnishes valuable data, because this compound has a 
yellow colour — nitrosamines are nearly colourless — and is moreover 
not easily to be identified as a nitrosamine by chemical researches. 
Utrecht. Org. Chem. Univ. Lab. 
Mathematics. — “Some Considerations on Complete Transmutation.” 
(Second Communication.) By Dr. H. B. A. BoCKWINKEL. (Com- 
municated by Prof. L. E. J. Brouwer. 
(Communicated in the meeting of October 28, 1916.) 
7. If we say that a transmutation of the form (1) 
7 a, (z) 
Tu =a, (#)u + —w+...4+ 
m! 
EL Oa ois SE 
n. 
is complete in a certain circular domain («) with centre w,, it does 
not imply that this domain (@) belongs to the given transmuta- 
tion as an invariable field; indeed, if 7 is complete in (a), it is 
certainiy also complete in a domain (a) << a, and probably in a 
domain (a!) > (a). Only the aggregate of functions which have in such 
a domain a transmuted determined by (1), is different for any 
new domain, such that the aggregate diminishes if the domain 
increases. If we want to make this more prominent, we shall say 
more fully: “the transmutation is complete in the domain (a) 
with corresponding domain (8), or something of the kind. As 6 
further increases and decreases monotonely with a, it is clear that 
functions with evactly the radius of convergence 2 have a trans- 
muted determined by (1) in any domain (a) < (a); the series con- 
verges for this wniformly in the domain (a’) (final paragraph N°. 4), 
Special attention should be paid to the fact that we call a trans- 
mutation only then “complete in a domain (a)” if it 7s im that domain 
determined by a series of the form (1) for all functions belonging toa 
certain circle (9). If this is not the case, the transmutation, which is then 
of course defined in another way than by the series (1), may produce 
a transmuted in the whole domain (a) for all functions belonging 
to (8), but this does not in itself give a reason to call it complete in 
