28 
other words we should have to succeed in dividing the functional aggre- 
gate of all functions satisfying the equation (24) by special charac- 
terising properties; and so also the aggregate of binomial series, in 
such a way that between the two kinds of sub-aggregates there 
existed a complete correspondence, such that functions of some sub- 
aggregate K could only be expanded in binomial series belonging 
to the sub-aggregate K and in no others. But the problem to find 
suchlike characterising properties will perhaps be very difficult, since 
it is required for it to derive the character of a function from that 
of the coefficients of the series representing it; a problem which 
already causes the greatest difficulties when it regards the more 
known power-series. 
