THE ideas of Du Bois-Reymond's Infinitarcalcul are of great and growing importance in all branches of the theory of functions.
With the particular system of notation that he invented, it is, no
doubt, quite possible to dispense; but it can hardly be denied that
the notation is exceedingly useful, being clear, concise, and expressive
in a very high degree. In any case Du Bois-Reymond was a
mathematician of such power and originality that it would be a great pity if
so much of his best work were allowed to be forgotten.
There is, in Du Bois-Reymond's original memoirs, a good deal that
would not be accepted as conclusive by modern analysts. He is also
at times exceedingly obscure; his work would beyond doubt have
attracted much more attention had it not been for the somewhat
repugnant garb in which he was unfortunately wont to clothe his most
valuable ideas. I have therefore attempted, in the following pages,
to bring the Infinitarcalcul up to date, stating explicitly and proving
carefully a number of general theorems the truth of which Du Bois-
Reymond seems to have tacitly assumed—I may instance in particular
the theorem of in. § 2.