PREFACE.
The favorable reception accorded to the first two volumee of this history has
encouraged the author to complete promptly the present third volume which is
doubtless the most important one of the series.
By a " form w we mean & homogeneous polynomial such as {=аз? + Ъху-\-сугу all
of whose terms are of the same total degree in x and y. The arithmetical theory of
forms has an important application to the problem to find all ways of expressing a
given number m in a given form ft i. ev to find all sets of integral solutions/ For this application we do not consider merely the given form f,
but also the infinitude of so-called equivalent forms g which Gan be derived from /
by applying linear substitutions with integral coefficients of determinant unity. It
is by the consideration of all these forms g that we are able to solve completely the
proposed equation / = tn. The theory needed for this purpose is called the arith-
arithmetical theory of forms, which is the subject of the present volume* This theory
is applicable to most of the problems discussed in Volume II, The present methods
have the decided advantage over the special methods described in Volume II in that
they give at once also the solution of each of the infinitely many equations
By thus treating together whole classes of equivalent equations, the methods
effect maximum economy of effort.
Enough has now been eaid to indicate that we are concerned in Volume III mainly
with general theories rather than with special problems and special theorems» The
investigations in question are largely those of leading experts and deal with the most
advanced parte of the theory of numbers. Such a large number of the important
papers are so recent that all previous reports and treatises (necessarily all very-
incomplete) are entirely out of date»