J.E.W.Wallis (1913).

For a general sketch of the period see Lavisse _Hist.

de France_, t.

II, and for an elaborate critical study of certain aspects of Charlemagne's reign (including the _Polyptychum_) see Halphen, _Etudes critiques sur l'Histoire de Charlemagne_ (1921); also A.Dopsch, _Wirtschaftsentwicklung der Karolingerzeit, Vornehmlich in Deutschland_, 2 vols.

(Weimar, 1912-13), which Halphen criticizes.
_B.