Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 169

B. The Modern Theory.
The beginnings of this theory are to be found in Nos. 110
and 111.
13. Frobenius. Ueber Relationen zwischen den Niiherungsbriichen von
Potenzreihen. Jour. fur Math., vol. 90, pp. 1-17, 1881.
This fundamental memoir marks an important advance. See
16(a).
14. Stieltjes. Sur la reduction en fraction continue d1une serie procedant suivant les puissances descendantes d'une variable.
Ann. de Toulouse, vol 3, H, pp. 1-17, 1889.
15. Pincherle. Sur une application de la theorie des fractions continues algebriques. Comp. Rend., vol. 108, p. 888, 1889.
16. Pade. (a) Sur la representation approchee d'une fonction par des
fractions rationnelles. Thesis, published in the Ann. de l'Ec.
Nor., ser. 3, vol. 9, supplement, pp. 1-93, 1892.
This very fundamental memoir is the best one to read for the
purpose of learning the elements of the theory of algebraic
continued fractions. The same point of view is taken as by
Frobenius in (13) and is more completely developed. The
thesis was preceded by the two following preliminary notes:
(a') Sur la representation approchee d'une fonction par des
fractions rationnelles. Comp. Rend, vol. 111, p. 674, 1890.
(a") Sur les fractions continues regulieres relatives a e".
Comp. Rend, vol. 112, p. 712, 1891.
(b) Recherches nouvelles sur la distribution des fractions
rationnelles approchees d'une fonction. Ann. de l'Ec. Nor.,
ser. 3, vol. 19, pp. 153-189, 1902.
(c) Aper9u sur les d6veloppements recents de la theorie des
fractions continues. Compte rendu du deuxieme Congres international des mathematiciens tenu a Paris, pp. 257-264, 1900.
Only a restricted portion of the field is here reviewed, and in
this portion the important work of Pincherle is overlooked.
17. Pade. (a) Sur les series entieres convergentes ou divergentes et
les fractions continues rationelles. Acta Math., vol. 18, pp.
97-111, 1894.
(a') Sur la possibilit6 de definir une fonction par une serie
entiere divergente. Comp. Rend., vol. 116, p. 686, 1893.
See also No. 26a, 76.
II. ON CONVERGENCE.
(For a resume of the criteria for the convergence of continued
fractions with real elements see PRINGSHEIM'S report in the Encyklopddie der mathematischen Wissenschaften, I A 3, p. 126, ff.)