Colloquium publications.

182

THE BOSTON COLLOQUIUM.

was first given by Heine in Crelle's Jour., vol. 42 (1851), p. 72.
See also his Handbuch, vol. 1, pp. 78-79, 197-200. Among
the numerous works relating to expansions in terms of Kugelfunctionen erster und zweiter Gattung may be mentioned:
91. Bauer. Von den- Coefficienten der Reihen von Kugelfunctionen
einer Variablen. Jour. fir Math., vol. 56, pp. 101-121, 1859.
92. C. G. Neumann. Ueber die Entwickelung einer Function mit imaginirem Argumente nach den Kugelfunctionen erster und zweiter
Gattung, Halle, 1862.
93. Thom6. Ueber die Reihen welche nach Kugelfunctionen fortschreiten. Jour. fiir Math., vol. 66, pp. 337-343, 1866.
94. Laurent. Memoire sur les fonctions de Legendre. Jour. de Math.,
ser. 3, vol. 1, pp. 373-398, 1875.
See the comments by Heine in vol. 2, pp. 155-157, also by
Darboux and Laurent in the same vol., pp. 240, 420.
Numerous memoirs relate to series in terms of the polynomials arising from the expansion of (1 - 2ax + a2)v. It suffices
here to refer to the Encyklopaidie der Math. Wissenschaften,
I A 10, ~ 31.
95. Frobenius. Ueber die Entwicklung analytischer Functionen in
Reihen, die nach gegebenen Functionen fortschreiten. Jour.
fur Math., vol. 73, pp. 1-30, 1871.
An interesting memoir.
96. Darboux. Sur l'approximation des fonctions de tres-grands nombres et sur une classe etendue de developpements en serie, Part
2. Jour. de Math., ser. 3, vol. 4, pp. 377-416, 1878.
97. Gegenbauer. Ueber Kettenbriiche. Wiener Berichte, vol. 80, Abth.
2, pp. 763-775, 1880.
98. Poincare. (a) Sur les equations lineaires aux diffSrentielles ordinaires et aux differences finies. Amer. Jour. of Math., vol. 7,
pp. 243-257, 1885.
This gives an important criterion for the convergence of series
of polynomials. See lecture 4.
(a') Sur les series des polynomes. Comp. Rend., vol. 56, p. 637,
1883.
99. On the series zA,(x - al)(x - a2). (x - a,,).
A series of this form is employed in Newton's interpolation
formula, Philosophike naturalis principia, book 3, lemna V.
See the Encyklopadie der Math. Wissenschaften, I D 3, ~ '8. A
similar use is made by