A treatise on the theory of Bessel functions, by G. N. Watson.

6

THEORY OF BESSEL FUNCTIONS

[CHAP. I

circular membrane of radius a with a fixed boundary* are to be determined
from the consideration that u vanishes when r = a.
This investigation by Euler contains the earliest appearance in Analysis of
a Bessel coefficient of general integral order.
1'4. The researches of Lagrange, Carlini and Laplace.
Only a few years after Euler had arrived at the general Bessel coefficient
in his researches on vibrating membranes, the functions reappeared, in an
astronomical problem. It was shewn by Lagrange- in 1770 that, in the elliptic
motion of a planet about the sun at the focus attracting according to the law
of the inverse square, the relations between the radius vector r, the mean
anomaly M and the eccentric anomaly E, which assume the forms
M =E-esinE, r=a(1 - e cosE),
give rise to the expansions
r c
E=M1 + S AnsinnM, -= l+ e62+ Z B,,cos??Af,
n=1 a n=l
in which a and e are the semi-major axis and the eccentricity of the orbit, and
A,= 2 G (-)m nn42m-n 1 En+2mn B - 2 E (_)m (n + 2mn). nl+2n-2 )n+2m
n =0 2fl+2m n! (n + im)! ' r=O 2) +2f n m (n + n) S
Lagrange gave these expressions for n = 1, 2, 3. The object of the expansions
is to obtain expressions for the eccentric anomaly and the radius vector in
terms of the time.
In modern notation these formulae are written
A,= 2J (ne)/n, B, = -2 (e/n) J'(ne).
It was noted by Poisson, Connaissance des Tems, 1836 [published 1833], p. 6 that
e dAn,
n de '
a memoir by Lefort, Journal de Math. xi. (1846), pp. 142-152, in which an error made by
Poisson is corrected, should also be consulted.
A remarkable investigation of the approximate value of A, when n is large
and 0 < e < 1 is due to Carlini+; though the analysis is not rigorous (and it
would be difficult to make it rigorous) it is of sufficient interest for a brief
account of it to be given here.
Cf. Bourget, Ann. Sci. de l'Ecole norm. sup. II. (1866), pp. 55-95, and Chree, Quarterly
Journal. xxi. (1886), p. 298.
t Hist. de I'Acad. R. des Sci. de Berlin, xxv. (1769), [published 1771], pp. 204-233. [Oeuvres,
II. (1869), pp. 113-138.]
+ Ricerche sulla convergenza della serie che serva alla soluzione del problema di Keplero.
(Milan, 1817). This work was translated into German by Jacobi, Astr. Nach. xxx. (1850),
col. 197-254 [Werke, vnI. (1891), pp. 189-245]. See also two papers by Scheibner dated 1856,
reprinted in Math. Ann. xvIn. (1880), pp. 531-544, 545-560.