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

1.5]

BESSEL FUNCTIONS BEFORE 1826

9

integration (cf. ~ 8'31). Laplace seems to have been dubious as to the validity
of his inference because, immediately after his statement about real and
imaginary variables, he mentioned, by way of confirmation, that he had
another proof; but the latter proof does not appear to be extant.
1'5. The researches of Fourier.
In 1822 appeared the classical treatise by Fourier*, La Theorie analytique
de la Chaleur; in this work Bessel functions of order zero occur in the discussion of the symmetrical motion of heat in a solid circular cylinder. It is
shewn by Fourier (~~ 118-120) that the temperature v, at time t, at distance
x from the axis of the cylinder, satisfies the equation
dv K /d2v 1 dvc
dt CD dx2+d 'xd ).where K, 7, D denote respectively the Thermal Conductivity, Specific Heat
and Density of the material of the cylinder; and he obtained the solution
-mt 2i g2 g94 4,1
1 22 22. 42 22.42. 2 62
where g = mCD/K and m has to be so chosen that
hv + K (dv/dx) = 0
at the boundary of the cylinder, where h is the External Conductivity.
Fourier proceeded to give a proof (~~ 307-309) by Rolle's theorem that
the equation to determine the values of m, hast an infinity of real roots and
no complex roots. His proof is slightly incomplete because he assumes that
certain theorems which have been proved for polynomials are true of integral
functions; the defect is not difficult to remedy, and a memoir by Hurwitz+
has the object of making Fourier's demonstration quite rigorous.
It should also be mentioned that Fourier discovered the continued fraction
formula (~ 313) for the quotient of a Bessel function of order zero and its
derivate; generalisations of this formula will be discussed in ~~ 5'6, 9'65.
Another formula given by Fourier, namely
a2 a4 a6 1 =
- 2 +... = - cos(a sin x) dx,
22 22.4222 2.42.62 V
had been proved some years earlier by Parseval~; it is a special case of what
are now known as Bessel's and Poisson's integrals (~~ 2'2, 2'3).
* The greater part of Fourier's researches was contained in a memoir deposited in the archives
of the French Institute on Sept. 28, 1811, and crowned on Jan. 6, 1812. This memoir is to be
found in the Mem. de l'Acad. des Sci., iv. (1819), [published 1824], pp. 185-555; v. (1820),
[published 1826], pp. 153-246.
t This is a generalisation of Bernoulli's statement quoted in ~ 1'2.
Math. Ann. xxxiII. (1889), pp. 246-266.
~ Mem. des savans etrangers, i. (1805), pp. 639-648. This paper also contains the formal
statement of the theorem on Fourier constants which is sometimes called Parseval's theorem;
another paper by this little known writer, Memn. des savans etrangers, I. (1805), pp. 379-398, contains a general solution of Laplace's equation in a form involving arbitrary functions.