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

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BESSEL FUNCTIONS BEFORE 1826

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John Bernoulli's equation * has resulted in the name of Riccati being associated
not only with the equation which he discussed without solving, but also with
a still more general type of equation.
It is now customary to give the namet Riccati's generalised equation to
any equation of the form
dy
-d =P + Qy + R2,
where P, Q, R are given functions of x.
It is supposed that neither P nor R is identically zero. If R=0, the equation is linear;
if P=0, the equation is reducible to the linear form by taking 1/y as a new variable.
The last equation was studied by Euler +; it is reducible to the general
linear equation of the second order, and this equation is sometimes reducible
to Bessel's equation by an elementary transformation (cf. % 3'1, 4'3, 4'31).
Mention should be made here of two memoirs by Euler. In the first~ it
is proved that, when a particular integral yj of Riccati's generalised equation
is known, the equation is reducible to a linear equation of the first order by
replacing y by yi + l/u, and so the general solution can be effected by two
quadratures. It is also shewn (ibid. p. 59) that, if two particular solutions are
known, the equation can be integrated completely by a single quadrature; and
this result is also to be found in the secondl| of the two papers. A brief discussion of these theorems will be given in Chapter Iv.
1'2. Daniel Bernoulli's mechanical problem.
In 1738 Daniel Bernoulli published a memoirF containing enunciations of
a number of theorems on the oscillations of heavy chains. The eighth * of
these is as follows: " De figura catenae uniformiter oscillantis. Sit catena AC
uniformiter gravis et perfecte flexilis suspensa de puncto A, eaque oscillationes
facere uniforrmes intelligatur: pervenerit catena in situm AMF; fueritque
longitudo catenae = 1: longitude cujuscunque partis FM = x, sumatur n ejus
valoris t ut fit
1 l 13 14 _ 1
1 - - -t + etc. = 0.
n 4+ n 4. 9n3 4 9. 16n 4.9.16.25 +
* See James Bernoulli, Opera Omnia, I. (Geneva, 1744), pp. 1054-1057; it is stated that the
point of Riccati's problem is the determination of a solution in finite terms, and a solution which
resembles the solution by Daniel Bernoulli is given.
t The term ' Riccati's equation ' was used by D'Alembert, Hist. de l'Acad. R. des Sci. de Berlin,
xix. (1763), [published 1770], p. 242.
+ Institutiones Calculi Integralis, ii. (Petersburg, 1769), ~ 831, pp. 88-89. In connexion with
the reduction, see James Bernoulli's letter to Leibniz already quoted.
~ Novi Comm. Acad. Petrop. vIII. (1760-1761), [published 1763], p. 32.
11 Ibid. ix. (1762-1763), [published 1764], pp. 163-164.
Y~ "Theoremata de oscillationibus corporum filo flexili connexorum et catenae verticaliter
suspensae," Commn. Acad. Sci. Imp. Petrop. VI. (1732-3), [published 1738], pp. 108-122.
** Loc. cit. p. 116.
tt The length of the simple equivalent pendulum is n.
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