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

2

THEORY OF BESSEL FUNCTIONS

[CHAP. I

And, in fact, this form of the solution was communicated to Leibniz by
James Bernoulli within a year (Oct. 3, 1703) in the following terms*:
"Reduco autem aequationem dy= yydx + xxdx ad fractionem cujus uterque
terminus per seriem exprimitur, ita
X3 X7 3311 X15 X19
__a1 - a5 + -- etc.
3 3.4.7 3.4.7.8.11 3.4.7.8.11.12.15 3.4.7.8.11.12.15.16.19
X4 X8 X12 16
-a - '+ - etc.
3.4 3.4.7.8 3.4.7.8.11.12 3.4.7.8,11.12.15.16
quae series quidem actuali divisione in unam conflari possunt, sed in qua
ratio progressionis non tam facile patescat, scil.
X3 X7 2x11 13X15
x3 x7 2x' 13x$
- + _____ +_ + - - - etc."
Y 3 3. 3.3.3. + 3.3.3.7 1 + 33 3.5.7.7.11
Of course, at that time, mathematicians concentrated their energy, so far
as differential equations were concerned, on obtaining solutions in finite terms,
and consequently James Bernoulli seems to have received hardly the full credit
to which his discovery entitled him. Thus, twenty-two years later, the papert,
in which Count Riccati first referred to an equation of the type which now
bears his name, was followed by a note+ by Daniel Bernoulli in which it was
stated that the solution of the equation~
axl dx + uudx = bdu
was a hitherto unsolved problem. The note ended with an announcement in
an anagram of the solution: "Solutio problematis ab Ill. Riccato proposito
characteribus occultis involuta 24a, 6b, 6c, 8d, 33e, 5f, 2g, 4h, 33i, 61, 21m,
26n, 16o, 8p, 5q, 17r, 16s, 25t, 32u, 5x, 3y, +, -, --, +, =, 4, 2, 1."
The anagram appears never to have been solved; but Bernoulli published
his solution]l of the problem about a year after the publication of the anagram.
The solution consists of the determination of a set of values of n, namely
- 4m/(2m + 1), where m is any integer, for any one of which the equation is
soluble in finite terms; the details of this solution will be given in ~~ 4'1, 4-11.
The prominence given to the work of Riccati by Daniel Bernoulli, combined
with the fact that Riccati's equation was of a slightly more general type than
* See Leibnizens gesamnellte FTerke, Dritte Folge (Mathematik), III. (Halle, 1855), p. 75.
t Acta Eruditorum, Suppl. vIII. (1724), pp. 66-73. The form in which Riccati took the
equation was
x'ndq =du+ uu dx: q,
where q= xn.
+ Ibid. pp. 73-75. Daniel Bernoulli mentioned that solutions had been obtained by three
other members of his family-John, Nicholas and the younger Nicholas.
~ The reader should observe that the substitution
b dz
z dx
gives rise to an equation which is easily soluble in series.
II Exercitationes quaedam mathematicae (Venice, 1724), pp. 77-80; Acta Eruditorum, 1725,
pp. 465-473.