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

88

THEORY OF BESSEL FUNCTIONS

[CHAP. IV

and so the series terminates with the term A,,z-qm if q has either of the values
~ 1/(2n + 1); and this procedure gives the solution* examined by Bernoulli.
The general solution of Riccati's equation, which is not obvious by this method, was
given explicitly by Hargreave, Quarterly Journal, vII. (1866), pp. 256 — 258, but Hargreave's
form of the solution was unnecessarily complicated; two years later Cayley, Phil. Mag. (4)
xxxvi. (1868), pp. 348-351 [Collected Papers, vII. (1894), pp. 9-12], gave the general solution in a form which closely resembles Euler's particular solution, the chief difference between
the two solutions being the reversal of the order of the terms of the series involved.
Cayley used a slightly simpler form of the equation than (2), because he took constant
multiples of both variables in Riccati's equation in such a way as to reduce it to
(5) d1 +1 2q-2=0.
4'14. Cayley's general solution of Riccati's equation.
We have just seen that Riccati's equation is reducible to the form
d + 12 _ c2z2-2 = 0,
dz
given in ~ 4'13 (2); and we shall now explain Cayley's- method of solving
this equation, which is to be regarded as a canonical form of Riccati's
equation.
When we make the substitution+ V = d (log v)/dz, the equation becomes
dz2
(1) gv c - 0;
and, if U1 and U2 are a fundamental system of solutions of this equation, the
general solution of the canonical form of Riccati's equation is
(2) C7 1, + C2, U,
where C7 and C2 are arbitrary constants and primes denote differentiations with
respect to z.
To express U1 and U2 in a finite form, we write
v = w exp (czq/q),
so that the equation satisfied by w is ~ 4'13 (4). A solution of this equation
in w proceeding in ascending powers of zq is
q-l (q - 1) (3q-)
q (q- I1) c q (q- 1) 2q (2q - 1) 22
(q- ) (3q- l)(q-1-) 3z
q(q - l) 2q (2q- l) 3q (3q - l)
and we take U1 to be exp (czq/q) multiplied by this series.
* When the index n of the Riccati equation is - 2, equation (4) is homogeneous.
t Phil. Mag. (4) xxxvi. (1868), pp. 348-351 [Collected Papers, vn. (1894), pp. 9-12]. Cf. also
the memoirs by Euler which were cited in ~ 4'13.
+ This is, of course, the substitution used in 1702 by James Bernoulli; cf. ~ 1-1.