762 
MR. J. W. L. GLAISHER ON RICCATI’S 
does not suggest the general formula, the terms of the finite series being written in- 
the reverse order. 
Certain formulae of Boole’s and Cauchy’s are also considered and extended in this 
section. 
The sixth section, which is the longest in the memoir, relates to the numerous 
symbolic solutions of the equation (l) and its transformations (3) and (4) in the cases 
in which they are integrable in finite terms. In this section these symbolic solutions 
are derived from the definite integrals considered in § V. ; and the various symbolic 
theorems to which they lead by comparing different forms of the results are examined. 
A great many symbolic solutions of the differential equations have been given by 
R. L. Ellis, Boole, Lebesgue, Hargreave, Williamson, Donkin, &c., and these 
are briefly noticed and connected with one another. It may be observed that the 
solution 
1 fZy+i /c 1 c“ + c 2 e _fl - 1 ' 
x dx \ x 
which has been several times independently discovered, seems to have been first 
published by Mr. Gaskin, who in effect gave it in a problem set in the Senate House 
Examination at Cambridge in 1839. 
The seventh section relates to the connexion between the results given in §§ I.-YI. 
and the formulae of Bessel’s Functions. Bessel’s equation 
dho 1 dw ( v v 
Tod-T-+ 1 — 2 
dx~ x dx \ or 
w= 0, 
may He derived from (1) by the simple substitutions 
u—xhv, — G a 3 = — 1 ; 
so that all the theorems relating to the solutions of (1) have analogues in the solutions 
of Bessel’s equation, which are deducible from them by these transformations. In 
this section the formulae in Bessel’s Functions which correspond to those considered 
in the memoir are stated in a convenient form for comparison. The number of such 
formulae is not great, and the substitution of x /(— 1) fora, which converts exponentials 
into sines and cosines, and a single series multiplied by an exponential factor into the 
sum of two series multiplied respectively by a sine and a cosine, changes considerably 
the appearance of the results, which, from an analytical point of view, are less simple 
when the differential equation is of Bessel’s form. The principal case considered 
in the theory of Bessel’s Functions is that of v = an integer : this corresponds 
to the case of p = an integer + f, which is generally excluded in this memoir, as 
it renders certain of the particular integrals infinite (§ I., arts. 1, 3). The case of 
