764 
MR. J. W. L. GLAISHER ON RICCATI’S 
only two terms, is the simplest form ; but as regards the expression of the results, 
both (1) and (3) are superior in every respect. The equation (3) was adopted as the 
standard form by M. Bach in his paper of 1874 (see § VIII.). 
The form 2q— 2 for the exponent in Riccati’s equation (4) was first employed, I 
believe, in Cayley’s paper in the ‘Philosophical Magazine’ for 1868, which has been 
already referred to. The use of the quantity q greatly simplifies the formulae relating 
to the solution of the equation. 
With the exception of § VII., the memoir was written about three years ago, the 
delay in communicating it to the Society being due to the fact that it seemed desirable 
to connect the results more closely with Bessel’s Functions. As the theory of these 
functions forms a distinct and recognised branch of analysis, and as the differential 
equations considered are transformable into Bessel’s equation by very simple changes 
in the variables, it was clearly of importance to examine with some care the connexion 
of the formulae with those of Bessel’s Functions, and it even seemed possible that it 
might be advisable to adopt Bessel’s equation as the standard form. For the reasons 
already stated it appeared that this was not the case, and that the analytical treat¬ 
ment of the subject was complicated by the change to Bessel’s equation. It is well 
known that the general integrals of the differential equations (l), ...(4) can be 
expressed in terms of Bessel’s Functions ; and Lommel has specially considered these 
solutions in several papers in the c Mathematische Annalen.’* In these papers, 
however, the points to which the memoir relates are not referred to. It therefore 
seemed sufficient to give in § VII. the connexion between the principal formulae, 
reserving for a separate paper, if it should appear desirable, the examination of the 
relations in which the series considered in the memoir stand to Bessel’s Functions 
with negative indexes and to the functions of the second kind introduced by Lommel 
and by Neumann. 
During the time that the memoir has been in manuscript I have published two 
extracts from it, viz. the theorem in § II., arts. 8, 9, in the £ British Association 
Report’ for 1880, and the theorem (50) and its proof (§ VI., art. 41) in the £ Proceedings 
of the Cambridge Philosophical Society’ for 1879. 
The differential equations (1), ... (4) present three distinct peculiarities, viz. (i.) 
they are finitely integrable only in special cases ; (ii.) they are satisfied by certain 
remarkable definite integrals, which have attracted attention quite independently of 
the differential equations; and (iii.) the solutions when finite admit of being exhibited 
in various symbolic forms. In reference to the third of these properties, it is remark¬ 
able how much attention has been devoted to the solutions of the equations in these 
finite cases during the last fifty years. The differential equations (1), . . . (4) have 
however been frequently discussed not as simple transformations of one another 
* Yol. ii. (1870), pp. 624-635 ; yoL iii. (1871), pp. 475-487; vol. xiy. (1879), pp. 510-536. 
