446 On Riccatis Equation and its Transformations. [June 16, 



The third section contains the six particular integrals of (2) and 

 (3), corresponding to those of (1), from which they are deduced. 



The fourth section relates to the particular cases in which the 

 differential equations admit of integration in a finite form. If a 

 differential equation is satisfied by an infinite series, and if for certain 

 values of a quantity involved in it the series terminates, then in this 

 case we may present the integral in a different form by commencing 

 the finite series at the other end and writing the terms in the reverse 

 order. These reverse forms in the case of (1), (2), (3) are given in 

 this section. 



The fifth section contains the evaluations of the definite integrals — 



f 30 m f 00 cos hx 



J^-^&S Jo("> + « a )"* B ' 



where m denotes any real quantity and n any positive quantity. 



— 4f 



These integrals have been evaluated when m is of the form -, 



& 2t±l 



and when n is a positive integer, but the general formulae are, the 



author believes, new. The results exhibit changes of form similar to 



those referred to in the account of the contents of the first section. 



Certain formulae of Cauchy's and Boole's are also considered in this 



section. 



The sixth section, which is the longest in the memoir, relates to the 

 different symbolic solutions of (1), (2), (3) 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 the fifth section ; 

 and the various symbolic theorems to which they lead, by comparing 

 different forms of the results, are examined. A great many symbolic 

 solutions of these differential equations have been given by Gaskin, 

 R. L. Ellis, Boole, Lebesgue, Hargreave, Williamson, Bonkin, and 

 others, and these are briefly noticed and connected with one another. 



The seventh section relates to the connexion between the results 

 contained in this memoir and the formulae of Bessel's Functions. The 

 equation (1) may, as is well known, be transformed into Bessel's 

 equation 



dhv _j_ 1 dw , A _ v 1 



day 1 x dx V x 2, 



by the substitutions u=x i w, p-j-^—v, a-— — 1, so that the theorems 

 relating to the solutions of (1) have analogues in the solutions of 

 Bessel's equation. 



The eighth section contains a list of writings which are closely con- 

 nected with the subject of the memoir, with short accounts of their 

 contents. 



