1881.] On Riccatis Equation and its Transformations. 445 



pL-ah^v=0 (3), 



dz z 



by the substitution x=— zi where <7=-. 

 J q * n 



It is well known that these equations admit of integration in a 

 finite form if p= an integer, n= an uneven integer, and q= the 

 reciprocal of an uneven integer. 



The memoir consists of an introduction and eight sections, the head- 

 ings of which are : (1) Direct integration of the differential equation 

 in series and connexion between, the particular integrals; (2) New 

 integration of the differential equation when p = an integer ; (3) 

 Transformations of the original differential equation ; Biccati's 

 equation ; (4) Special forms of the particular integrals in the cases 

 in which the differential equations admit of integration in a finite 

 form ; (5) Evaluation of definite integrals satisfying the differential 

 equations ; (6) Symbolic forms of the particular integrals in the cases 

 in which the differential equations admit of integration in a finite 

 form ; (7) Connexion with Bessel's Functions ; (8) Writings con- 

 nected with the contents of the memoir. 



In the first section six particular integrals of the equation (1) are 

 obtained and the relations between them are examined. When p is 

 not an integer all the six particular integrals extend to infinity and 

 the relations between them present no special peculiarity. When p is 

 an integer two of the series terminate, and we thus obtain two par- 

 ticular integrals of (1), which contain a finite number of terms. The 

 series terminate in consequence of the occurrence of zero factors in 

 the coefficients of the terms, but if they be continued, zero factors 

 occur also in the denominators, so that, after a finite number of terms, 

 the series may be regarded as recommencing and extending to in- 

 finity. If the terminating series are supposed to recommence in this 

 manner, so that all the series extend to infinity, then the relations 

 between the particular integrals are the same as when p is not an 

 integer ; but if the series are supposed to terminate absolutely when 

 the zero terms occur, the relations are quite different. As the finite 

 portions of the particular integrals satisfy the differential equation, it 

 is more natural to regard the series as terminating absolutely, and on 

 this supposition the relations between the particular integrals exhibit 

 a remarkable diversity of form, according as p is or is not an integer. 



The second section contains what is believed to be a new form of 

 the solution of (1) in the case of p= an integer. It is shown that 

 this equation is satisfied by the coefficient of JiP +l in the expansion of 

 e a jw+xk) j n ascending powers of h. The six particular integrals 

 given in the first section and the relations connecting them are also 

 obtained by different expansions of this expression. 



