766 
MR. J. W. L. GLAISHER ON RICCATI’S 
Che headings of the eight sections, with the numbers of the articles which they 
contain and the pages, are as follows : 
§ I. Direct integration of the differential equation in series, and connexion between the particular 
integrals. Arts. 1-7 ; pp. 766-774. 
§ IT. Integration of the differential equation when p = an integer. Arts. 8-15 ; pp. 774-779. 
§ III. Transformations of the original differential equation. Riccati’s equation. Arts. 16, 17; 
pp. 779-782. 
§ IY. Special forms of the particular integrals in the cases in which the differential equations admit 
of integration in a finite form. Arts. 18, 19; pp. 783, 784. 
§ Y. Evaluation of definite integrals satisfying the differential equations. Arts. 20-28, pp. 784-797. 
§ YI. Symbolic forms of the particular integrals in the cases in which the differential equations 
admit of integration in a finite form. Arts. 29-42 ; pp. 798-819. 
§ YII. Connexion with Bessel’s Functions. Arts. 43-48 ; pp. 819-822. 
§ VIII. Writings specially connected with the contents of the memoir. Pp. 823-828. 
§ 1 - 
Dio 'ect integratiooi of the differential equation in series, and connexion between the 
'particular integrals. Arts. 1-7. 
1. The most direct method of integrating the differential equation 
d?u 2 p(p+l) 
— — a-«=- —u 
dx~ x z 
( 1 ), 
and obtaining the relations that exist between the different particular integrals, appears 
to be as follows. 
Let 
u=% A r x m+r , 
the summation extending to all positive integral values of r ; then, substituting in 
the differential equation, we have 
(on + r-\-p) (m -f r—p — 1) A,.—a 2 A,._. 3 -= 0, 
whence, putting r— 0 or 1, 
m— —p or p-\- 1. 
Taking the first root, the equations giving Ao, A. t , A G . . . are 
2(1 — 2p) A 3 — a 2 A 0 = 0, 
4(3 — 2p)A 4 —« 3 Ao = 0, 
6(5 — 2p) A 6 —a 3 A 4 = 0, 
