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XVIII. On Riocati’s Equation and its Transformations, and on some Definite 
Integrals which satisfy them. 
By J. W. L. Glaisher, M.A., F.R.S., Fellow of Trinity College, Cambridge. 
Received and read June 16, 1881. 
Introduction. 
The present memoir relates chiefly to the different forms of the particular integrals of 
the differential equation 
dh 6 
— — am— 
dr-. 
p(p + 1 ) 
u 
( 1 ). 
and to the evaluation of certain definite integrals which are connected with this 
equation. Transforming (1) by assuming u=x~ p v, it becomes 
that is, writing n —1 for 2 p, 
Tv_2p 
dx 2 x 
— — a~v= 0 
dx 
( 2 ), 
d-v n — 1 
dx} x 
ch 
dx 
— a~v=0 
(3). 
and this equation may be transformed into 
a~z~ J -v=0 
(4), 
by the substitution x—~7d, where q=~- The equation (4) may be regarded as the 
standard form of Riccati’s equation (see § III., art. 17). 
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, 
respectively. 
The contents of the memoir are as follows : 
In the first section (§ I.) 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 
integrals extend to infinity, and in this case the relations between them present no 
special peculiarity. When p is an integer, two of the series terminate, and we thus 
obtain two particular integrals of (1) which contain a finite number of terms. The 
