ACE. 
17. 
ACE. [709 
710] 17. 
q ; viz. the 
, = 33, 
!, = 33, 
5, =33, 
II 
CO 
w 
in the first 
is therefore 
3ion. In the 
is a triple 
ondition. In 
; in each of 
5, where the 
al reasoning: 
the number 
nadric curve: 
it writing it 
the number 
ingly infinite 
a diminution 
3, in fact, the 
4 constants, 
= 5 constants; 
le number is 
710. 
ON A DIFFERENTIAL EQUATION. 
[From Collectanea Mathematica: in memoriam Dominici Chelini, (Milan, Hoepli, 1881), 
pp. 17—26.] 
In the Memoir on hypergeometric series, Crelle, t. xv. (1836), Kummer in effect 
considers a differential equation 
(a'z 2 + 2b'z + c ) dz 2 (ax 2 + 2hx + c) dx z 
z 2 (z — l) 2 x 2 (x — l) 2 ’ 
viz. he seeks for solutions of an equation of this form which also satisfy a certain 
differential equation of the third order. The coefficients a, b, c are either all arbitrary, 
or they are two or one of them, arbitrary ; but this last case (or say the case 
where the function of x is the completely determinate function x 2 + 2bx + c) is scarcely 
considered : a', b', d are regarded as determinable in terms of a, b, c; and z is to 
be found as a function of x independent of a, b, c : so that when these coefficients 
are arbitrary, the equation breaks up into three equations, and when two of the 
coefficients are arbitrary, it breaks up into two equations, satisfied in each case by 
the same value of £ ; and the value of z is thus determined without any integration : 
these cases will be considered in the sequel, but they are of course included in the 
general case where the coefficients a, b, c are regarded as having any given values 
whatever. 
Writing for shortness X = ax 2 + 2bx + c, in general the integral 
f Ndx 
J DVX ’ 
where D is the product of any number n of distinct linear factors x — p, and N is 
a rational and integral function of x of the order n at most, and therefore also the 
integral 
[NVXdx [XXdx 
D JdJx’ 
C. XI. 
3