[908 
909] 
69 
3N. 
gh 
- 36 
g'h' 
- 36 
V 
- 36 
w' 2 
- 36 
h' 
+ 72 
yn 
+ 72 
2 h' 
- 36 
hh' 2 
- 36 
h 2 
- 36 
■h’ 2 
- 36 
k 3 
+ 72 
Ï 3 
+ 72 
l'h' 2 
-144 
I'hh' 
-144 
hh' 
-144 
lh 2 
-144 
g'h 
- 36 
g' 2 h' 
- 36 
+ 288 
-936 
gig’Z 
- 
1 
fg 2 g' 2 h' 
+ 
6 
fg 2 g' 2 h 
+ 
6 
fgg'h' 2 
+ 
24 
f 2 gg'h 2 
+ 
24 
ffgg’hh 
+ 
12 
f 3 h' 3 
- 
64 
f' 3 h 3 
- 
64 
f 2 g 2 h 2 
+ 
54 
f' 2 g' 2 h' 2 
+ 
54 
ff'hh! 2 
+ 
96 
ff' 2 h 2 h' 
+ 
96 
+ 372 
- 129 
±2866 
two invariants of 
909. 
ON A PARTICULAR CASE OF RUMMER’S DIFFERENTIAL 
EQUATION OF THE THIRD ORDER. 
[From the Messenger of Mathematics, vol. xx. (1891), pp. 75—79.] 
The general form of equation in question is 
x 
x 
_ 3 
2 
x 
x j 
+ x' 2 
A B G 
(x — l) 2 x (x — 1) X” 
' A' B' C'\ 
(<-i) 5+ <(i-i) + ?j 
= o, 
here x is a function of t; and A, B, G, A', B', C' are numerical constants. For 
various given values of A, B, C, and values determined thereby of A', B', G', the 
equation admits of a solution in the form x = rational function of t\ the theory in 
reference to the cases considered by Schwarz is considered in my paper “ On the 
Schwarzian Derivative and the Polyhedral Functions,” Camb. Phil. Trans., t. xiii. (1883), 
pp. 5—68, [744]. But the theory is considered in a more general and exhaustive 
manner in Goursat’s memoir, “Recherches sur l’equation de Kummer,” Acta Soc. Sci. 
Fennicce, t. xv. (1888), pp. 47—127. I consider here one of the solutions given by 
him, viz. writing 
P = 4i - 5 , X= №, 
Q = 5t - 4 , Y = Q s , 
R = 8t*-llt + 8, Z=-(t-l) 2 R 2 , 
so that, identically, X + F+ Z = 0; then the solution is expressed by either of the 
equivalent equations 
X t 2 P s 
x ~~Z~ (t - l) 2 P 2 ’ 
, F_ (f _ 
x v ~ Z (t-lfR 2 ‘