[737 
738] 
125 
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738. 
NOTE ON A HYPERGEOMETRIC SERIES. 
\í, vY- 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 
pp. 268—270.] 
In the memoir on hypergeometric series, Schwarz, “ Ueber diejenigen Falle, &c.,” 
Crelle, t. lxxv. (1873), pp. 292—335, the author shows, as part of his general theory, 
that the equation 
d-y dy a Qg 
+ 
-y=°> 
dx 2 xx dx x. 1 — x 
which belongs to the hypergeometric series F(\, — b x )> algebraically integrable, 
having in fact the two particular integrals 
y 2 = J(cl — oPx®) ± \/(— a 5 + ax*), 
where a is a prime sixth root of —1, a 6 +l = 0, or say a 4 — a 2 +l=0 (see p. 326, 
a being for greater simplicity written instead of S 2 , and the form being somewhat 
simplified). 
It is interesting to verify this directly; writing first y = f(Y) and then x = X 3 , 
the equation between Y, X is easily found to be 
d 2 Y 
\X 2 y dY 
-*c 
dY 
+ 
3 y 
8 Y' 2 = 0, 
dX 2 1 -X 3 * dX 2 \dX) ‘ l-X 3 
and the theorem in effect is that that equation has the two particular integrals 
Y=f(P)±f(Q), 
P and Q being linear functions of X: in fact, 
P= a - a:X, 
Q — — a 5 + a X.