m 
Hm 
DETERMINATIO SERIEI NOSTRAE PER AEQUATIONEM DIFFERENTIALEM ETC. 
217 
[89] F[a, fi, afi, 1 — x) 
= -ng^h^-T) (log«+¥(a-l)+V(6-l)-a V(0)j P(a, <5, 1, *) ' 
_ H(a + g—l) | A CU6 
II(a—l)II(6 —l) i 
2.1.2 
+ [A+B+C 
-f- etc.} 
■XX 
a.a+l.a+2.6.6+1.6 + 2 3 
_____ — t % 
. 3 .1.2. 3 
ubi 
A=l + |-2. B 
— U — 2u 
a -f-1 g 1 a ' 
C ' — ¿TT2+rb — *’ etC ’ 
Ita e. g. pro a — \, fi = £ obtinemus (cfr. form. 52, 71) 
[90] F{hh 1. 1 — *) 
= logrV^ • T’ * > ^ 
— ii 2 • ( 2 + i) 524- ( 2 + -H- A) 
1.1.3.3.5.5 3 
X 
4.4.6 . 6 
+ (2 + ^+^+^:‘:^xX+ etc., 
= — — {logi, <#)~h\ x “h-rli x?> 
i 1 0 2 5 0 1 r> 5 —U 1 3 9 4 2 3 9 I afn I 
6 5 5 3 6 0 * 10485 JIO* —J— CUv. j 
Denique casum tertium, ubi 7 est integer positivus unitate maior, seorsim 
tractare haud necesse est, quum sit 
F{ a, fi, a-\-fi-\-\—y, \—x) — —7, 6 + 1 — 7, a+6 + l —7, l—x) 
transformatioque seriei _F(« + 1— 7, 6 + 1— 7, a+ 6 + 1— 7, 1 — x) pro 7 + 1 
ad casum primum sponte reducatur. 
47. 
Transimus ad alias transformationes, inter quas primum locum obtineat 
Hinc fit dx = —adeoque 
y—1 (y—0 
substitutio x 
dP 
— —^(1—yf, differentiando denuo fit 
y dP 
a dP 
d dx — 
ddP , /. x4 ddP , V 3dP 
— "H 1 — y) T^~ 2 ( 1— y) di 
(i—y) 2d d^ + 2 ( 1— ^) dP ’ adeo( l ue 
dP 
dy“ V } dy 
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