211] 
ON A THEOREM RELATING TO HYPERGEOMETRIC SERIES. 
269 
a.a + l.a + 2./3.j8 + l./3 + 2 
1.2.3.7 + |.7 + f-7 + f 
a . a+1 . /3 . /3 + 1 
+ 1.2.7 + ^ . 7 + f 
+ 1 
. 1 
7 — a . 7 — ¡3 
I.7 + I 
7 — « . 7 — a + 1.7—/8.7 — /3 + 1 
1.2.7 + 4.7 + I 
7 — a.7 — a+1.7 — a + 2.7 — /3.7 — /3 + 1.7 — /3+2 
1.2.3.7 + ^.7 + 1.7 + f 
2a . 2a + 1.2a + 2.2/8. 2/3 + 1.2/3 + 2 
1.2.3.2 7 .2 7 + 1 . 2 7 + 2 ~ ' 
2a . 2a + 1.2/3.2/3 + 1 7 — a — /3 
+ I.2.27.2 7 +1 ' 1 
2a. 2/3 7 — a — /3.7 — a — /3 + 1 
~ 1.2t~ ' 12 
7.7+1.7 + 2 
7 + i-7 + t-7 + f’ 
+ 1 
7 — a — 6 . y — ol — /3+I.7 — a — /3 + 2 
1.2.3 
; 
It may be observed that the function on the right-hand side is, as regards a, a 
rational and integral function of the degree 3, and as such may be expanded in the 
form 
J.a.a + l.a + 2 
+ i? a . a + 1 .7 — a 
+ Ca . .7 — a.7 — a + 1 
+ D .7 — a. 7 — a + 1.7 — a + 2, 
and that the last coefficient D can be obtained at once by writing a = 0; this in fact 
gives 
D 7.7 + 1.7 + 2 = 
7 — /3.7 — /3 + I.7 — /3 + 2 
1.2.3 
7.7 + 1.7 + 2 
7 + 2 • 7 + I • 7 + f ’ 
and thence 
ta _ 7 — /3 ■ 7 — /3 + 1.7 — /3 + 2 
- 1.2.3.7 + 4.7 + f*7 + f ’ 
which agrees with the left-hand side of the equation : and the value of the first 
coefficient A may be obtained in like manner with a little more difficulty ; but I have 
not succeeded in obtaining a direct proof of the equation. The form of the equation 
shows that the left-hand side should vanish for 7 = - 2, which may be at once verified. 
Grassmere, August 25, 1858.