738] 
NOTE ON A HYPERGEOMETRIC SERIES. 
127 
viz. this is 
We have 
and 
hence 
■ (a — a 5 ) X 
1 + X + X 2 * 
QP' 2 + PQ' 2 = a 10 (- a 5 + aX) + a 2 (a - a 5 X), 
— a 3 — a 15 — (a 7 — a 11 )X, — (cl — a 5 )X ; 
PQ = — a 6 + (a 2 + a 10 ) X — a 6 X 2 , = 1 + X + X 2 ; 
_1 (OP' 2 + PQ' 2 ) — ( g ~ aS ) % 
PQ W + ^ } ~1 + X + X 2 ’ 
and the sum of the two parts is =0. 
Similarly as regards the second equation, the second part 
HTZÏ FQ (P<2 ' + PQ) - FQ ‘ 
IS 
P^{(PQ + FQ)X-PQ}. 
Here PQ' + P'Q is a (a - a?X) - a 5 (- a 5 + aX), which is = 1+ 2X ; and PQ being 
= 1 + X + X 2 , the term in 
is 
(1 + 2X) X — (1 + X + X 2 ), = - (1 — X)(l + X) ; 
hence, outside the { } writing for PQ its value = 1 + X + X 2 , the term is 
-sx(i + x + X‘Ki-xhi + x) = _ 3X(1 + X)> 
1 — X 3 
which is the value of the second part in question ; the first part is 
(PQ' + QPJ-PQP'Q= (1 + 2X) 2 — (1 + X + X 2 ), = 3X (1 + X) ; 
and the sum of the two terms is thus = 0.