• of the Hyper geometrical Series. 239 



Put a?=I, we have 

 ( 7l - 1 )(%2) 1 = (n + w 1 + m 2 -2)x 1 + (w 2 ~m 1 4-l)(%o)i 



. . n 2 + n(5m 2 — 1) + m 2 2 + m x m 2 — 2m 2 /1A . 



= (Xo)i , • (10) 



by aid of (7). 



l^ us ri 2 + n(3m 2 — l)+m 2 2 + 77? 1 m 2 --2m2 /1f , N 



v 2 = ^ t fr — . • (11) 



n(n — 1) v 7 



and o 



^ 2 = v 2 -v 1 J 



_^ 2 m 2 (n 2 + m 1 n + m 2 ) 



or, we may write 



c W?(n + «)(n + / 3) (1) 



:* n 2 (n — 1) v y 



Multiplying (9) by # and differentiating again, we find 



{l-x) {x± + m^ +{m 1 + m 2 -2)X 2 +(2m 2 — m 1 )xi + (m 2 —m 1 + l)xo} 

 + (n— 2)^— (71 + 2772! + w 2 -3)x2 — (2w 2 — m 1 )%i~(m 2 —m 1 + l)xo = 0. (14) 

 Putting a?=l, we have 

 (n-2)(x 3 )i=(n + 2m 1 + m 2 -3)( %2 ) 1 + (27722-77?!) (^Oi 



+ (2722-772! + 1) (% )l, 



or, by aid of (7) and (10), 



v 3 = c 3 {?2 3 + n 2 (7m 2 — 3) + n (6tt2 2 2 + 67Bi772 2 — 15m 2 + 2) + m 2 3 



+ %m Y m 2 2 + 2m l 2 m 2 — 7m 2 2 — Gm^g + 6m 2 }-4-w(?2 — 1) (72 — 2) . (15) 



Hence, since fi z = v 3 — Sv^— Vi 3 , we have after some reduc- 

 tions 



_ c 3 xj3(n + *)(n + {3Xn + 2a)(n + 2l3) 

 P»- » 8 (»-rl)(n-2) ' ' ' U j 



Differentiating (14) after multiplication by a?, we find 



(l-^)|X5+K + l)X4+(2m l + m2-2)x3+(3m 2 -2) %2 



+ (3m 2 — 2m 1 + l)xi+(w 2 — m 1 + l)xo} + (w— 3)^4 



— (71 + 3772! + m 2 — 3)X3~(3w 2 - 2)% 2 —(3772 2 — 2772! + 1)^! 



- (7722-772! + 1) %0 = (17) 



Putting * =3=1 j we have 



(n - 3) ( X4 )x = (n + 37?ii + m, - 3) Gfc) i + (3m 3 - 2) ( %2 ) j 



+ (3m, — 2m 1 + l)(xi)i+ (wia-mx+lXxoJi. 



