of the Hyper geometrical Series. 



241 



Pi = v 6 — 5 Vi/^4 — 10 vfps — 10 j/ x 3 ^2 — V 



_ c 5 m 2 (n* -f ti^ 4- 77? 2 ) (t? 2 + 2m 1 n + 4m 2 ) 

 7i 5 (n-l)(>i-2)(n-3)(7i-4) 



X {n 4 + ?i 3 (10?7i 2 +12m 1 + 5) +w 2 (10m 1 m 2 + 127/i l 2 ) 

 + n(10m 2 2 + 24m 1 m 2 ) + 24m 2 2 }. . . 

 To determine the value of n, m u and m 2 , let us write 



A=A t 3 2 /^ 3 , A=/*V/*s 8 i ^3=^5/(^3^2) ; 



and to render the elimination easier, let us put 

 e = w 2 -f nmx + m 2 , ' 



m 2 e = 2 2 , V (22) 



e-\-m 2 — z x . 

 Then, from (13) and (14), 



(21) 



A-(Mt ( ^-^ 



(23) 



From (19) and (14), 



From (21), (13), and (13), 



ft = (^p^ {^ + 5 ^+^(10n--24) + 12(* 1 »-n«* l )}. (25) 



These are linear in 2 2 ; collecting all the terms of ^ 2 on the 

 left, we can rewrite (23) to (25), 



*2#1 



(n-2V 



71-1 



— ^4 



7l 4 + 4^ 1 2 -7l 2 < 2r 1 ), 



(26) 



zA/3 2 {n ^n-3) _ (3h _ 6) J = »*+!!■ + e^-^), ( 27 ) 



z 2 {g 3 (n ""^" 4) -(10n-24) 1 = n« + 5n'+12fa»-n« g| ). (28) 



Multiply (26) by 3, and (27) by 2, and subtract the results 

 from (28) : 



{ 



'•Jfl 



(n-3)(n-4) „„ (n-2) 2 



1 



3/8, 



!L_2L_(i0n_24)l=-2» 4 + 5n 3 , . (29) 



^k (W - 8) _ ( r 4) - 2 /3^ "-^r 3) -(4n-12)U-n 4 + 3n3. 



(30) 



