228 DR. W. F. SHEPPARD ON 
we have 
(B) ft™, 2/+ 1) P »'. <122> 
(o (fr», 2 Kli[ fe-..< 123 ) 
The first of these has been given, for /= 0, in “Reduction,” § 15 (v.), p. 362 ; the 
notation of P 2t differing, however, in a factor \m. 
(D) (E) Writing 
P 2t -1 — [i m > 2^) E 2t_i, Q‘2t-2 = 2£ l] — 
we have 
(10 ' Lfrn, 2/) f «-. (124) 
(E) (fa>. 2/-l]^-V, = .(125) 
(vi.) The A’s are obtained from the E’s by means of (IV.) of § 6 ; e.g., for advancing 
differences, m.s.e. of E^ = m.p.e. of E ; and A j u 0 . We can use either the values of 
the E’s given by (ill)—(115) or those given by (116)-(120); for the former we 
require the m.pp.e. given in (86), (87), (89), (90), (92), and for the latter the value 
of the m.p.e. of two differences of order f as given in § 20 (x.). Using the former 
method, we get the following results :—- 
(A) a, = 8 T 1 (- y-i (s, j) U +1 o') (,, + j t j) 
’ -j ’ K ' J> (s+ } + \, 3 )(m , 3 + \y 
= (2 'j, j) P { -m+j+ 1, 2j+1 ; 1, 2j+2} 
= j) (m-j-l)\(2j+l) \/{m+j )! 
= ( 2 i,y)/(w, 2/+1];.(126) 
(B) A - / y-ft |2^'+ 1> 2/1+1} [ fm, 2s +1) / J , a 
2k ~ si k { } {2k+l,2s+l}[±m,2k+l) [2S + 2/C ’ S + k) 
= (4 k, 2k) F { —n + k, n + k + 1, 2 k + ^ ; 1, 2 k + 1, 2l' + f} 
= (4&, 2k)/(m, 4&+1];.(127) 
u *\ M _ %" / {2 k, 2k} (^m, 2s] (2s + 2k — 2, s + k— l) 
V ' “- 1 S = U j {2k, 2s} {^m, 2k] 2s + 2k 
= (4& —2, 2& —1)/(4&) . F{ — n + k, n + k + 1, 2k —; 1, 2k + 1, 2k + };} 
= {U-2, 2k-l)/(m, 4^—1] ;.(128) 
(1) A m _, = Y (-)-* ft™. 2«) (2s + 2&—2, a + *-l) 
s = t {2A:, 2s| 2/:) 
= (4& —2, 2k—1) F {—n + k, n + k, 2k—\ ; 1, 2&, 2& + |-} 
= (4&—2, 2&—l)/(m, 4&—l] ;. 
(129) 
