226 
DR. W. F. SHEPPARD ON 
(E) ^ (lc+s-2, 
s = 1 
k+f-2)(k-s, k-f) 
(2k + 2/— 3, 2fc-2) (n, 25-1] «*. 
(2k + 2s-3, 2k-2)(n, 2/-l] M 4 
(105) 
s-k 2f {2k-\, 2/-1} (4m, 2s-l] * 2s _ 2 
s-f2k {2k-\, 2s -1} (4m, 2/-1] 
(ii.) From (i.) we obtain 
(A) fl/.j 
(./+/> . 7 - 1 ) (m,/+i) 
(2//-1) (m,/+l)’ 
. (105a) 
. (106) 
(B) 
(D) 
(E) 
a . / u_; {2fc, 2/} [4m, 2fc + l) 
2/ ’ 2 *“^ ' {2£, 2&}[4m, 2/+1)’ • ' 
. _ / _ u-/ {2/c-l, 2/-1} (4m, 2fe] 
02/ - 1>2A - 1 1 ' {2jfe-l, 2ife-l} (4m, 2/]’ • 
, y c _ f {2k-l,2f-l}^m,2k) 
2/ ~ 1 ’ 2 *- 1 ~ 1 ; { 2 / v - l , 2 / 1 - 1 } [| m , 2 /) ’ ' 
, — (_y-f {%k — 2, 2/— 2} (4m, 2/>—l] 
02/ - 2 ’ 2ft - 2 ~ 1 ; (2& —2, 2A —2} (4m, 2/-1] * 
(107) 
(108) 
(109) 
( 110 ) 
(iii.) Also, by putting/* — j in (101) and f — k in (102)-(105), 
(A) 
<1 
II 
S ~A h (2/+1,/) (m, s+l) AS 
= A (s, i)-7~ -}—X-—(A X> 
s =i J 0+s+l,.?) (wi-,^ + 1) 
• • (HI) 
(B) 
o 
£ 
II 
CM 
w 
S = ’ l {2& + 1, 2&+l} Tim, 2s+l) « 3s „. 
g = fc (2A+1, 2s+l} [4m, 2& + 1) 
• • (112) 
(C) 
1-24—1 = MO 2 * \ 
_ s = n {2&, 2&} (|-m, 2s] 
s = 4 {2&, 2s} (4m, 2&] ^ 
. . (113) 
(D) 
E 2 *_! = 
{2A 2&} [4m, 2s) Ms-u, 
“.r fc {2i,2*}[4m,2A) *’ * * * • 
- - (H4) 
(E) 
1-24—2 = 
! v {2^—1, 2&—1} (4m, 2s—l] a 2s _ 2 
'..*{24-1, 2s —1} 0,24-1/ *' ' 
• ■ (H5) 
(iv.) It has been pointed out in § 11 (iv.) that the differences of order j all have 
the same improved value. It follows that (112)—(115) are particular cases of 
(ill), expressed in terms of central differences; the proper values being taken for 
m and for/, and u 0 being altered to u_ n for (112) and (113) and to u_ n+i for (114) and 
(115). We can verify this by expressing the E’s in terms of the differences of 
order/. For (A) we have 
+ ... 
AX = {(l + A)-l} s j A j u 0 = A j u s _j-(s-j, l) AX-j-i 
