224 
DR. W. F. SHEPPARD ON 
(vii.) Writing 
Elf.. O . 1-1, et.p... , a(a + l) . (3 ((3+l) ... , 
F{<x> 8> +•■■■> = 1 + 7^-. + rGli).i,(i, + l).. + 
where a is a negative integer, it should be noted that 
and that, if 
then* 
F{-n, 0 ; 1, *} = . . 
cl — — n, + —?r + /3 + y+1, 
(93) 
FI O ,,. , ,L v i _ D-ff. ra][x-g. »] _ [V-'-r. «][x-y, »1 
1 ,fty ’ ’^’ xf [>,»] [x.»] “ [*. *][*»] 
(94) 
(viii.) For the central-difference formulae it will be convenient to write, if r and s 
are both even or both odd and s ^> r, 
{», r} = (ty (,+r - «)(»’* <+ * r) . 
r + J 
{r. = (|) r 
(r+1) (%r + % 8 > r ) 
so that, if k ;> f, 
Sob, i of, | \ — [/+t> (&>./) _ [«/+!> &+t] (&+1,/+l) 
{ 2 &+ 1 , 4 /+ 1 } 2 /+2 2/+1 
{ 2 £, 2 /} 
L/ + A> A] (lc,f) _ [/+f, A— l] (A— I,/ — l) 
2/+1 
2 / 
and, if k s, 
{2A+1, 2s+ 1} = 
[s 4- A] _ [s + •g - , k + 1 ] 
(2k+2)(s,k) (2s +1) (s, k) ’ 
{2&, 2s} = [s + 2 , ^] 
(2k+l) (s, k) 
(95) 
(96) 
(97) 
(98) 
(99) 
( 100 ) 
(ix.) The successive steps are as follows. The formulae for the e’s (the improved 
values of the differences) in terms of the differences have already been found in 
“ Reduction ” and “ Fitting ” ; they depend on certain theorems as to the coefficients 
when l.cc. of moments or sums are expressed in terms of differences. From these 
formulae we get the 6’ s, and also the E’s ; and thence we get, in each case, the 
progression formula supplied by (77). This formula is not really necessary, but it is 
useful for checking. The A’s, i.e. the m.ss.e. of the E’s, are found from (86)-(92), 
using (IV.) of § 6 ; this results in certain hypergeometric series, to which we apply 
(93) and (94). We then get expressions for the A’s, by (80). From these, by (40), we 
* ‘Proceedings of the London Mathematical Society,’ 2nd series, x., 474, 
