DR. W. F. SHEPPARD ON 
230 
P) 
(E) 
l Vi = 
= 
g = k 
X ■) 2 V/-1,2i/-1/ u ' t 5 CUt+h 
<7 = 1 
g = k 
I] 2 U-2f-2,2g-2 rT ^ j + 4 
L 1<7 = 1 
£ = 
£ = — hm 
t = 
£ — —%TYl 
■ . (139) 
. . (140) 
The expression in (136) differs slightly from that given for AA; 0 in “ Fitting,” (15) and 
§ 5 ; see Appendix III. 
(ix. ) Finally, we want to express the e’s in terms of the us. This is done by means of 
the general theorem in § 10 (iii.), that the coefficients of the y s in any e are related 
to the m.pp.e. of this e and the e’s in the same way that the vs are related to the e’s. 
Thus we find that—- 
(A) 
If 
r = m 
A f v 0 = 2 p r u r , 
r — 0 
then 
Pr = 2 (r, g)\ f , g \ . 
<7 = 0 
• (141) 
(B) 
If 
r = n 
= 2 p r u r , 
r = — n 
then 
g = k 
P r = 2 [r, 2 g)\ 2f , 2 j \ ... . 
g = o 
• (142) 
(C) 
If 
r = n 
/jl 2f ~ l v 0 = 2 p r u r , 
r = — 7i 
then 
g = k 
Pr ~ ^ (A 2>g i] +2/-l,2y-l 5 • 
<7 = 1 
. (143) 
P) 
If 
<> 2/ ~p = 2 p r u T , 
r = — 7i+l 
then 
g = k 
Pr = 2 [? 2g—l)\ 2f -l,2g-l> 
<7 = 1 
■ (144) 
(E) 
If 
r = n 
/ Ji § 2 f- 2 v i — 2 p r u r , 
r = —n+ 1 
then 
g = k 
Pr = ^ 2^ — 2] V 2 f-2,2g-2- • 
<7 = 1 
• (145) 
For a comparison of these formulae with those given in “ Fitting,” see 
Appendix IV. 
22 . Extent of Improvement (Central Differences ).—A question of practical 
importance is the extent to which the use of these formulae actually reduces the 
m.s.e. of some selected quantity, such as, for the cases marked (B), u 0 or S 2f u 0 . The 
m.ss.e. of the various improved values are found from (131)—(135), by putting g =f 
Comparing these with the m.ss.e. of the original values, for the central-difference 
formulae (which are the important ones for practical use), we obtain the following :— 
/m m.s.e. of _ 1 % k f{2 1, 2/ } \fm, 2£+ l){ 2 (4 1, 2 1) 
m.s.e. of tf f u Q (4 f 2f) t=j\{2t, 21) [^m, 2/+l)J (m, 4/5+1] 
= 1 ‘P^ + l f {2 1, 2f} \ 2 
(m, 4/+ 1] t =/ 4/+1 \{2f 2/} J 
{m 8 -(2/+ l) 2 } {m 2 —(2/+3) 2 } ... {m 2 - (2t- 1) 2 } . 
{m 2 — (2/+2) 2 } {m 2 — (2/+ 4) 2 } ... {m 2 — (2£) 2 } 
