REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
223 
shall require the following m.pp.e., which can be obtained from ordinary difference 
formulae. 
m.p.e. of A f u 0 
and 
A 9 u 0 
= (-Y' 9 (f+g,f), . 
(86) 
„ S 2f u 0 
? ? 
<f% 0 
= i-Y' 9 (2/+2gr,/+gr),. 
(87) 
„ S 2f u 0 
? ? 
iuS 29 ~ l u 0 
- o,.. 
(88) 
,, fx8 2f ~ l u {i 
5? 
= (-Y'’(2f+2g-2,f+g-l)l(2f+2g), . 
(89) 
„ W-Hn 
?? 
8 2 ^u, 
= (-V' , (2f+2g-2,f+g-l), . . . . 
(90) 
1 ? 
^~ 2 u h 
= 0,. 
(91) 
„ ^8 2f ~ 2 u h 
? J 
= (-Y'’(2f+2g-4.,f+g-2)l{2f+2g-2). 
(92) 
(iii.) For advancing differences we shall have 
= A /u 0 , €f EE A f v 0 . 
The formulae will be marked (A). 
(iv.) For central differences the two cases of m odd and m even must be considered 
separately; but it will be found that, when the formulae relating to v 0 , S 2 v 0 , ... 
(m odd) and to Svi, S 3 v 4 , ... (m even) are properly expressed, they are practically 
identical in form, as also are those relating to /u8v 0 , /j.S 3 v 0 ... (m odd) and to 
/jlVi, /uS 2 Vi, ... (m even) ; and the latter correspond to the former with certain inter¬ 
changes of ( ] and [ ). We therefore, for /x8v 0 , /ul8 3 v 0 , ... and [xv h _, /uS 2 ih, ... , replace 
0, E, A, X, by <p, I, M, /x, with the appropriate suffixes. 
(v.) For m = 2n+\ it will be seen from (88), taken with (XIV.) of §6, that the 
differences of even and of odd order can be treated independently. The <fs will be 
u 0 , S 2 u 0 , ... 8 2n u 0 in the one case and /u.Su 0 , iuS 3 u 0 , ... iuS 2n ~ l u 0 in the other. We shall 
denote these by <^ 0 , S 2 , ... S 2n and § 1} 4? ••• 4»-i respectively, and shall take j to be 
2k or 2^+1 for the former and 2k—1 or 2k for the latter. The subscripts of the 6 ’s 
and the (p’s will be modified accordingly ; i.e.. 0 2 f t2k will mean the coefficient of — S 2k in 
i e 2 f) 2 k- 2 , and similarly for i- The formulae for the two cases will be marked 
(B) and (C) respectively. 
(vi.) Similarly for m = 2 n we see from (91) that differences of odd and of even 
order can be treated separately. The 8’s are ... S 2n _ l in the one case, and 
$ 0 , 8 2 , ... S 2n _ 2 in the other, where S 2f _i = 8 2f ~ l u h 8 2f _ 2 EE ix8 2f ~ 2 u h ; and j is taken to be 
2k—\ or 2k for the former and 2k—2 or 2k— 1 for the latter. Also 0 2 /-i,2*_i means 
the coefficient of — § 2k _i in (e 2 /-i) 2 *-s 5 and similarly for <p 2 f- 2 , 2 k- 2 • The formulae will be 
marked (D) and (E). 
2 i 
VOL. CCXXI.-A. 
