REDUCTION OF ERROR BY LINEAR COMPOUNDING. 
231 
/q\ m.s.e. of /u.8 2f by _ _ if _ \ {2t — 1, 2f— 1} (j-m, 2tf 2 (it — 2, 2t — l) 
m.s.e. of u^ f - l u 0 (4/—2, 2/— l) i =/ l{2£— 1, 2t — \) (^m, 2f]\ (m, 4i — l] 
= 4/ 4£ — 1 f {2*-l, 2/— 1} 1 2 
(m, 4/-1] * = / 4/-1 1 { 2/-1, 2/-1} I 
{m 2 — (2f+ l) 2 } {m 2 -( 2/+3) 2 } ... |m 2 -(2£-l ) 2 } . / u7 ^ 
{m 2 —(2y) 2 } {w 2 — (2jf+ 2) 2 } ... {m 2 — (2£ — 2) 3 } 
/j-^\ m.s.e. of _ _1 * = f {2£ —1, 2/— 1} [^m, 2£)| 2 (it —2, 2t — l) 
m. s. e. of 8 2f - (4/- 2, 2/-1) t = / 1 { 2t — 1, 2t-l \ [|m, 2f)\ (m, it -1 ] 
!_ 1 %* it — 1 [ [2t-l, 2/-1}! 3 
" (™, 4/— l] «=/ 4/— 1 t{2/—1,2/— 1}J 
|m 2 -(2/) 2 } {m 2 — (2/+2) 2 } ... {m 3 -(2*-2) 2 | . , , 
| w 2_(2/ + i) 3 } | m 2-(2/+3) 2 } _ {m 3 — (2it — l) 3 } ’ V ' 
(-p\ m.s.e. of /uS 2 f- 2 Vi _ if— 2 J {2t — 2, 2/—2} (%m, 2t—1]\ 2 (it — i, 2t — 2) 
m.s.e. of fi.8 2f - 2 u h ( if—i , 2f—2)t=f l{2£ —2, 2£ —2} (^m, 2f— l]j (m, 4£ —3] 
= 4/-2 %* it-S \ {2t-2, 2f-2} \ 2 
(m, if— 3] < =/ if— 3 1 { 2f— 2, 2/—2}j 
{m 2 -(2/) 2 } {m 2 -(2/+2) 2 } ... {m a -(2£-2) 8 } / 14g \ 
{m 2 -(2/-l) 2 } {m 2 — (2f+ 1) 2 } ... (m 2 -(2^3) 2 } 1 ' 
23. Smoothing .—When we have a table containing a very large number of us, a 
common method of procedure is to use a formula involving a limited number of terms 
and to apply it to successive sets of the us for the purpose of obtaining a table to be 
substituted for the original table. Thus we might use a formula involving 2n +1 
terms, and apply it to u 0 , u u u 2 , ... u 2n for finding a new value w n , then to u 1} u 2 ,u 3 , ... 
u 2 n+i l' or finding a new value w n+1 , and so on. These values having been obtained, a 
differenced table would be formed ; but, as by hypothesis the true differences of order 
exceeding j are negligible, the table would only go up to differences of order j. There 
are two cases to be considered. 
(i.) If our object is to obtain as accurate values as possible for the w s, consistently 
with our using only the specified number of us for each, the most accurate values 
would be the vs given by the formulae considered in this and the preceding papers. 
It should, however, be observed that the differences of the w s are not the same as the 
A-Ty, S 2f v 0 , etc., occurring in those formulae. Suppose, for instance, that we replace u 0 
by its improved value v i} obtained by means of the (B) formula involving u_ n , u_ n+l , 
•.. u n , and replace u x by the improved value v x obtained in a similar way. The 
resulting value of v x — will involve the 2n + 2 us from u_ n to M n+1 ; but it will not be 
the same thing as the improved value Vi obtained by the (D) formula involving these 
us, and its m.s.e. will therefore be greater than that of the latter. 
(ii.) If our object is to obtain a smooth table of the w's as a whole, we could do this 
by obtaining as accurate values as possible for the differences of the tv's of order j. 
VOL. CCXXI.-A. 2 K 
