138 
Miscellanea 
Assume now that this operation has heen performed q times and that as a result the following 
equation is identically true : 
{m - [m - q + \)Y , , fwi - (m - o + 1)1 ^ 
^— — 2 >±- = {m - q) ] ^ 
4ot [ m + {m - q) 
Operating as before, we have 
; . (m - (m - q + \) m - (m - o) )^ 
+ (m - q - 1) i ~ — i — — ' . - } + etc. 
( m+ (m - q) m + {m - q - \)\ 
{m + (m - q + 1)}^ , |w - (m - q + 1) - 
! 5^ — - — = (m - q + I) + ' 5^ — - — ~ — . 
4m 4m 
n/r li- 1 • u (m - {m - q + 2)]'^ , . , . , , {m - (m - q + 1)}^ , 
MuJtiplying by -{ ^ V and inserting the value of ^ ^ — : — — we ha 
^ ^ \m+ {m - q + l)i ^ 4m 
{m - (m - g + 2)}^ , (m - (m - q + 2)]^ 
^ = (^m - q + 1) ■{ " 
4m ^ [m + {tn - q + I) 
, , (m - (m - q + 1) m - (m - q + 2)]^ 
+ {m - q) ] 5— — 3 — ^ 7 
[ m+ {m - q) m + (m - q + 1)J 
m + (m - q) rn + (m 
: , (m - (m - o + 1) m - (m - q) m - (m - o + 2)) ^ 
+ {yn - q - \) { — ^— — ' . =^ . ^ \ 
'(m + (m~q) m + (m - q - 1) m + (m - q + 1)1 
+ etc. 
Hence if the expansion be true for q + 1 it will be true for q + 2. But it has been shown true 
for 1 and 2 and it is therefore true generally. 
Tf • i i"^ - (m - q + 1)}' ^ ^ 
Lt now in the expansion tor ^ — we put q = m - 1, we get 
m f in \^ f m m - « / m m - 1 m - 2\2 „, , 
7 = 1 +2 r ■ +3 J . J, . 7, + . . . = f{m). 
4 \m +1/ \wt + 1 m + 2/ \« + 1 m + 2 wi + 3/ \ ' 
Collecting these results together we may now write the probable error of „{i\y 
= .67449 ^-^^{2^(«0-l-|;j^ 
Values of the function (20 [m ] - 1) have been calculated for values of m from m = 1 to m = 20 
and are given in the table below. 
m 
2^ (m) - 1 
m 
20("i)-l 
1 
1-5000 
11 
4-2282 
2 
1-9444 
12 
4-4100 
3 
2-3100 
13 
4-5845 
4 
2-6265 
14 
4-7.527 
5 
2-9094 
15 
4-9151 
6 
3-1673 
16 
5-0723 
7 
3-4059 
17 
5-2248 
8 
3-6289 
18 
5-3730 
9 
3-8390 
19 
5-5172 
10 
4-0383 
20 
5-6578 
After m = 20, the approximate value 
2(/) (m) - 1 = Vl-570796(m + -375) 
given by Stirling's Theorem is close enough for practical purposes, e.g. m = 20, it gives 5-6573 
or 5-6578. 
