L. ISSBRLIS 
135 
For example if we require the value of 1712223; we commence with, 
?123456 — ^ ah"^ cd'^' ef) 
— ''12 (''34'66 + *'35''46 + '"ss'as) + '''l3 (*'24''56 + ''25''46 + ^'26''45) 
+ *'l4 (^'23*56 + ''25 ''36 + ^26 ^'as) + '''l5 ('''23*46 + *'24''36 + ^26 ''34) 
■ + >'l6 (''23»'45 + '"24 ''35 + ''•25 ''34) (7). 
Identifying 4 with 1, 5 with 2 and 6 with 3 we find at once 
gi22232 = 1 + + ^''23^ + 2/'31^ + 8''i2r23''"31 (8)- 
3. We note first that (/i" which in the more usual notation for distributions in 
one variable is ^^ll^i '^ is known to have the value 1 .3.5 ... (n — 1) when n is even. 
As regards »S (r^jfc,; ... r^j,.), if all the n variables are made identical, each term 
becomes unity and the number of terms is the same as the number of ways of break- 
ing up an even number [n) of objects into (ji/2) pairs. This last number is clearly 
n\ n-2\ 4! / 
2! w - 2! 2! n - 4! ■■■ 2! 2!/^"'"''' 
which also reduces to 1 .3.5 ... (n — 1); thus equation (<>) is correct for this par- 
ticular case. 
Secondly let us consider the value of qv<--^2- The mean value of x.^ for a given 
value of a"i is J'i20'2*'i/cti, let 
— ^ JO ~!~ jSi. c) « 
"CTj 
Then the distribution of for a given value of .Tj is itself normal and its Z'th 
moment is zero for an odd k and 
1.3.5... (/^-l)(ia2)''-/2 
for an even k where 10-2 is the standard deviation of 2 within the x-^ array so that 
i<^2^ = (1 - ria^) (72^. 
'?i""^2= , Mean yaXue {x-^-'^x.^ 
K Mea 
n |,Ti«-i Mean ^r,2 ^- x^ + X.^ 
= rj29i«= 1.3.5... (»- 1)^12 (9). 
The method employed in the original proof of equation (1) is not convenient 
for generalisation and we will now prove the equation 
^1234 = ''12 ^"34 + ''13 ''24 + ''14 ''23 
by the method that leads to the general case. 
Putting as above = — .^1 + X^ , 
0-1 
•^Z ~ *'l3 ~ ^1 + A3, 
0"l 
