L. ISSERLIS 
137 
the summations in each line extending to all possible permutations of the suffixes 
2, 3, 4, ... n. The last Une for example being 
^^^^M {r^^a. Mean {X^X, ... ZJ + r,,G, Mean {X^X.X, ... ZJ + ... 
+ ?'i„a„ Mean (Z^Zg ... Z„_i)}. 
Now we have seen that Mean (Z2Z3) = (rgg - /laJ'i.s) (^2<^-3- Similarly, 
Mean (Z2Z3Z4Z5) = {^a^) do-g) da^) da,) dq^^is) 
= M M M M [(l''23) (As) + (l'"35) (l''24) + (l»-25) (l»'34)J 
= (»'23 - «"l2''*13) (>'45 - ^'U'l'l&) + (''3.5 - ''l3»'l5) (''24 - ^12'''l4) 
+ ir-z5 - n-Jib) ('% - *'i3n4). 
and our assumption of the truth of equation (6) up to (n — 2) variables will enable 
us to write down the mean value of every product of Z's occurring in (11 ). 
Dividing by a^CTg ■ ■ ■ ^^n we have, remembering that Mean x-^ja-^ is 1 . 3 . 5 . . . (>i — 1) 
(?i23...« = (''i2»'i3 ••■ ••• (" - 1) 
+ S {}\a^-uru ••• (^•"g - >'ia»-,^)} 1 .3.5 ... {n - 3) 
+ ^ {'•io»-ih*-i. ••• 'S' [(/'a^ - v,aVi/3) ('^5 - '-.v'-ia)]} 1 .3.-5 ... [n - 5) 
+ 
+ '5 - - r.yf.i) (/-.p - ri.rip) ...]}.! (12), 
where S' refers to permutations of a/Sy ... only, and ;S' to permutations of all the 
suffixes a, b, c, ... a, y i.e. all the suffixes 2, 3, 4, ... n. 
It is clear that when the right-hand member of (12) is completely expanded 
no terms can survive which contain as a factor more than one correlation coefficient 
with suffix unity. This is easily verified in simple cases, and if in the general case 
a term ri^.'^r^^r^,^ ... survived, this term would reduce to /,a^ when we identified 
the characters a, 2, 3, ... n, which contradicts the value 1 .3 .5 ... (n — 1) we 
have already found for it (equation (9)). 
The value of the right-hand member is therefore easily found by neglecting all 
terms containing more than one such factor. 
Hence on the assumption that (5) is true for all values of n up to (n — 2) we find 
?i23...« = {>\aS' {)\firysr,p ...)}, 
but this is exactly the formula we wished to establish for it is obvious that 
S {Vahi'cd ■■■ '''hi) where ahr ... A: is a permutation of 12 ... >i is equivalent to 
S{r,aS' {>\,r,,...)} 
where a, a, (8, y ... is a permutation of 2, 3, 4, ... n. Thus our formula which has been 
proved true for 4 variables is seen by induction to be true in general. 
5. Formula (6) can be exhibited as a multiple definite integral: Let A denote 
the determinant whose A"th row consists of the elements 
(''lA-j *'2A-5 ••• /,-, • • • *'A-t-l, 7.- ' ••• frik) 
and let A^^^. denote the cofactor of the element in the /;th row and Ith column. 
