138 High Product- Moment Coefficients 
Let x'=^ "I + 2S A,, , 
1 
and z = ^ 6-2" , 
(277)2 a2 ... CT^-v/A 
then ••• «ia^2 ■•• a^nZf^a^ifZj^a ••• f^^n = (raft >"u J (13), 
J — 'JO J — -Ji J — 'Xj 
where a, b, c, d, ... v, v are the suffixes 1, 2, 3, ... n in any possible order. 
It is clear that (13) will enable us to write down the value of the multiple in- 
tegral Pe'^dx^ ... (h'„ where P is any polynomial in x^, Xj, ... x„ on Q a positive 
quadratic form. 
In fact, let 'Laj,j,Xj,^+ lY^a ^^^x jfir ^. (a^^ = a,jj) be a positive, definite, quadratic 
form, then 
IF = I I ... I Xi'^i x{-- ... x^"" exp - \ (2«j,j,Xj,2 + 2Y,aj,^Xjfir,^ dx/Ix^ ... dx„ 
1 
-OC . —00 
(277)2 (To... a,, V^ 
CO J— GO J— OO'-'l 
exp - A (2A,,^-^J + 2SA,, '''^') r/^r^r^^^ - 
= S [ra&^frf ••• where abc ... /^A; is any permutation of the aj f + ... + a„ 
suffixes of which are equal to 1, «2 are equal to 2 and so on. 
Let D denote the determinant of the quadratic form and Dp,, the cofactor of 
0^,1 the two multiple integrals will l)e identical if 
^j>Q = o-i2o-2^ ... O-pCT, ... cr„2ADp,. 
Hence r,/ = [D^J/D.„,D,, and (t,^ = D^^/Z) while A ^ D-~^ID,,D,, ... D„,„ 
H 
SO that If = ^4 2f a!,^c. - A,;, (in 
2)2+2 
where a, h, ...h, /,• is a permutation as above, and m = aj + +••• + «« is even. 
W = 0 when is odd. 
As an illustration of this result: 
00 ,'00 CO 
(M«2yy2 22+ A^a;2//2) 
exp — I {ax^ + + + 2fyz + 2gzx •+ 2hxy) dxdydz 
= ^-"^li^- M (8FGH + 2AF^ + 256^^ + 2CH^) + ^% N [2GH + ^^), 
where A, B, C, F, G, H are the cofactors of a, b, c, f, g, h in 
a, h, g 
A = 
9^ f' c 
