L. ISSERLIS 139 
A cognate result is discussed by Mr Arthur Black in the Transactions of the 
Cambridge Philosophical Society*. Black's integral is Ve~^dzi ... dx^^, where F 
J n 
and U are any quadratic functions, the only restriction on U being that it should 
be essentially positive. Other particular cases have been dealt with in the paper 
previously quoted, and for the case of two variables several results are given by 
Mr H. E. Soperj. 
For reference we add a table of values of the reduced product-moment coeffi- 
cients that occur frequently in formulae for probable errors and similar work. 
3. 
3ri2. 
it^^ = 
1 + 2r,^. 
71-23 — 
+ 2ri2ri3. 
1v = 
15. 
'7p2 = 
15^12- 
71^2- = 
Iv'i' = 
9ri2 + 6/12''. 
(Zl"2-3 = 
3 (fia + 2>-23fi2 + 2/-i3ri22). 
9'l<23 = 
3 (f23 + f2/-i2'/-i3)- 
<l\i 3:1^1 = 
1 + 2r,s' + 2r,,' + 2^12' + 8 
''12 ''23 ''31 • 
qi» = 
105, (717 2 = 105?-i2, ?l0 2- = 
15(6^12^4-1). 
'7l^2» = 
15 (4.12^ + 3ri2). 
(714 24 = 
3 (8ri2* + 24ri2' + 3). 
'7l'^23 = 
1 .3 ... A - 1 (y-gg + Ari2ri3). 
A even. 
'71^22 3 = 
].3.5...A[(A-l)ri2V,3 + , 
'i3 + 2ri2r.^3]. 
A odd. 
For the case of two variables we add the following formula which is easily proved 
by the methods employed in this paper. 
gi«2,. = 0 (n + v) r- + (I) 0 (2) ifj {u + v - 2) r'-^ (1 - r^) 
+ i/. (4) ip{u + v-i) r^-" (1 - r2)2 + ... J 
the series terminating. Here 
iP(2m)= 1.3.5 ... (2m- 1) 
g^jjjj (v\ _v{v-l) ... {v ~ m+ I) 
mj ml 
* Vol. XVI, 1898, pp. 219—227. 
f Biometrika, vol. ix, p. 101. 
X Thia is virtually' the formula (xxxii) employed by H. E. Soper, l ea. corrected for some misprints. 
