184 Miscellanea 
This result gives many particular cases if we identify two or more of the variables. For instance 
Wean value of dp^i dp-i-'= (p-^^.y! -pi^Pi^ ^ , 
where p^i is the second moment coefficient with regard to x, i.e. in the usual notation for one 
variable and so forth. Also 
Mean value of (dp^if = ^^ (Pi' - Pi'')- 
1>. iSimilarly if there are 6 independent variables x^^, x.^, ... x^, x^. 
The mean value of r//>34 
= ^ - PiiuP-os - IhiMPvi - Pvii(,Pzi + '^PriPuPi(.\> 
, N - 271 
where . ^ ~ N - 2 ' 
Particular eases are 
Mean value of = ^4 Oh" - '^P^' Pi' ^ "-''i'''' 
or Mean value of (dfx'„f = {/c - + 2/x'./), 
and Mean value of dp^.^dp.y.dp^^ =^ [Pi^.^^ - Pxi'-i P^x - Pt.ih P12 - P23 + ^Pn P23 P31I 
C. Let there be 8 independent variables x^^, x.^, ... x-, x^. 
The mean value of dp^.-, dp.,,! dpr,^ 
= ^4 [ PX23l P567H + i^l256 P347S + /^127S /'m56 - Pl2H PsC P'S " Pr67H Px2 p3i 
- Piz-oa P3A P7>^ - Pn-a PiiP-M - Pi-i7s Pn Pss - P-Mi^o Pio P-s + '^PiiPsiPsc Piil 
+ ^ [Pl >3Vo67>i + i'l234 /'dots + ^^1256 ^3478 + Pl2-S PiiiG ' PviUbi Pit, " 2^123478 Pb6 
- Pl25(i'S Psi " i''34567S 1^12]' 
where cj) = 2 - ix' + Sx" + ^x'l'^ - Ox'7". 
1'= _ 1 + 3x' - 2x" - -Sx'lii + -ix"/", 
and ' x" = {N - 3n)l{N - 3). 
When the sampled population is infinitely great 
= X = X' = X" = 1. <^ = 1 - 2/«. 
As a particular case, 
Mean value of (f//v)* = 4 (^Px^ - + + ^ iPi^ + '-^Pi'' - ^Pi" Pi')' 
or in the notation usual for a single variable, 
Mean value of ((//x',)* = ^ (3/.,' - Gpifi'-r 3/x'./) 4 {^', -I- S/x'^- - 4/„m',,). 
When the sampled population is normal tiie results of (A). (B) and ((") can be immediately ex- 
pressed in terms of the correlation coefficients r,2, ... r-^. by means of the formulae established 
on p. 138 in the current issue of this journal. 
