H. E. Soper 101 
of y from its mean value in the array the distribution of y is normal ; its second 
moment is p 02 — — -J and its higher moments follow the same laws of reduction 
P-io 
as above. Hence 
p uv = mean x u y v 
= mean [x w x mean y v for given x) 
= mean \x" x mean ^— x + y'^j for given a 
n 8 11 
= mean \x u x mean — -x + v ,- — — x v 1 y 
1 \pia v pf? 
v.(v—l) Pn°~ s » o /, \ P 
+ | 2 ."^p 2 **~ 2 2/ '-+••■ J for given x 
and so remembering that mean y , mean etc. vanish, and mean i/' 4 , mean y' 6 , etc. 
reduce by the above formulae we have 
^"UJ^ 0 1.2 [pj Pu+V - 2 " 
\p, m / ~ i . -z \p. 
+ 1.2.3.4 [pj P**»;'°\P« p w 
6\(v-(5)\\pJ r™-'-^ p 20 
It follows that if u + v is odd p uv is zero, that is 
P30=Pn = P\2 =Pos = 0, 
PSO =2hl = #32 = #23 = PU = #05 = 0. 
If u + v is even it is convenient to divide by suitable powers of p 20 and p a2 , and 
putting p for ._ : exhibit all the reduction formulae together as follows: 
Vp 20 VjO 02 
Pio/p-20 2 = Pnjp02 = 3, Pnjpjpo-? =Pv,llh<?pJ = 3/3, PnfjPsoPw = 1 + 2/0 2 , 
P«o/W =Poe/Po* =15, Psr/P-JpJ = PnlpJpJ = lop, 
pj. P»p<a = Pnlp*>Poi = 3 + 1 2p\ pn/psrpj = 9/3 + 6/3' (xxxiii). 
If in like manner the numerator and denominator of the expression (xvii) 
for the deviation in r in terms of the deviations in the moments be divided by 
^Pio-^Po-2 and we write 
_ dp 10 _ dp^ „ _ dp nl _ dp_ n2 dp n 
>JpJ P-20 ' Vp 02 ' " #02 ' VpaVpoa 
it becomes r + (Jr=-r^. — — ^ + , J — ^' , , ^ (xxxiv). 
V( 1 + a, - 0l «) x V(l + A - /3, 2 ) V ; 
