424 
The Incomplete Moments of a Normal Solid 
Applying the above to our former example we have 
x-yzd:vdy 
k I'll 
'XI J — OD 
= 2\2\ HKV^+ 0.A'r + {0,, T\ - + {0,,T\ ~ 
+ ... 
+ o\i\HK\i^ 0^r + (00, n + {Oo.n + 
+ 2rl\O\HKll + 0,r + {0,,T\ ~ + {d,, T), ^, + 
^Q + 0.A'r + {0,0rr/r; -I) 
21 
.(128). 
+ {0oA'T,T,' -20,T,T,') 
+ (0.A'T,T,'-H0,ToT,') 
+ |l + ^/ /• + ( ^/ T,') I? + (0; T,T,')^^+... 
+ 2ri^l + 0,r + (0, Z T/} + {0,T,T,') ~ + 
The further simplification of the last two terms is obvious. 
12. The determination of the constants of a whole distribution from 
those of a quadrant. 
The formulae developed in the preceding pages have many uses, but one of the 
most useful and interesting is their application to the inverse problem of deducing 
the constants of a complete normal solid from those of a given quadrant. Thus 
the records of a body of soldiers might show the age and height distribution, but 
if recruiting were limited to those over 5' 8" in height and over 18 years of age 
we should have the data of a quadrant only, bounded by a plane at 18 years of 
age and another at 5' 8". The problem is to determine from measurements of this 
quadrant the constants for the whole population. 
If we express the moment coefficients about the general mean, with standard 
N 
deviations as units, as niio, nio,, m.2Q, etc. and write — = \ we have from the pre- 
ceding paper 
771 
m,o = - \ (BA + vKB), 
nio, = -X (rEA + KB), 
= - X (hHA - r (1 - r') x + r^kKB) + 1, 
moo = - \ {r%HA -r{l-r-)x + kKB) + 1, 
= - \ (rhHA - (1 - r^) x + rkKB) + r. 
