274 
PROFESSOR KARL PEARSOK, MATHEMATICAL 
arrays. If o-g and (t\ be the means of unloaded uncles and nephews, we note that 
they are arrays owing to common parentage, and hence their array standard 
deviations* will be o-g a/ 1 — p“ and o-'g \/l — p“, p being the standard deviation of 
parent and offsjDring. As before we find : 
= a-l + (tI {1 — p') 
-p^) 
(xxviii.). 
If, as will probably be the case, there be no secular change between uncles and 
nephews, then cr = cr', cr,, = cr'a, erg = cr'g, and accordingly 7'' = Hcrl/cr; whence, using 
(xxiv.), we have : 
r' = r X R X 
(xxix.). 
If we could assume cr„ = 0 -^ and o- = cr, this result would reduce to the very 
simple form : 
?•' = 7’ X R. 
Now the assumption o-g = o-g is, I think, legitimate, for the distribution for an 
unloaded array of nephews or uncles should be sensibly that of an array of brethren. 
But the equality of cr„ and cr„, which would now involve that of a and cr, is a much 
more doubtful point. cr„ and cr„ mark indeed quite different systems of loading. 
Both, it is true, are of the form 
S {nnW) / N - {S (7^7^'M) / N jq 
but in the case of brethren 71 = ^ [n — 1) or td has perfect correlation with 71 , while 
in the case of uncles and nephews n is only imperfectly correlated with n. The 
intensity of this correlation depends upon the correlation between the sizes of arrays 
of uncles and nephews, a quantity which may be very small, or not, according to the 
nature of the record. Hence it appears necessary in applying the method to make 
some attempt to appreciate the value of (Xa as well as cr^. If this be done R can be 
found from (xxix.), if not directly. This value of R is not without importance for 
the inheritance of characters latent in one or other sex. 
We have thus reduced the correlations of individuals to a calculation of the corre¬ 
lation of arrays. 
(6.) P7'opositio7i V.—To Ji7id a 77ieasiire of the effect of 77ii7igling imcorrelated 
7nate7dcd iviili co7'related material. 
The imjDortance of this investigation lies in the fact that death, restraint, or other 
* Or, again, tlie means of tlie staiidard-deviations of the arrays. 
