ERROR TO CASES OP NORMAL DISTRIBUTION AND CORRELATION. 117 
Hence the values of lU + mM' -f- n-N' + . . . are normally distributed; and 
therefore the values of ZL + wiM + + • • • are normally distributed. 
§ 14. Correlated Normal Distributions .—If L, M, N, , . . R are the measures of 
coexistent attributes A, B, C, . . . G ; and if the values of L, in every class dis¬ 
tinguished by particular values of M, N, . . . R, are distributed normally with constant 
mean square about a mean value Li /a (M — Mj) -{■ p(N — Nj) -f- . . . -j- p (R — Rj), 
where Li, Mj, Ni, . . . Ri are the respective mean values of L, M, N, . . . R taken 
separately, and [x, v, . . . p are constants : then the distribution of L is said to be 
correlated with the distributions of M, N, . . . R. 
If the distribution of R is normal ; the distribution of Q correlated with that of R ; 
the distribution of P correlated with those of Q and R ; and so on, for L, M, N, . . . 
P, Q, R : then the distributions of L, M, N, . . . P, Q, R may be said to be mutually 
correlated. We require to find, in this case, the distribution of Ih -f- -f- . . . 
+ ^jP -h -h rR, where I, m, n, . . . p, (j, r are any constants. 
For convenience, consider only the case of four attributes L, M, N, R. From the 
definition, we see that L — Li is equal to p (M — Mj) + n (N — N,) + p (R — Rj) + L', 
where L' is independent of M — Mj,. N — Ni, and R — Rj, and is distributed normally 
with mean value zero. Similarly M — Mj is equal to F (N — N]) fi- p (R — Rj) + M', 
where M' is independent of N — N, and R — Ri; and N — Ni is equal to p" (R — Rj) N', 
where N' is independent of R — Ri; the values of M' and of N' being distributed nor¬ 
mally with mean values zero. Since M' is independent of N — Nj and R — Ri,and N — Nj 
is equal to p''(R — Ri) + N', it follows that M' is independent of N' and R — R, ; 
and similarly L' is independent of M', N', and R — R,. Thus the distributions of 
L', M', N', and R — R, are mutually independent. Also each of the measures 
L — Li, M — Ml, N — Ni, R — Ri, is a linear function of the measures L', M', N', 
R — Ri ; and therefore 1{L — Lj) -f- m{M. — Mj) -f- (N — Ni) + 7’(R — Rj) is a 
linear function of these measures It follows, from § 13, that the values of 
?(L-Li) + ?n(M — Ml) + 7 i(N — Ni) -f- r(R — Rj) are normally distributed ; i.e., 
the values of Ih + mM -}- nN -p rR are normally distributed. The argument 
obviously applies to any number of correlated distributions. 
This result might also be obtained by the second of the two methods given in the 
last section. 
II. Theory of Error. 
§ 15. Distribution of linear function of errors of random selection .—Let the 
individuals comprised in an indefinitely great community be divided into any number 
of classes A, B, G, ... , and let the numbers in these classes be proportional to 
a, (3, y, ... , SO that a-fyS-)-y+.. . = 1. Suppose a random selection of 7i 
individuals to be made, and let the numbers drawn from the different classes be 
respectively na, n^', ny, . . . , so that a' + yS' + y'fi- . . . =1. Then a — a, (3' — /3, 
y' — y, . . . are the em'ors in a, ^8, y, . . . We require to investigate the distribution 
