ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 139 
cos (tt - TOY') = + {Im + cos D + mm'¥} 
/ {{To? -f 2lmah cos D + mW) + 2Vm'al> cos D + vfiy')]^ 
= cos D', 
and therefore X'OY' = tt — D'. Hence the solid is the correlation-solid of the 
distributions of IL -j- inM and /'L -{- m'M., OX' and OY' being taken as axes. 
Thus the correlation-solid of the distributions of L and M is the same as the 
correlation-solid of the distributions of IL -f mM and Z'L wi'M, where I, m, V, m' are 
any constants whatever.^ 
Fig. 9. 
r 
It may be noted that if Dj and D., are the divergences of the distribution of /L+wM 
from the distributions of L and of M, we have D = Di D 2 . Or, generally, if the 
divergence may be supposed to be either positive or negative, and if L, M, N are 
measures connected by a linear relation /L + mM + = 0, their divergences 
D, D', D" from one another are subject to the relation D -{- D' -h D" = 0. 
§ 26. Correlation-Solid for Normal Distributions. —(i.) Now suppose that the 
distribution of L is correlated with that of M, i.e., that the values of M are 
distributed normally with mean square b'^, and that for any i^articular value of 
M the values of L are distributed normally with constant mean square /3“ about 
a mean value Li + X (M — Mj), where X is a constant. Then (§ 14) we may write 
L — Li = X (M — Ml) + L', where L' is a measure whose values are distri¬ 
buted normally with mean square /3" about a mean value zero, this distribution 
being independent of that of M. Hence the mean square of L — Lj is 'tdb'^ + yd“, and 
the mean product of L — Lj and M — Mi is W ; so that, if a' is the mean square of 
L — Li, we have X = ajb . cos D, yS" = siir D. Thus for any particular value of 
(M — Mi)/7> sin D the values of (L — Li)/a sin D are distributed normally with mean 
* We must, of course, allow for the possibility of two solids, which really are identical, appearing to 
be the “ reflexions ” of one another. 
T 2 
