114 MR. \V. F. SHEPPARD OH THE APPLICATIOH OF THE THEORY OF 
General Theorems Relating to Normal Distributions. 
§ 12. Mean Squares and Mean Products of Composite Measures .—Let A, B, 
C, . . , E, F, G be a number of attributes, all of which exist in every member of a 
community ; and let the measures of their respective magnitudes be denoted by 
L, M, N, . . . P, Q, It. Let the mean values of L,' M, N, . . . P, Q, R be re.spectively 
Li, Ml, Nj, . . . Pi, Qi, Ri; let the mean squares of their deviations from their respec¬ 
tive means be a^, b", c", . . . e"^,/'^, f-\ and let the mean product of the deviations of 
any two L and M from their respective means be denoted by S (L, M). Then, what¬ 
ever the relations amongst the distributions may be, 
(i.) The mean value of -p ?uM + . . . -f- ?’R, where /, m, n, . . . r are any con¬ 
stants, is IL^ -f- 7nMi + nMi rRi ; and the mean square of its deviation 
from its mean is 
l-a? -p m%- + rdc- + . .. + rdf -p 2bnS (L, M) + 2bhS (L, N) + 2»i/hS (M, N) + ... 
(ii.) The mean product of the deviations of Ih -p 7?iM -p nN -p . . . -p ?’R and 
Z L -p M -p H N -p . . . -p rTv from their respective means is 
ll'd- + mm'lr -p 7in'c' -p . . . + rr'g- -p [hn -p Idn) S (L, M) 
-P {In' -P I'n) S (L, N) + {mn -P m'n) S (M, N) + . . . 
As we shall often require to use these last two expressions, it will be found 
convenient to express the mean squares and mean products in the form of a table, 
thus :—- 
L M 
H 
&c. 
L 
Cl’ 
S (E, M) 
S (L, Xj 
S (L, M) 
S (M, H) 
N 
S (L, N) 
S (M, N) 
c- 
i 
Ac. ! 
! 
§ 13. Independent Normal Distributions.—M the different values of L, in the class 
distinguished by particular values of M. N, . . . P, Q, R, are distributed in the same 
way, whatever these particular values may be, the distribution of L is said to be 
independent of the distributions of M, N, . . . P, Q, R. 
