ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 
115 
If the distribution of Q is independent of that of R; the distribution of P 
independent of those of Q and II; and so on, for L, M, N, . , . P, Q, R ; then the 
distributions of L, M, N, . . . P, Q, R may he said to be mutually independent. 
Now suppose that each distribution, taken separately, is normal; we require to find 
the distribution of /L + mM + nN -f • • • 4* pR + ’"R? where I, m, 2 ^, <1, 
are any constants. 
Consider first the case of two measures L and M. Let their mean values be 
Li and Mj, and let their mean squares of deviation from the mean be ctr and Let 
L = Li + ax. M = Ml + by. Then the values of x and of y are distributed normally 
about mean values zero with mean squares unity, and the distribution of x is inde¬ 
pendent of the distribution of y. Take two lines OX, OY at right angles to one 
another, and on OXY as base-plane construct the solid of frequency of values of 
X and y, these values being measured parallel to OX and OY respectively. Let OZ 
be draAvn at right angles to OXY; and let Ki and Ko be two planes whose equations 
referred to OX, OY, OZ as axes are la. x -p mb = and la .x mb .y=^> 
respectively, where and have any values. Then the portion of the solid lying 
between Ki and K 2 includes all elements representing individuals for which 
/a. X -p mb . y lies between and ; and therefore the number of these individuals 
is proportional to the volume of this portion of the solid. Denote this volume by V. 
Since the distribution of x is independent of the distribution of y, the sections of 
the solid of frequency by planes parallel to OZX are figures which when projected 
on OZX are orthogonal projections of one another with regard to OX ; in other 
words, the solid is a projective solid. Since the values of x are distributed normally 
with mean value zero and mean square unity, it follows from (iii.) of § 6 that the 
sections by planes parallel to OZX are normal figures whose semi-parameters are 
unity, and whose central ordinates lie in OZY ; and similarly the sections by planes 
parallel to C)ZY are normal figures whose semi-parameters are unity and whose 
central ordinates lie in OZY. Hence, by § 9 (i.), the solid is a normal solid ; and 
therefore it may be regarded as a projective solid whose principal sections are 
parallel and perpendicular to the planes Kj and K 2 . Through OZ draw a plane at 
right angles to Kj and K.,, cutting them in ordinates WjRi and WgRo, and cutting 
the solid in a normal figure S. Then the volume V is proportional to the area 
WiRiR- 2 W 2 of the figure S. Also OW, = ^,ls/lkC~ + OWL = + nrb\ 
Hence the number of individuals for which la.x -P mb.y lies between and I 2 is 
proportional to the area, comprised between ordinates at distances ^J\/l“ar p 
and ^oJ\/V‘a- + rrvlr from the median, of a normal figure of semi-parameter unity; 
and therefore, by § 2, it is proportional to the area, comprised between ordinates 
at distances and ^2 from the median, of a normal figure of semi-parameter 
\/ZW-j- In other words, the values of la.x mb.y are distributed normally 
with mean square l'a“ ?n"6“ about a mean value zero, and therefore the values 
Q 2 
