138 
l\R. W. F. SHEPPARD OH THE APPLICx\.TIOH OF THE THEORY OF 
Part IV.— Application to Normal Correlation. 
( 1 .) Correlation-Solid of Two Attributes. 
§ 25, Correlation-Solid in General. —Let the values of L and of M, the measures of 
two coexistent attributes A and B, be distributed in any manner whatever. Let 
Li and Mi be the means, and cr and IT the mean squares of deviation from the mean. 
Then we know that the mean value of (L — Li) (M — Mj) is less than ab. Let 
this mean value be ah cos D ; then the angle D will be called the divenjence of the 
two distributions. 
Take two lines OX, OY, including an angle tt — D, and on OXY as base-plane 
construct the solid of frequency of values of (L — Li)/a sin D and (M — Mi)/6 sin D, 
these values being measured parallel to OX and OY respectively. Thus if we draw 
O.r at right angles to OY, and Oy at right angles to OX, and if on Ox and Oy 
respectively we take ON' = x', ON" = x”, and On’ = y', On" — y", then the 
portion of the solid included between planes through N' and N" at right angles to 
ON'N" and planes through n' and n" at right angles to Onn" includes all the 
elements representing individuals for which L lies between Lj -j- ax and L^ -f- ax', 
and M between Mj + hy' and Mj -f hy". This solid will be called the correlation- 
solid of the two distributions. The ordinates are supposed to be measured on such a 
scale that the total volume of the solid is unity. 
Let L' = ^L -j- mM, M' = Vh m'M, and let the means, mean squares of devia¬ 
tion, and mean product of deviation of L' and M' be respectively L',, M'l, a'\ 6 '“, and 
a'h' cos D'. Then 
L' = IL, + mM'i, M' = I'L, 
a'^ = /“cr -j- 2 hnah cos D + m~T, 
h'~ ■= V'-ar -|- 2 l'mab cos D -f- m~h', 
a'h' cos D' ~ ll'a/ 4- ijoi' -p I'm) ah cos D -j- mm'C. 
Let Wll be any ordinate of the correlation-solid, the co-ordinates of AY with 
regard to OX and OY being x cosec D and y cosec D ; and let ax' = lax -p mhy, 
h'y' — I'ax 4 m'hy. Then VVR is proportional to the number of individuals lor 
which L = Li -p ax and M = -p hy, and therefore it is proportional to the 
number for which L' = L'j -p a'x', M' = M'j -p h'y. Through O draw the lines 
OY', OX', whose equations referred to OX and OY as axes are lax -p mhy — 0, 
I'ax -p m'hy = 0 ; and draw WN parallel to Y'O, meeting OX' in N (fig. 0). Then 
ON sin X'OY' = {lax -p mhy)l{l'~cr -j- 2lmah cos D -p m“h~f = x ; and similarly 
NW sin X'OY' — y'. Hence the solid is the solid of frequency of values of 
(L' — L'i)/a' sin X'OY' and (M' ~ sin X'OY', these values being measured 
parallel to OX' and OY' respectively. Also 
