EREOR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 149 
The value of V is different for different values of D. But, if x and y are the 
abscissae of the standard normal figure corresponding to the class-indices a and /3, it 
is easily seen that V depends solely on x, y, and D. Hence, if we choose a. and /3, 
and observe V, D is (theoretically) determined. 
Let the errors in the values of X and of Y be ^ and y. Then the observed value 
of V is the proportion of individuals for which L exceeds X -j- f and M exceeds 
Y 7). Let the actual nubmers coming from the four classes (A), (B), (C), and (D) 
be n (Y -f xfj), n (Y'-f f), n (Y''+ f'), and n (Y'"-l- f''); thus i// + f-{- f'+ f "= 0 . 
Let the areas of the sections of the standard solid by the planes NWHtj and nWRf 
(§ 29) be r and A, and let these areas be divided by WE, in the ratios of 
1 y : 1 ~ 7 and 1+8:1 — 8 respectively. Thus V and A are equal to the ordinates 
of the standard figure corresponding to abscisste x and y (class-indices a and /3); 
while y is the class-index of Y in the class for which L = X, and 8 is the class-index 
of X in the class for which M = Y, these being the class-indices corresponding to 
abscissae {y — x cos D)/sin D and {x — y cos D)/sin D in the standard figure. 
The erroneous values X + ^ and Y + are obtained bj^ transferring n {xfj + xfj") 
individuals from (A) and (C) to (B) and (D), and n (xjj + xfj') from (A) and (B) to (C) 
and (D). The first transfer takes place (to our order of approximation) in the class 
for which L = X, and the second in the class for which M = Y; so that the propor¬ 
tion appearing to fall in (A) is 
V + '/'-i(i-y)('/' + ^") — Hi - S) f) 
= Y + -H1 + y).i(l + 8 ).v// — ^(1 +y).^(l — 8 ).+ 
— i (1 - y) • Hi + ^) • H i (1 - y). Hi - S) 
= Y + 
Let WE = Z. Then the error in Y produces (§ 29) an error — in D, and 
therefore the probable error in D, as determined by this method, is 
Q . © / yn, 
where 
0^--uz-’-[(Y(1+^.r+8)H V'(I^.I^)Hw'(lAH.T+8)Hw''(T^y. 
- {Y.TTy.T+S-YM‘-H-^i^S-V".l-y.T+S+YYYH.l-S]“J. 
Since Y + Y' + Y" + Y'" = 1 , Y + Y' = i (1 - j8), Y + Y" = i (1 - «), this 
probable error can be expressed in terms of Y, a, j8, y, 8 . But the above is the most 
symmetrical form, and the most convenient for calculation. 
(4.) By taking a number of difterent values of a and j8, and observing the 
corresponding values of Y, we get a series of values of D ; and then we can take the 
weighted mean of these, the weights being assigned in such a way as to make the 
probable error as small as possible. 
