152 
^IR. W. F. SHEPPARD OH THE APPLICATION OF THE THEORY OF 
Bi = A” (1 + x“) + 2 AB (1 + cos D) cos D 
+ B- (I + iy") A (Asc + By) Z sin D cos D + Z- sin- D, 
and 
@2 = V(1 -V). 
Thus the “discrepancy” is Bie difference between N as calculated by finding the means, 
mean squares, and mean product, and V as found by direct observation ; and the 
probable discrepancy is Q (Bi Bi)-/\/ n. 
In adopting this metliod we are testing both the normal distribution of each 
measure separately and also the normal correlation of the two distributions; and 
therefore it is not necessary to test first whether the separate distributions are 
normal. 
( 2 .) Suppose that we are satisfied that the separate distrihutions are normal, and 
that we require to test whether, on this assumption, they may he regarded as 
normally correlated. Then B, in ^ 3 I (5.), will denote the proportion for which L 
exceeds the value found to correspond to class-index a, and M exceeds the value 
found to correspond to class-index The discrepancy is (§ 3U (3.)) the difference 
between the errors — Zd and xfj — — 7 ) ('/' + — g (1 — 8 ) (’A + (This 
difference, by § 31 (2.), may be written in the form (f> — ^ (I ~ y) X ~ i ~ ^) x'-) 
The mean square of the discrepancy is Z' (B" — sird D)/ri, where B" has the value given 
in § 30 (3,); so that the probable discrepancy is Q.Z (B^ — siid D)-/\Ai. When this 
method is adopted, the sum of all the discrepancies in any row or in any column 
of the table of double classification is zero. 
(3.) In some cases we are not able either to calculate the averages, average squares, 
and average product, or to test whether the separate distributions are normal. We 
must then determine D by some other method, and proceed as in (2.). Suppose, for 
instance, that D is determined by the double-median-classification method of § 27. 
Then, as in ( 2 .), the discrepancy is the difference between the value of V, calculated for 
particular class-indices a and /3, and the observed value of V for these class-indices ; 
and the probable discrepancy is Q.^h/A??, where <J>- has different forms according as 
a and /3 are positive or negative. If a and /8 are both positive, it may be shown 
that 
cp-^ = D(7r~D)Z-^-2(7r-D)Z{T(l -a).i(l-7) + I(f -S)} 
- 2 DZV A 277Z{T(1 _ y) W a i(l - S) W'} a ©-ZA 
B" having the value given in § 30 (3.), and W and W' denoting what V would 
become if we put /3 = 0 and a = 0 respectively, without altering the value of D. 
