150 MR. W. F. SHEPPARD OH THE APPLICATIOH OF THE THEORY OF 
§ 31. Relative Accuracy of the Different Methods. —By means of § 5 it may be 
shown that, with the notation of § 20 and § 30 (3.), 
= a {ir(l ~ y) + 1 A(1 - 8) cos D], 
0 - 0.1 r (1 - y) cos D + I A {1 - 8)}, 
0-2 0 = a'{ V + Z sin D cos D + |-r(l — y)a;-l-^A(l — 8)y cos' D], 
0-11 =: ah{Y cos D + Z sin D + |-r(l — y)a: cos D + ^ A (1 — S) y cos D], 
0 - 0,2 = {V + 21 sin D cos D+i'r(l — yjcc cos' D + i A (1 — 8)?/]. 
Thus from § 20 we have the following table :— 
Li 
Ml 
a 
h 
D 
y ' 
Li 
ah cos D 
0 
0 
0 
« {2 r (1 - 7 ) + i (1 — ^) cos D} 
Ml 
b^ 
0 
0 
0 
Mir(i-7) cosD+ iA(l-o)} - 
a 
I 2 
■g ab cos^ D 
— sin D cos D 
\a {7i sin D cos D -f i F (1 — 7 ) re 
+ ^ A (1 — ^) 2 / cos-D} 
b 
— \b sin D cos D 
^ 5 {Z sin D cos D + ^ F (1 — 7 ) a; cos- D 
■t 2 ^ (1 “ 2 /} 
D 
sin^D 
- [Z sin^D+i{iF (1-7) rc 
-F ^ A (1 — ^) ?/} sin D cos D] 
V 
1 
y(i-yj 
Let the errors in Lj, Mj, ct, h, D, be &>, oj', p, p , 6. The error in V is i//; if we 
write this = ^F (1 — y) (w + xp)!u -B ^A (l — 8) (w' + yp')l^ ~ 'Z‘0 (f), then it may 
be shown by the above table that the mean products of (f) with co, co', p, p, and 9 are 
zero. By writing in the one case A = 0, y= — 1,Z = 0, and in the other F = 0, 
8 = — 1, Z = 0, we see that xp + aod xp xp' are of the forms F (w + xp)/a + y 
and A (w' + yp')lh + f, where the mean products of y or y' with oj, oj , p, p, and 6 , 
are zero (c/. § 23). Hence we obtain the following results :— 
(1.) Suppose that we fix on definite values X and Y of L and M, and that we 
require the proportion of individuals for which L exceeds X, and M exceeds Y. If 
we determine Lj, Mi, a, b, and D from the averages, average squares, and average 
product, and then calculate the value of V, the resulting error is i-r(l—y) 
(co + xp)jct fi- ^A(1 — S) + yp)ff ~ The mean square of this error is less 
