132 MR. W. F. SHEPPARD ON THE APPLICATION OF THE THEORY OF 
(3.) As an extension of this last result, let X, Y, U, .. . be values of L correspond¬ 
ing (in the complete community) to class-indices a, y, . . . , and let the corresponding 
abscissae in the standard figure be x, y, u, . . . Then X = Lj -f- ax, Y = Li -f ay, 
U = L,+ au, . . . ; and therefore 
Li = (ZX + mY -f pU + ...)/ (Z + m + ^9 +.. .) 'I 
a = {l'X + mY ^ p'JJI{l'x^ my ^ p'u +‘ ' 
where I, m, p . . I', m , p'. . . are any quantities which satisfy the conditions 
lx + my + pu -h .,. = 0 
V m' p = 0 
SujDpose that we fix on the values of a, yS, y, .. . beforehand, and choose I, m, p,.. 
V, m', p,. . . to satisfy (ii.), and then observe the values of L whose class-indices in 
the collection of n individuals are a, /3, y, . . . If the errors in these values are 
r), 6, .. the resulting errors in Lj and in a will be (/f + my -\-p9 l{l-\-m -\-p + • • •) 
and {V^ -|- m'y p'9 + ...)/ {J!x -|- my + p'u + .•.); and therefore the probable 
errors in Lj and in a, as deduced from (i.), are Q.E/ \/?i and Q.H / \/7i, where 
(iii.). 
For any particular values of a, yS, y, . . ., the values of I, m, p, . . V, m', p , . . . can 
be chosen so as to reduce E“ or H“ to a minimum, 
§ 23. Relative Accuracy of the Different Methods. —Now let w and p be the errors 
in Li and in a as obtained by the average-OMd-average-square method; i.e., the 
errors due to taking them as equal to the average and the standard deviation of the n 
individuals. Also let the class-index of X, in the n individuals, be a + ^, the true 
class-index of X being a. Then, with the notation of § 19, the mean values of oid 
and of 2ap6 are respectively — (vj — v\)/ n and — {v 2 — v'o + But, by § 5, 
Vi = az, v\ ■= — az, V 2 = ^ {I — cl) cr + a~xz, F., = ^(I + a) a' — a~xz. Also the 
error ^ in X is due to the error 6, and is equal to — ad j'2z. Thus we have the 
following table of mean squares and mean products of errors, the divisor n, as 
usual, being omitted :— . , 
