If a series u1 , u2 ,... u2n + 1 constitute a random sample from a normally distributed population, they any linear function A = S 12n + 1 (
a
r
u
r ) will also be normally distributed; moreover its mean will be zero if S(
a
r ) = 0, and its variance will be equal to that of the original population if S (
a
r2 ) = 1. Any other liner function B = S 12n + 1 (
b
r
u
r ) will be distributed independently of the first if S(
a
r
b
r ) = 0, and in this case the sum of the squares, x = A 2 + B 2 , will be distributed so that the chance of exceeding any particular value of x is
e
-x/e , where c is the mean value of x, equal to twice the variance of the population sampled.