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224.   NATURAL INHERITANCE.

being Q ; the bar, always carrying the index with it, moves under some restraint up and down a fixed frame YMY', the mean position of Q being M : the movements of the index relatively to the bar and of the bar relatively to the frame being quite independent. For any given observed position of q, required the most probable position of Q (which cannot be observed) ; it being known that the probable error of q relatively to Q in all positions is b, and that of Q relatively to M is c. The ordinary law of error is to be assumed.

If in any one observation, MQ = x, Qq = y, then the law of error requires

22 + y2   (12)

P b2    

to be a minimum, subject to the condition

x + y = a, a constant.

Hence we have at once, to determine the most probable values of x, y,

C2   b2   b2 + c2   (13)

and the most probable position of Q, measured from M, when q's observed distance from M is a, is

c2
bL + c2

a.


It also follows at once that the probable error v of Q (which may be obtained by substituting a -x for y in (12)) is given by

which it is important to notice, is the same for all values of a.

1 1   1, orv=   be

= +

,U2 C2 62'   VP + e2   . . . (14)