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 rela- 

 tively to M is c. The ordinary law of error is to be assumed. 



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

 requires 



fj*2i A2 



I t/ /I C) \ 



& b' z ' 



to be a minimum, subject to the condition 



x + y = a, a constant. 

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







x - y - 

 ~ ~" 



n i\ 



and the most probable position of Q, measured from M, when q's ob- 

 served distance from M is a, is 



c 2 

 b* + c 2 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. 



