08 NATURAL INHERITANCE. [CHAP 



lar Hue drawn from its starting point, and each shot 

 will have a Prob: Error that we will call b. Do this 

 for all the AA compartments in turn ; b will be the 

 same for all of them, and the final result must be to re- 

 produce the identically same system in the BB com- 

 partments that was shown in Fig. 7, and in which each 

 shot had a Prob: Error of q. 



The dispersion of the shot at BB may therefore be 

 looked upon as compounded of two superimposed and 

 independent systems of dispersion. In the one, when 

 acting singly, each shot has a Prob: Error of a ; in 

 the other, when acting singly, each shot has a Prob: 

 Error of 6, and the result of the two acting together is 

 that each shot has a Prob: Error of q. What is the 

 relation between , b, and q ? Calculation shows that 

 q* = a 2 + b 2 . In other words, q corresponds to the hypo- 

 thenuse of a right-angled triangle of which the other two 

 sides are a and b respectively. 



(2) It is a corollary of the foregoing that a system Z, 

 in which each element is the Sum of a couple of inde- 

 pendent Errors, of which one has been taken at random 

 from a Normal system A and the other from a Normal 

 system B, will itself be Normal. 1 Calling the Q of the Z 

 system q, and the Q of the A and B systems respectively, 

 a and b, then q 2 = a 2 + b 2 . 



1 We may see the rationale of this corollary if we invert part of the 

 statement of the problem. Instead of saying that an A element deviates 

 from its M, and that a B element also deviates independently from its M, we 

 may phrase it thus : An A element deviates from its M, and its M deviates 

 from the B element. Therefore the deviation of the B element from the 

 A element is compounded of two independent deviations, as in Problem 1. 



