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lar line 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 reproduce the identically same system in the BB compartments that was shown in Fig. 7, and in which each shot had a Prob: Error of q.

The dispersion of the shot cut 13B 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 b, and the result of the two acting together is that each shot has a Prob: Error of q. What is the relation between n, b, and q ? Calculation shows that q2 = a2 + b2. In other words, q corresponds to the hypothenuse of a right-angled triangle of which the other two sides are ct 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 independent 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.' Calling the Q of the Z system q, and the Q of the A and B systems respectively, a and b, then q'= a'+ b2.

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 n element also deviates independently from its am, we

may phrase it thus : An A element deviates from its al, and its M deviates from the n element. Therefore the deviation of the n element from the A element is compounded of two independent deviations, as in Problem I.