204 On Correlated Averages. 



When A 2 = 576, A = 24. Substituting the value of A in the 

 expressions for the coefficients, we find for the sought quantic B, 



2^ ! 2 + 4^ 2 2 + 6^ 3 2 + 4^ 4 2 - 2 x/ 2 ^ x 2 - 2 s/ 6 # 2 # 3 - 2 V 12 # 3 # 4 . 



To verify and illustrate this result, let it be employed to 

 answer the question, What is the most probable value of ,i\ 

 corresponding to assigned values of a; 2 , #3, sg±. Differentiating R 

 with respect to x x and equating to zero, we have 2&\ — V2 x 2 = ; 

 importing that to any assigned deviation of the second variable 

 the most probable corresponding deviation of the first variable 

 is v^2 times less ; each deviation being reckoned in units of 

 its own modulus. 



The truth of this follows at once from the datum that 



p 12 = -7=. To give another proof of that proposition : — the 



mean positive error of the actual deviations of the first decade 

 of digits from its mean value 45 is the modulus for such a 

 decade, say c, -f- Vtt. Xow let there be added a second 

 random decade. Its deviations being in the' long run as often 

 positive as negative, the sum of the two decades is the same 

 as that of the first (the deviations of the first being exclusively 



Q 



positive). Thus to an actual mean positive deviation — y= of 

 the first quantity corresponds the same actual mean deviation 



c 



= of the second quantity. But the modulus of the second 



quantity (the sum of two decades) is ^2 times that of the 

 first. Therefore, in units of modulus, to x x assigned corre- 

 sponds, as the most probable value of # 2 > a deviation less in 



the ratio —7=. Whence, by the Galtonian theorem, to x<> 

 assigned corresponds as the most probable value of &i a devia- 

 tion less in the ratio — —-. Which is the proposition here 



deduced from our formula. 



The case of Jive variables follows in respect of the signs of 

 the first minors the analogy of the case of three above illus- 

 trated ; the case of six is analogous to that of four. And so on. 



