480 Mr. G. U. Yule. On the Significance of Bravais Formula 



tion this " characteristic relation " must become the " equation of 

 regression " which gives the means of any a^-array, as only in this 

 way can Snd* be made a minimum, i.e., zero. 



It might be said that it would be more natural to form a " charac- 

 teristic relation " between the absolute values of the variables and 

 not their deviations from the mean. This may, however, be most 

 conveniently done by working with the mean as origin until the 

 characteristic is obtained, and then transferring the equation to zero 

 as origin. It would be much more laborious and would only lead to 

 the same result if zero were used ab initio as origin. 



We may now proceed to the discussion of the special cases of two, 

 three, or more variables. The actual formulee obtained are not, it 

 will be found, novel in themselves, but throw an unexpected light 

 on the meaning of the expressions previously given by Bravais* for 

 the case of normal correlation. 



(1) Case of Two Variables. Since x and y represent deviations 

 from their respective means, we have, using S to denote summation 

 .over the whole surface, 



S() = 8(2,) = o. 



'The characteristic or regression equations which we have to find are 

 of the form 



Taking the equation for x first, the normal equations for a\ and l\ 

 are 



SO) = Na 1 +6 1 SQ/) 1 



' 



N being the total number of correlated pairs. From the first of 

 ihese equations we have at once 



a z = 0. 

 From the second 



"To simplify our notation let us write 



SO 2 ) = N^ 2 . 



so*/) = 



<TI and <?2 are then the two standard-deviations or errors of mean 



* " Memoires par divers Savants, " 1846, p. 255, and Professor Pearson's paper 

 .on " Regression, Heredity, &c." ' Phil. Trans.,' A, vol. 187 (1896), p. 261 et seq. 



