for Regression, <J*c., in the case of Skew Correlation. 47 $ 



X=a + lY 

 be the equation to BR. Then for any one array 



Hence, extending the meaning of S to summation over the whole 

 surface 



But in this expression S(w<r) is independent of a and 6, it is, in fact y 

 a characteristic of the surface. Therefore, making S(ncF) a minimum 

 is equivalent to making 



a minimum. That is to say, we may regard our method in another 

 light. We may say that we form a single-valued relation 



x = a + by 



between a pair of associated deviations, such that the sum of the 

 squares of our errors in estimating any one x from its y by the 

 relation is a minimum. This single-valued relation, which we may 

 call the characteristic relation, is simply the equation to the line of 

 regression B/R. There will be two such equations to be formed 

 corresponding to the two lines of regression. 



The idea of the method may at once be extended to the case of 

 correlation between several variables x it x^ x z , &c. Let n be the 

 number of observations in an array of a?i's associated with fixed 

 values X 2 , X 3 , X 4 , &c., of the remaining variables, let a^ be the 

 standard deviation of this array, and let d be the difference of its 

 mean from the value given by a regression equation 



Xi = a 13 X;j + ^3X3 4-014X4 + ...... 



Then, as before, we shall determine the coefficients a 12 , a 13 a 14 , &c., so 

 as to make Snd z a mini mum. But this is again equivalent to- 

 making 



a minimum for 



S {a?! - (a ia ajj + a 18 ar a + Wi + ....) } 8 = S 

 Hence, we may say that we solve for a single- valued relation 



between our variables ; the relation being such that the sum of the 

 squares of the errors made in estimating aj x from its associated 

 values a?2, 3-3, &c., is the least possible. In the case of normal correla- 



