108 MCEWEN AND MICHAEL. 



between any particular value of y and its average value {y — yt) 

 must be multiplied by the coefficient expressing the regression of w 

 on y in group 4, i. e., by Ri, and subtracted. Hence — R^{y — 74)1 

 is the total correction due to the position in group 4 of the nu values 

 of w common to groups 1 and 4, where the subscript outside of the 

 bracket signifies that only the values of y in group 4 corresponding to 

 values of lo in group 1 are to be summed. Similarly — Rh^{y — yb)u 

 and — Re^iy — ye)! are the total corrections due to the rus values of tv 

 common to groups 1 and 5, and the 7iie values of lo common to groups 

 1 and 6. Finally, the sum of all these corrections divided by their 

 number, Ni = iiu -\- ??i5 + nu, gives the correction to the average 

 A*, which is the second term in brackets of equation (16). In the 

 same way the corresponding expressions of equations (17) to (21) are 

 obtained. 



Introduction of the six unknown regression coefficients, however, 

 requires six additional eciuations, which are readily obtained. Any 

 regression coefficient, for example i?i, is by definition 



_ S(^ - A) (.r - xQ 



where A denotes that each observed value of the dependent variable 

 in group 1 is corrected to a constant value of the remaining indepen- 

 dent variables. Since, in this demonstration, y is the only remaining 

 independent variable, this correction for any particular value of to 

 common to groups 1 and 4 is evidently given hy d — Ri{y — 74)1 

 for the corresponding value of y, whence the part of the numerator of 

 equation (22) due to all values of w common to groups 1 and 4 is 



MA' +[d- R,{y - 74)1] - A}4 {.r - Xi}4- 



Likewise the part of the numerator due to all values of w common to 

 groups 1 and 5, and also to groups 1 and 6 are respectively 



• 2{.4^+ [0 - 7?5 {y - y5)i] - A} 5 [x - Xi}5 and 

 SU^ + [/ - R^ {y - yM - AK {x - Xije. 



Rearranging and combining these three terms of the numerator, 

 equation (22) becomes 



