102 MCEWEN AND MICHAEL. 



each value of tv in the {u\ x) series as nearly as possible to the value 

 it would have had if y, z, etc., had constant values arbitrarily chosen. 

 Regarding the average of the original values in each group of the {to, x) 

 series as a first approximation, the second is obtained by applying to 

 each value of h', corrections derived from the relation of w to y, w to z, 

 etc., indicated by the original series of group averages, (w, y),(w,z), etc. 

 A second approximation to the relation of w to y is then obtained by 

 introducing corrections based upon the second approximation in the 

 {w,x) series and first approximations in the remaining series. The 

 process is thus continued until second approximations are obtained 

 to the relation of w to each of the remaining independent variables. 

 By means of these second approximations in the {w, y), (w, z), etc., 

 series a third approximation to the functional relation of lo to x is 

 obtained, and so on. It seems reasonable that such successive approxi- 

 mations would result in convergence to values of w corresponding 

 to a variation of only one independent variable at a time. This is 

 confirmed by the following analytical demonstration, which also 

 yields a practicable method of making and checking the computations. 

 For clearness the analytical demonstration is given for the special 

 case of three independent variables and three groups of each, but the 

 same reasoning applies to the general case of any number of variables 

 and groups. Arrange the values of the independent variable x in 

 ascending order, segregate them into three groups, and let Xi, Xo, and 

 X3, be the average of x in the three groups respectively. Let A\ 

 B', and C' be the corresponding original averages of the dependent 

 variable w, and A, B, arid C be the required values Oi these averages 

 corresponding to constant values of the two independent variables, 

 y and z. Denote the number Oi entries per group by Ni, N2, and N3. 

 This notation, together with that for the y and ? v'ariables, is presented 

 in tabular form as follows: 



1 All values of the independent variable, y, in any one y-group are thus 

 assumed to equal the average within that group, and similarly for the remaining 

 independent variables, z. etc. To state it otherwise, in correcting for the 

 effects of any variable, y, each value of w in each y-group is assumed to cor- 

 respond to the average value of y within that group (see p. 105). 



