RELATION OF VARIABLES. 105 



I = I^ + — (ngia + W93C + und + ?i96/) (15) 



A 9 



where 7141 = nu, thi = nis, etc. 



These nine equations, together with equations (4), (5), and (6) 

 defining the quantities a, c, d, f, g, and i, determine the nine unknowns, 

 A, B, C, etc. They can be solved simultaneously, but as a rule, labor 

 is saved by employing a method of successive approximation, the 

 details of which are given in table 2. 



If the process of successive approximation be continued, as indicated 

 by lines 1, 2, 3, and 4, 6, 8, and results in convergence ^ to definite 

 limiting values, these values will evidently satisfy equations (4) to 

 (15). In the third approximation the procedure indicated by lines 

 5, 7, and 9, involving first difl^erences, affords a numerical check on 

 the computation of A A'", AB"', etc., and A^a"', A^c"', etc., of lines 

 4, 6, and 8. It is possible to continue checking each computation in 

 this way until convergence is attained. But, beginning with the 

 fourth approximation a further saving of labor is effected by computing 

 first differences (lines 10, 12, and 14) and checking these results by 

 second differences (lines 11, 13, and 15), since all differences converge 

 to zero. For a numerical illustration see page 113. 



B. THE CASE WHEN VARIABILITY WITHIN THE GROUP IS TAKEN INTO 



ACCOUNT. 



As stated on page 102, the variability, for example in a (?<', x) 

 group, due to the range in value of x in that group, and the correlation 

 between w and .r in that group, is neglected in the foregoing solution. 

 Justification for this neglect depends upon the nature and magnitude 

 of the variability, which, in turn, d&pends upon the range in value of 

 the independent variable; magnitude of the change in the dependent 

 variable due to a given change in the independent one; degree of 

 correlation between the independent variables; and number of groups. 

 When a large amount of data is at hand it is usually possible to classify 

 it into a correspondingly large number of groups with respect to each 



1 No general criterion for convergence has been worked out, but it evidently 

 depends upon the closeness of correlation between the independent variables. 

 Of the ten problems to wliich the method has thus far been applied, ranging 

 from the relation between dew point, humidity, and minimum air temperatures 

 to the relation between body length, tail length, and foot length in mice, the 

 greatest number of approximations required was fifteen. 



