100 MCEWEN AND MICHAEL. 



method happened to be devised. It had its origin in our collabora- 

 tion on prol)lems concerning the quantitative relation between varia- 

 tions in the number of certain marine organisms and fluctuations in 

 the elements of their environmental complexes. Attempts to elimi- 

 nate the effects of correlation between the environmental elements 

 b}' the method of multiple correlation and that of least squares, 

 combined with various subsidiary expedients, proved highly unsatis- 

 factory. The reason is that, at the outset of the mathematical reason- 

 ing, assumptions of either a linear or some other definite type of 

 regression, or of the functional form of the observation equations 

 must be introduced for which no justification is afforded by the data 

 themselves. After a fairly exhaustive study of the literature, which 

 failed to provide any practicable and rigorous way of handling such 

 problems, we were led to devise one which culminated, in part, in 

 this method of successive approximation to group averages. x\lthough 

 the central idea is the product of our collaboration, the mathematical 

 demonstration and the practical process of making the computations 

 are primarily due to the senior author. Furthermore, the particular 

 problems whose study led to developing this method are too complex 

 to afford suitable means of illustration. For this reason the simpler 

 problem of the relation between temperature, precipitation, and yield 

 of wheat in South Dakota is used, a study of which, by means of 

 multiple linear correlation, has been published by Blair (1918). 



The mathematical demonstration, while close and rigorous, is 

 neither abstruse nor difficult. In the case when variability within 

 the group is neglected (section 3 A) it involves nothing beyond the ele- 

 ments of algebra. But, in the case when this variability is taken 

 into account (section 3B) the demonstration also presupposes knowl- 

 edge of linear regression, so that some readers may prefer to follow 

 through the concrete process of computation given in section 4 illus- 

 trating the first case, before turning attention to the analytic demon- 

 stration in the second case. 



3. Mathematical Demonstration: 



a. the case when variability within the group is 



neglected. 



When the change in the dependent variable, w, corresponding to a 

 given change in one independent variable, say .r, is negligibly influenced 

 by the magnitude of the constant values to which the remaining 



