RELATION OF VARIABLES. 99 



least squares or the method of moments may be used to determine 

 the numerical values of the constants appearing in the mathematical 

 expressions. But in many, if not most, cases in practice the forms 

 of the functions are quite unknown and must be determined solely 

 from the data at hand. 



The object of this investigation is to devise a general method of 

 obtaining the relation between a dependent variable and each of the 

 mutually correlated independent ones without being compelled to 

 employ an assumed or predetermined mathematical function. This is 

 accomplished by appl3'ing to the observed values of the dependent 

 variable successive corrections based upon each value of all the inde- 

 pendent variables. In this way is obtained a series of averages of 

 the dependent variable corresponding to a series of averages of each one 

 of the independent variables^ in turn, and corrected to a constant 

 value of each of the remaining ones. Perhaps this will be more intel- 

 ligible if stated in the concrete terms of the wheat problem. In this 

 particular case the method is that of obtaining a series of averages 

 of the wheat jdeld, corrected to a constant rainfall, corresponding to a 

 series of temperature averages; and a similar series of averages of the 

 wheat yield, corrected to a constant temperature, corresponding to a 

 series of rainfall averages. The averages thus obtained define, approxi- 

 matelv, the functional relation desired. 



The idea of defining a function by means of a series of correspond- 

 ing values of dependent and independent variables is utilized in cer- 

 tain problems of higher mathematics (Fredholm, 1900; 1903; Bocher, 

 1909). But, in pure mathematics, it is possible to pass to the limit 

 and obtain an infinite series of pairs of corresponding values, which 

 defines the functional relation uniquely. In objective science this 

 is impossible, and, although various well-known methods of inter- 

 polation are available for approximating thereto, one is between 

 the two horns of a dilemma. It is ol)vious that the efl'ect of acci- 

 dental variations is reduced to a minimum for any given number of 

 observations when the number entering into each average is a maxi- 

 mum, but this also reduces to a minimum the number of averages 

 upon which definition of the functional relation depends. Stated 

 otherwise, the greater the number of averages for a given number of 

 observations the more precisely will the functional relation be deter- 

 mined; but, owing to the larger effect of accidental variations, the 

 less reliable will be the result. One must therefore use his judgment 

 in classifying the data, and should test the reliability of the results. 



In closing this section, it may be of interest to mention how the 



