RELATION OF VARIABLES. 113 



S, = S5-6 + ^^-^^^ (S5-6 - S4-5) (47) 



Ye - y4 



Se = S5-6 + ^^^^^' (S5_6 - S4-5) (48) 



Ye - y4 



Equations (23) to (28) are thus replaced by equations (41) to (48) 

 whose solution depends solely upon the relation between the averages 

 A, B, C, etc. Thus, if S4, Ss, and Se be substituted for i?4, Ro, and Re 

 in equation (16) the last expression will be found to involve only d and 

 f as unknowns. For S4-5 and S5-6 of equations (45) to (48) are 



. , , . , E - D d , E - F - / 



respectively denned as = and — 



Yh - Yi Yb- Yi Y& - Yo Ye - Ys 



4. Illustration of Method by Solution of a Particular 

 Problem Concerning the Relation between Temperature, 

 Precipitation, and Yield of Wheat in South Dakota.^ 



From 1891 to 1917 the mean air temperature for the month of 

 June in South Dakota varied from 60.4° F to 73.4° F, while the total 

 precipitation during the months of May and June varied from 3.5 to 

 11.6 inches, and the yield of wheat (harvested in August) varied from 

 4.0 to 17.0 bushels per acre. In attempting to ascertain what effect, 

 if any, temperature and precipitation had upon the yield of wheat, 

 Blair (1913; 1915) appHed the method of simple linear correlation to 

 the portion of the data then available, and found a strong negative 

 correlation between temperature and yield, and a somewhat smaller 

 positive correlation between precipitation and yield. But, he also 

 found a high negative correlation between temperature and precipi- 

 tation, which, in 1918, led him to bring the data up to date and to 

 consider the question: "how much of the apparent relation between 

 precipitation and yield is really due to the influence of precipitation, 

 and how much is due to the simultaneous influence of temperature; 

 and, similarly, how much of the apparent relation between tempera- 

 ture and yield is due to precipitation." (Blair 1918, p. 71). He 



1 We desire to express our obligation to Mr. H. H. Collins who has made 

 the computations involved in this illustrative problem and in many others. 

 Without his aid publication would have been materially delayed. 



