376 "Correlation of Averages" for Four Variables. 



Similarly for the Plymouth group it is found to be '086. 



The results may be verified by equating -r to any of its 

 other values, as 

 1 



or 



-1 £? ?23 , ?24 921 



A * A +p2Z A +p24 A +p2i A' 



1 -1 P* , ?34 , ?31 , ?32 



A A rd4t A ri>± A roz A 



I have used the first of these equations, and with the same 

 result as before, i. e. '099 and "086. 



Verification may also be obtained by means of the equation 

 originally used *, i. e. 



1 



A 2 



7*2 



?23 



?24 



?32 



Pb 



#34 



?42 



?43 



P* 



= either of the other principal 

 minors. 



By this method I find -r for Naples to be '103, and for 



Plymouth *087, which coincides to two places of decimals 

 with the results already determined. This tests very satis- 

 factorily the consistency of the arithmetical work. 



It now remains to find the values of p 1 p 2 > & c -? which may 

 readily be done by dividing each of the proportionate coeffi- 

 cients by -r. Performing this operation for each of the 



twenty coefficients their values are determined to be as 

 follows : — 





Pv 



P2- 



403 



Pv 



3-88 



4-28 



Pi- 



2-67 



2-7S 



?12- 



?I3- 



191 



Qu- 



?23" 



3W 



Q 3 ± ■ 



1 



Naples 



2-03 



-1-45 



-•65 



-2-51 



-109 



-106 



•099 



Plymouth ... 



1-73 



4-85 



-1-42 



1-70 



-•41 



-3-11 



-1-39 



- -82 



•086 



The quantics therefore are : — 

 For Naples, 

 2-03a? 2 + 4-03/ + 3-882* + 2'67w 2 



— 2(T45ay— 1'91^ + -65zw + Z'hlyz + l-(%w; + 1-06; 

 * See Phil. Mag. Aug. 1892, p. 196. 



