362 



MEMOIRS OF THE AMERICAN ACADEMY. 



It may sometimes be convenient to consider relatives each one of which is of the 



form 



a, a -|- b,/ + c># + <M 4" e * c - 



where a, , b, , c, , d, , etc. are scalars, and •,/, k, I, etc. are each of the form 



m,ic -f- n t v -\- o,w -f- etc. 



where m,, n, , o,, etc. are scalars, and u, v, w, etc. are elementary relatives. In all 

 such cases (155) will give a multiplication-table for i,j, Jc, I, etc. For example, if we 

 have three classes of individuals, Ui, U2, U3, which are related to one another in pairs, 

 we may put 



Ui:ui 



1 



Ui : u 2 





Ui:u 3 



k 



u 2 : Ui 



/ 



u 2 ru2 



m 



u 2 :u 3 



n 



u 3 :ui 







u 3 : u 2 



P 



u 3 :u 3 



9 



and by (155) we get the multiplication-table 



1 



• 



k 



I 



m 



n 







P 



9 



1 



J 



k 



% 



• 



3 



k 





 













 



1 



m 



n 































P 



9 



























I 







% 











I 











o 







m 







3 











m 











P 







n 







k 











n 











9 



















% 











J 













o 



p 





















m 











p 



9 



. 







k 











w 











? 



