I 



1 



and then if w and w' are each either u or v, we have 



(155.) 



(w':wi6"" w 



II. 



This gives us the following multiplication-table, whore the multiplier is to he cut red 

 at the side of the table and the multiplicand at the top, and the product is found in 

 the middle : — 



(156.) 





c 



t 



P 



$ 



c 



c 



t 











t 











c 



t 



p 



P 



s 

 











s 







P 



8 



The sixteen propositions expressed by this table are in ordinary 1 nguage a* 



follows : 



achers 



The colleagues of the colleagues of any person are that | -son 



The colleagues of the teachers of any person are that person's t< 



There are no colleagues of any person's pupils ; 



There are no colleagues of any person's schoolmates; 



There are no teachers of any person's colleagues ; 



There are no teachers of any person's teachers ; 



The teachers of the pupils of any person are that person's colleagues ; 



The teachers of the schoolmates of any person are that person's teachers ; 



The pupils of the colleagues of any person are that person's pupils ; 



The pupils of the teachers of any person are that person's choolmates; 



There are no pupils of any person's pupils; 



There are no pupils of any person's schoolmates ; 



There are no schoolmates of any person's colleagues ; 



There are no schoolmates of any person's teachers ; 



The schoolmates of the pupils of any person are that person's pupils ; 



The schoolmates of the schoolmates of any person are that person's schoolmates. 



This simplicity and regularity in the multiplication of elementary relatives mu- 

 clearly enhance the utility of the conception of a relative as resolvable mto a sum of 

 logical quaternions. 



i 



