114 Mr Todd, A particular case of 



we can easily shew that if there exists an}^ unity in the order 

 other than + 1, then there exists a unity of positive integers other 

 than + 1 of which any other unity of positive integers is- a positive 

 integral power. For suppose that T is any unity of the order 

 other than + 1 : then by definition of a unity it follows that the 

 three numbers 



-r, i/r, -i/r, 



will be unities of the order also : and of these four it is plain that 

 one will be positive and greater than 1, i.e. it will be a unity of 

 positive integers. 



Now take any number k> 1; then there will be only a finite 

 number of F's for which k >T >1, since for any such F we must 

 have kXoO, K>y>0, k> z>0. Hence there must be a unity 

 of positive integers which is greater than + 1 and less than any 

 other ; let this one be 7. 



Suppose that F is any unity of positive integers which is, if 

 possible, not a positive integral power of 7. Then we shall have 

 F >,7, so that we can assume that F is intermediate in magnitude 

 between 7^ and 7^+S where p is some positive integer. But by 

 the last part of Dirichlet's Theorem we know that 



F/7^ - 



is also a unity of the order, i.e. we have found a unity of the order 

 which is less than 7 and greater than + 1, which contradicts the 

 assumption that 7 was the least unity greater than + 1. Hence 

 F must be a positive integral power of 7. Finally Ave have the 

 result that, if F is any unity of the order, it can be expressed in 

 the form 



where 7 has its previous significance and jo is any positive or 

 negative integer or zero. For if F is any unity of the order, other 

 than + 1, the numbers 



-F, 1/F, -1/F 



also will be unities, and one of these will be positive and greater 

 than 1, and so will be expressible in the form 



where g- is a positive integer. Hence F can be expressed in the 

 form 



where p is some positive or negative integer or zero. 



The result obtained can be put into an interesting geometrical 

 form as we shall proceed to shew. 



