112 Mr Todd, A 'particular case of 



mentioned there is one and only one fundamental nnit}'^ and in 

 cases (ii) and (iii) + 1 are the only roots of unity which the order 

 contains. In case (i) the theorem is easy to prove. In case (ii), 

 if P + 2bt + c = is the equation satisfied by "^j the unities of the 

 order are essentially the same as the solutions of the Pellian 

 Equation 



x"" - (b- - c) y^ = ± 1, 



and Dirichlet's results can be deduced from the theory of this 

 equation. In other cases the proof of the theorem is much more 

 difficult. Mr Todd is concerned with the case in which ^ is the 

 cube root of an integer — which comes under the heading (iii) 

 above. If ^ = n, the general theorem ass( -rts (a) that the order 

 of ^ contains an infinity of unities, (b) that they are all expressible 

 in the form 



where 7 is a particular one among them and m is a positive or 

 negative whole number, and (c) that every number of this form is 

 a unity of the order. Mr Todd's essay contained an elementary 

 proof of (6) and (c) ; the proof of (c) does not essentially differ from 

 that given in text-books, though this was not known to him at 

 the time, but the proof of (6) appears to be new and forms the 

 subject of the following note. — H. T. J. N.] 



If ^^ — - n, and T =x + y^ + 2'^^ is a member of the order of ^, 

 then 



r^ = nz + x"^ + 2/^2^ 



SO that r satisfies the cubic equation 



\x — t, y, z 



\ nz, x — t, y =0 ; 



I ny, nz, x — t 



*4 



hence it follows that F is a unity of the order if and only if x, y, z 

 satisfy the Diophantine equation 



= af^- ny' + n^z^ — Snxyz = ±1 (i). 



It will be the object of this short note to give a simple elemen- 

 tary proof of the fact that, if the existence of unities is assumed, 

 then every unity of the order of ^ can be expressed in the form 



