1892.] 



Presents. 



453 



modulus : that is, that there are numbers whose successive powers 

 have for their rests the complete set of numbers less than, and prime 

 to, the modulus. A primitive root may be said to generate by its 

 successive powers the complete set of rests. It is also known that in 

 general, when the modulus is any composite number, though primitive 

 roots do not exist, there may be laid down a set of numbers, which 

 will here be called generators, the products of powers of which give 

 the complete set of rests prime to the modulus. 



The principal object of Part I is to investigate the relations which 

 must subsist among any such set of generators ; to determine the 

 most general form that they can take ; to show how to form any such 

 set of generators, and, conversely, to furnish tests for the efficiency, 

 as generators, of any given set of numbers. Other results which 

 are obtained as instrumental in effecting these objects, such as the 

 determination of the number of numbers that belong to any exponent, 

 may also possess independent interest. 



The object of Part II is to make, for complex numbers, an investi- 

 gation which shall be as nearly as possible parallel to that of Part I 

 for real numbers. - Much of the work of Part I maybe applied imme- 

 diately to complex numbers; of the rest some will need slight modifi- 

 cation, and some will need replacing by propositions leading to corre- 

 sponding results. Of those cases which thus call for independent 

 treatment, the most noticeable is that of the modulus (l + i)*-, which 

 is the complex analogue of the real modulus 2 A . 



The work is put in the form of a series of propositions, and is 

 started almost from first principles. The early part is consequently 

 elementary, but the advantages of completeness and ease of reference 

 may be more than sufficient to compensate for this. A large number 

 of illustrative examples are given. These will sometimes, perhaps, 

 assist in elucidating the symbolical proofs which they follow : in all 

 cases they will help to maintain clearly the actual arithmetical mean- 

 ing of the results arrived at, a meaning which may easily seem obscure 

 if it be noticed only in its symbolical and generalised form. 



The Appendix contains tables of indices for complex numbers for 

 all moduli whose norms do not exceed 100. 



Presents, Maij 5, 1892, 



Transactions. 



Baltimore : — Johns Hopkins University. Circulars. JSTo. 97. 4to. 



Baltimore 1892. The University. 



Catania : — Accademia Gioenia di Scienze Natural!. Bullettino 



Mensile. Pasc 23-25. 8vo. Catania 1892. The Academy. 



Cracow : — Academie des Sciences. Bulletin International. Fevrier, 



1892. 8vo. Gracovie 1892. The Academy. 



