320 ' Sir Norman Lockyer. Note on the 



operations, namely, S + Sj + + S N-1 , obeys these conditions. This 



sum is called the complete group, and all other groups are its sub- 

 groups. 



The first six sections of this paper are devoted to the detailed estab- 

 lishment of this purely algebraic view of the subject. At times the 

 modification in treatment from that adopted in the standard treatises 

 on the subject, such as Burnside's ' Theory of Groups of Finite Orders,' 

 is slight. Where the modification would be of no sufficient interest it 

 has been simply omitted, and the theorems when wanted have been 

 assumed as part of the general knowledge of the subject. Only so- 

 much reasoning has been given as will establish the principles of the 

 Algebra of Groups of Finite Order, viewed as an algebra independent 

 of any interpretation, however vague. 



The more special object of this paper follows directly from the 

 changed point of view from which the Theory of Groups is here 

 regarded. The idea of the group is no longer so absorbing ; the set 

 takes its place as the fundamental general entity which has to be inves- 

 tigated. A group is a special type of set. Accordingly in this paper 

 some of the general properties of sets are investigated. A set of opera- 

 tions has numerous groups associated with it, and these groups have 

 many relations with each other which this paper cannot pretend to 

 have exhausted. The fundamental idea of this part of the paper 

 (cf. 7) is the formation from a set H of an unending series of other 

 sets, here called the successive powers of H, and in the notation of the 



algebra written H 2 , H 3 , This series is called the power sequence 



of H. Any group which contains H also contains its power sequence. 

 The power sequence is proved to have a periodic property (cf. 9} 

 which introduces a curious analogy to recurring decimals. This 

 periodicity is the foundation of the rest of the paper. It governs the 

 relations to each other of the various allied groups and sets. The 

 periodicity is expressed by an equation of the form 



fjn+sm+q __ ~Qn+q > 



where m is called the period of H, and n the characteristic, and s and 

 q are any integers including zero. The number of theorems relating to 

 m is very large. 



" Note on the Enhanced Lines in the Spectrum of a Cygni." By 

 Sir NORMAN LOCKYER, K.C.B., F.E.S. Eeceived January 20 

 Kead February 2, 1899. 



(PLATE 6.) 



When engaged in the classification of stars, according to their 

 photographic spectra, in 1893* I came across two sets of lines of 

 * c Phil. Trans.,' A, vol. 184, p. 675. 



