[ 187 ] . ' srr't ..-- ^^^ 



XXVI. On some new Theorems in Arithmetic. 

 By J. J. Sylvester, F.R,S.^ 



LET St(ff, b,c,...k, I) denote, as is not unusual, the complete 

 sum of the products of the elements {n in number) 

 a^bfCj... k, I, combined in every possible way i and i together. 

 Let Si(«, hjCy . . .k,T) denote the sum of the products of the same 

 elements combined i and i together, but so that all combinations 

 are excluded in which any two consecutive elements as a and h, 

 or b and c, ... or X: and /, appear simultaneously. S^ may be 

 termed a complete sum of zth products, and Sj a sum of products 

 of anakolouthic elements, or briefly an anakolouthic sum of iih 

 products. If we expand the continued fraction 



lab k I 



p-i- p-\- p+ ' ' ' p+ p' 



it will be easily found to take the form 



|Q"-^ + SV^-3 + S^2P''"^+ &C. 



p- + S,p--' + '^^--'-\-kc. ' 



where S'^ is intended to denote the anakolouthic sum of the tth 

 products of 6, c, . . . /, and S^ the anakolouthic sum of the zth pro- 

 ducts of a, b, c, , . , I. 



It is this fact, and the close relation of reciprocity in which 

 the generating continued fraction for anakolouthic sums stands to 

 ordinary continued fractions (a reciprocity which becomes more 

 apparent when p is made unity), which gives a peculiar import- 

 ance to the theory of anakolouthic sums of the kind denoted by S j 

 otherwise we might be t'^ mpted to embark upon a premature 

 generalization, extending the force of the term anakolouthic so as 

 to denote by S a sum of products in which no three consecutive 

 elements came together, S a sum of products in which no foul- 

 consecutive elements came together, and so on; these more 

 general forms of anakolouthic sums may hereafter merit andreward 

 attention, but my present business will be exclusively with a 

 statement of sonK3 remarkable properties which have accidentally 

 fallen under my observation, of anakolouthic sums of the kind first 

 mentioned, and referring to elements formed in a manner pre- 

 sently to be explained, from the natural progression of numbers. 

 In order to familiarize the reader with the construction of nako- 

 louthic series, I subjoin the following examples : — 



^ ' * Communicated by the Author. 



