288 Mr. Sylvester's Elementary Researches in the Analysis of 



The second being derived from the first by omitting every 

 alternate term ; similarly, below the lines are derived each from 

 its antecedent. 



m = 7, we have a .b .c . d . e .f. g 



a.c.e.g.b.d.f 



a . e .b .f. c . g . d 

 A very little consideration will serve to prove that in this 



way, m being a prime number^ — - — , cyclothemes may be 



formed, such that no element will ever be found more than 

 once in contact on either side with any other; whence the rule 

 for obtaining the diplothematic total to any prime-number 

 modulus is apparent. 



Ex. gr. to modulus 7 the total reads thus: — 



1st. a.bb.cc. dd.ee .f f. g g . a "| 

 2nd. a .c c .e e . g g . b b . d d ,J r f. a > 

 3rd. a . e e . b b .ff. c c ,g g . d d . a J 

 and no more remains to be said on this special case. 



Let us now return to the theory of even moduli, and show 

 how to apply what has been just done to constructing a syn- 

 thematic total to a modulus which is the double of a prime 

 number. 



Suppose the modulus to be six, the number of synthemes is 

 five. Let the six elements, ar, b, c, d, e,/, be taken in three 

 parts, so that each part contains two of them; let these parts 

 be called A, B, C, where A denotes a b, B, c d, and C, ef. 



Now the duads will evidently admit of a distinction into two 

 classes, those that lie in one part, and those that lie between 

 two; thus a b, c d, ef will be each unipartite duads, the rest 

 will be bipartite. 



The unipartite duads may be conveniently formed into a 

 svntheme by themselves; it only remains to form the four re- 

 maining bipartite duad synthemes. 



Write the parts in cyclothematic order, as below : 



ABC. 



It will be observed that each part may be written in two po- 

 sitions ; thus 



A may be expressed by 7 or by 



c d 



B 



c j 



d '" c 



e f 



