The K-partitions of the R-gon, 119 



posture numbers are f and g, keep their postures which we 

 see in Fig. 2 and in $$ v while the other two exhaust all 

 their possible combined changes of posture, which are fg 

 changes, each once counted. It is clear that Fig. 2 will 

 assume fg different forms, by variations of posture in only 

 two of the distributed primes. Next let only two primes 

 keep their postures, while the three, whose posture-numbers 

 are fg and h, assume all their possible fgh combined pos- 

 tures, each of the fgh once. We looking on shall have seen 

 in Fig. 2 fgh different configurations made by 3 primes, 

 combined with the unchanged attitude of the two others. 

 If z and j be the posture-numbers of these two, they could 

 assume each of their ij combined postures in succession, 

 and in company with every single one of the three fgh. 

 That is, Fig. 2, without change in the distribution or 

 order of its five primes, can take fghij= II2 different con- 

 figurations, all different partitions of the R-gon so far 

 constructed. 



But Fig. 2 is only one of 126*120 distributions upon H 

 of 3Bi and its 120 permutations, in which all the primes all 

 through use the same base ties. Wherefore, after insertion 

 of 3Bi into H, we can obtain in all, by changes of order and 

 posture of its so distributed primes, 126" 120 III, and no 

 more, differently partitioned R-gons, where 126 is the 

 partition factor of 3Bi, 120 its permutation factor, and IIi its 

 posture factor. 



The first factor is always given by theorem O, the 

 second by first lessons in Algebra, and the third is elemen- 

 tary in the theory of the Polyedra. We are about to see 

 that 126-12011!= 544,320,000. 



11. The next thing requisite, and enough for our pur- 

 pose here, is a rule for the posture-number of a prime 

 reticulation. 



