The K-partitions of the R-gon. 125 



the problem is completely solved. And it is evident that 

 whatever be our data, as irreducible, as belt, and as tape, it 

 is impossible that 2 of our constructions can be alike, unless 

 the primes are twice distributed in the same way, by the 

 guidance of the same permutation of the diagonals that 

 may be split to receive the primes. And this, twice, is 

 clearly impossible, because the number M of those permuta- 

 tions is exactly given by theorem Q, and none of them is 

 twice used ; vide Art. 7. 



It is impossible, also, that anyone of our constructions 

 should be the reflected image in any position of another ; 

 because the reflected image of H has never been used. 



If in a partitioned R-gon there is no irreducible H, the 

 R-gon is either a belt in which a tape is or is not distributed, 

 or it is a pair of collateral submarginals in which a tape is or 

 is not distributed, or it is a tape. 



The tape in every case can be dropped out, and the belt 

 that remains can have all its possible configurations enumer- 

 ated by the above method, after which the tape can be 

 inserted again in every way possible. 



Thus all the possible different k-partitions can be deter- 

 mined, both symmetric and asymmetric. But all the faces 

 must be exactly given in the sense in which the faces of our 

 18 belts were given in any single belt of them. 



Hence it is quite correct (Art. 6) to say that no k-par- 

 titioned R-gon can have more than one irreducible H, one 

 belt, or one tape, although each of the three may have many 

 equivalents, due to permutation and altered posture of the 

 primes, and to the equal right, which like primes, dC 2 C 3 . . . 

 D!D 2 D 3 . . . (Art. 5) have to every possible admission into 

 the equivalent belts. 



The number of distinct equivalent belts that we have 



used in the pages preceding (Art. 11) is 9-120' X (IJt + IL 



+ . . +n9) + 9-6o-x(nio + n 11 + . . +nis); and these are all 



given with any one of them that may be first handled 



J 



