The K-partitions of the R-gon. 117 



8. The number of belts that we have handled is 18. 

 We have virtually multiplied every one of these by its par- 

 tition-factor, which is 126, the number M of the permutations 

 of the (/£ = )5-partitions, of (x=)^. That is, we have con- 

 structed 



18-126 = 2268 



34-partitions of a 62-gon by the distribution upon the irre- 

 ducible H of our 18 unpermuted belts. 



In 9 of these belts which have 5 different units (Art. 5) 

 there can be made ju=I20 permutations of the 5 primes 

 without altering the posture of any one under its marginal 

 triangles. 



In the other 9, in which two of the 5 units are identical) 

 these two can be made or supposed to wear their marginal 

 triangles alike, after which the 5 primes can undergo ^ = 60 

 permutations with postures unchanged under marginal tri- 

 angles. In either nine every permutation of its primes 

 makes it a belt unused before ; so that the actual number 

 •of belts is 9-120 + 9-60= 1620. Here }i = 120 is the permu- 

 tation-factor of each of the first nine, and ^i = 6o is the 

 permutation-factor of each of the second nine. The 

 partition-factor is the same M = 126 for each of the eighteen 

 belts. Hence 



1 26,(9- 1 2 ° + 9'6o) = 204 1 20 



is the number of 34-partitioned 62-gons obtained by distri- 

 bution upon H of all the 18 permuted belts unaltered in 

 posture of their primes. 



9. We have next to consider the changes possible in 

 our belt 3Bi (call it 3Bi,i), by an alteration not in the 

 position as to order of a prime in it, but in the way in 

 which the primes wear their marginal triangles. Thus A 

 in 3Bi can wear, as B wears, its triangle above, giving to 3Bi,i 



