The K-partitions of the R-gon. 113 



the x=$ primes of the belt 3Bi into the k=$ split and 

 unsplit dark diagonals of H, is not less than M, that of all 

 the permutations of the 5 -partitions of the number 5, zeros 

 and repetitions being allowed in the partitions. 



By my theorem £1 {vide the coming volume of Reprint 

 of Educational Times), this number is the k ih co-efficient 

 in (i + i)***- 1 , i.e., the 5 th coefficient in (1 + i) 5+s_1 , which is 

 gSy6: r2'3'4= 126. This M=i26 is all the answer we 

 can get if our x=5 primes can be exhibited in no other 

 belt besides one 3Bi. This can be the case only when the 

 primes are 5 squares in the belt, which out of it are 5 6-gons, 

 that are capable of only one order and posture in a belt 3Bi, 

 having their marginal triangles all creased under and 

 forming not a belt, but a tape, like figure 8. Such primes, 

 having only two marginal triangles, are excluded by defini- 

 tion of a belt in Art. I. 



There are many belts. There is no second 6-gon A 2 , nor 

 second 9-gon B 2 , having 3 marginal triangles ; but there are 

 three 14-gons (C) and three 15-gons (D) that all have 

 6 marginal triangles. These are the 2-zoned d and the 

 monozones C 2 C s , Figs. a,c,e; also the 3-zoned Di, the 

 asymmetric D 2 and the monozone D 3 , Fig. b,d,f. 



To secure the construction of all our partitioned R-gons> 

 these eight primes must all alike contribute to form a belt 

 of five, the belts so formed being equivalents. 



6. We have to use the 18 equivalent belts following : 



JBi, ABCiD,D 2 ; J3 4 , ABCiD a D 8 ; JBt, ABGD 3 D i; 

 B2, ABCaDJ),; $5, ABC 2 D 2 D 3 ; 3Be, ABC 2 D 3 D i; 

 »,, ABC 3 DxD 2 ; Bj', ABC 3 D 2 D 3 ; B,, ABQAA; 



nine belts in which is no repeated prime ; and 



Bio, ABCiDxDx; 3Bi8, ABdD 2 D 2 ; 3B 16 , ABdD 3 D 3 ; 

 »u, ABQDxDx; Bu, ABC 2 D 2 D 2 ; JS17, ABC 2 D 3 D 3 ; 

 »u, ABCsDxDx; Bis, ABC 3 D 2 D 2 ; Bis, ABC 3 D 3 D 3 ; 



nine in each of which is a repeated prime. 



