The K-partitions of the R-gon. 



121 



J3i, 



AB Ci D x D 2 







2 , 2 , i5'io*6o; 



n x 



$2 



ABC 2 DiD 2 







2'2 , 3o , io"6o; 



n 2 



JBb, 



ABQ^D, 







2 , 2'3o , io'6o; 



n 3 



3B±, 



ABCiD 2 D 3 







2'2-i5'6o-3o; 



n 4 



J&5, 



ABC 2 D 2 D 3 







2'2'3o - 6o , 3o; 



n 5 



We, 



ABC 3 D 2 D 3 







2 , 2'30'6o , 3o ; 



n 6 



3^7, 



ABd^D! 







2-2'15'30-IOj 



n 7 



JD8, 



AB C 2 D 3 D x 







2'2'3o # 30 , io ; 



n 8 



3B9, 



ABC 3 D 3 Di 







2'2-3o-3o-io; 



n 9 



= 36000 

 = 72000 

 = 72000 

 = 108000 

 = 216000 

 = 216000 

 = 18000 

 = 36000 

 = 36000 



810,000 



jBio, 



ABQDi D x 







2*2'i5'io'io; 



IIio= 6,000 



3Bn> 



AB C 2 Dx Di 







2 , 2'30'IO*IO j 



Er u = 12,000 



jBi2 3 



ABCgDxDi 







2"2"30'io'io; 



iii 2 = 12,000 



3Bis> 



ABdD 2 D 2 







2*2*i5'6o'6o ; 



iii 3 = 216,000 



3Bi4, 



ABC 2 D 2 D 2 







2 , 2"3o*6o , 6o; 



1114=432,000 



3Bi5> 



ABC 3 D 2 D 2 







2 # 2'3o*6o*6o \ 



1115=432,000 



jBl6, 



AB Ci D 3 D 3 







2'2-i5'3o-3o; 



IIi 6 =54,000 



3Bl7 3 



ABC 2 D 3 D 3 







2'2-30'3o-3o; 



ITi7io = 8,ooO' 



JDl8j 



AB C 3 D 3 D 3 







2 -2'3o-3o-3o; 



n 18 = 108,000 



1,380,000 



By our subsequent changes of postures in 126 times 

 distributed primes in the 120 times or 60 times permuted 

 belts, we multiply every posture-factor in this table either 

 by 126-120 or by 126*60. 



It follows that the sum S of all the partitions of the 

 R-gon thus far constructed is 

 1 26* 1 20'8 1 0000 +126-60- 1 3 80000 = S or S = 22,680,000,000. 



We need not distress ourselves about the undercreased 

 triangles in this hurly-burly of changing base-ties. The 

 pretty primes are nimble and well drilled. There is no fear 

 of damage to their wings in these thousands of millions of 

 evolutions. 



12. In Fig. 4 and 5 are seen different postures of the five 

 distributed primes of the unpermuted 3Bi. 



In 4, D 2 uses its base-tie 13 (Fig. d) ; in 5 it uses 12 of 

 (d) ; the third triangle in (d) being hid in 4 and seen in 5,, 

 while the second triangle in (d) is seen in 4, but hid in 5. 



