The K-partitions of the R-gon. 123 



Since each of these T variations of our distributed tape 

 will be combined with all the preceding S configurations, 

 without changing anything in the latter, we shall obtain 



TS = 441708960 x 22680000000, 

 = 1 00 1 79 5 92 1 2 800000000 



different partitions of our R-gon. The (19+ i5)-partitioned 

 (39 + 23)-gon (Art. 3) has become by addition of the 

 6 faces and 9 summits of the tape, a 40-partitioned 71-gon, 

 and we have formed T.S of these asymmetrical propyramidal 

 71-gons, no one of which is either the repetition or the 

 reflected image of another. That is, if M be a million, we 

 have formed of them 



ioM 3 +i7959M 2 +2i28ooM. 



If the submarginals in H, standing on the dark edges 43 

 and 21, be detached, the first is seen to be the prime A, and 

 the second is a square under four marginal triangles ; and 

 evidently neither of these could change the figure H by 

 undercreasing another triangle. The marginal diagonal of 

 that A should begin at 4. 



The submarginals on the dark edges 65 and 31 are 

 monozones, each of which can, by undercreasing its other 

 two triangles, place its zonal trace in three positions, and 

 thus the two together can, still occupying 65 and 31, give 

 to H 3*3 = 9 configurations. 



Further, these four submarginals can occupy the edges 

 43, 21, 65, and 31 in 24 different ways, and by the same 

 changes of two concealed marginal triangles turn H into 

 24-9 = 216 ^equivalent H's, all of which have been virtually 

 handled by us, and have each given us the same number 

 T.S of 40-partitioned 71-gons. Thus we have constructed 

 more than 2,000 millions of billions of them, namely, 



2i6.T.S = 2i63M 3 + 879i89M 2 + 9648ooM. 



