122 The Rev. Thos. P. Kirkman on 



In 4, D 1 uses its base-tie 14 of (b) ; in 5, D : uses 12 of 

 (5). In 4, d uses 52 of (#) ; in 5, Q uses 54 of (a). In 5 

 A and B have each made a revolution about its base-tie, 

 used in 4. B, in 5, should have a small marginal triangle 

 on the left at the fracture between 1 and 1'. 



The above S results would be nearly all that is required; 

 had we not to give an account of a third datum (Art. 6) 

 which is the tape, Fig. 8, of three triangles and three rect 

 angles that carry no marginal triangle. 



All the S partitions above made have each (Art. 3) 33 

 diagonals, i.e., places to receive, after splitting the proper 

 diagonals, the 6 units of the tape, which, after each distribu- 

 tion of them, will have added six faces to the 34 in each of 

 the S partitions. 



Instead of 5, afiySs, we have now 33 places to name, to 

 fix, and to split; in these we include the bases of the 23 

 marginal triangles in H, and in each of the 18 distributed 

 belts ; for by splitting such bases we cannot introduce a 

 new submarginal, as the tape carries no marginal triangle. 



13. Our distribution of the tape can be exactly effected 

 in 33IR3 different ways, which is the number of permuta- 

 tions of the 33-partitions of 6, without altering the order or 

 the posture of any prime in the tape. Such prime, out of 

 the tape, has two marginal triangles that in the tape are hid. 



This 33 1R 6 is by theorem £2, Art. 5, the 33 rd , which is also 

 the 7 th , coefficient of (1 + i) 6 ** -1 , or 



38-37-36-35-34-33 =2;6o68l . 

 r2'3'4'5-6 



This is the partition-factor of the tape. Its permutation 

 and posture-factors are 20 and 8, so that we have 160 tapes 

 to distribute, which are all one (Art. 6). 



The product of the three factors is 



20-8-276068 1 = 44 1 708960 = T. 



