n8 The Rev. Thos. P. Kirkman on 



a summit more above and one fewer below. Make that 

 change, and call the new figure 3Bi, 2 . This 3Bi,2 is a 19 th 

 equivalent belt, which we have to distribute, like 3Bi,i,. 

 126*120 times. 



If we now make any other such a change of posture in 

 any one, any two together, or in every prime at once in 

 38i,i, the new figure will be 3Bi, 3 , which we have to dis- 

 tribute I26"I20 times. 



The same is true if the belt in which we change one or 

 more postures of its primes, but not their order, be any one 

 3B C of the first nine belts that are capable of 120 permuta- 

 tion of their primes. We distribute 3B c ,i, 3B C >2, &c. 



If the belt handled be 3B ft , say JB^a, one of the nine that 

 can have only 60 such permutations, the new belts 3B fc ,2, 

 36^,3, &c, can each be distributed, like 3B*,i, on ty 126-60 

 times. 



A prime in a belt can change its posture under its mar- 

 ginal triangles by using a different base-tie. 



Definition : A base-tie of a prime is a line not drawn* 

 but conceived as drawn, bisecting two bases of its marginal 

 triangles that have or have not a common summit, the 

 prime being complete, i.e., not in a belt concealing creased 

 triangles. In a belt the base-tie bisects two non-marginal 

 diagonals. 



The number of different ways in which a prime can 

 wear in a belt its marginal triangles is the posture-number of 

 the prime. 



The product of all the posture numbers of its primes is 

 the posture-factor W e of a belt 3B e . 



10. The changes of posture of its primes may be con- 

 sidered to take place either before their distribution or after, 

 as follows : — 



Consider our first belt % x distributed in Fig. 2 by Art. 6. 

 Let now all the primes so distributed but two, whose 



