REFERENCE TO THE ENUMERATION OP POLYHEDRA. 253 



to the formation of the polyacrons as far as the ii- 

 acrons we are only concerned with the first and second 

 processes. 



Consider the entire series of w-acrons, say A, B, C, &c., 

 and suppose that the w-acrou A gives rise to a certain 

 number^ say P, Q, R, S of (w+ i)-acrons^ the {n-\-i)- 

 acron P is of course derivable from the ?2-acron A, but it 

 may be derivable from other w-acrons, suppose from the 

 /i-acrons B and 0. Then in considering the {n+ i)-acrons 

 derived from B^, one of these will of course be found to be 

 the (w+i)-acron P, and it is only the remaining [n+i]- 

 acrons derived from B which are or may be {n+ i)-acrons 

 not already previously obtained as (w+i)-acrons derived 

 from A. And if in this manner, as soon as each {n+ i)- 

 acron is obtained, we apply to it the process of subtraction 

 so as to ascertain the entire series of w-acrons from which 

 it is derivable, and, in forming the (/^^- i)-acrons derived 

 from these, take account of the {n+ i)-acrons already pre- 

 viously obtained and found to be derivable from these, we 

 should obtain without any repetitions the entire series of 

 the {n+ i)-acrons. 



For merely finding the number of the {n+ i)-acrons, a 

 more simple process might be adopted : say that an w-acron 

 is ^-wise generating when it gives rise to p {n+ i)-acrons, 

 and that it is g-wise generable when it can be derived 

 from q {n+ i)-acrons; and assume that a given «-acron is 

 (2/1 + 2/2 + 2^8 + ^^0 "^^^6 generating, viz. that it gives rise to 

 ?/i (7^+ i)-acrons which are i-wise generable, y^in+i)' 

 acrons which are 2-wise generable, and so. on ; these form- 

 ing the sum 



where 2 refers to the entire series of the w-acrons, it is 

 clear that every m-wise generable (^+i)-acron will in 

 respect of each of the w-acrons from which it is derivable 

 be reckoned as 1, that is, it will be in the entire sum 



