256 MR. A. CAYLEY ON THE A FACED POLYACRONS, ETC. 



octacrons, which have no tripleural summits, the passage 

 back to the heptacrons is by the second process : thus for 

 the octacron 44445555 we have but one kind of tetra- 

 pleural summit 4555; but as opposite pairs of summits of 

 the basic quadrangle are of diflPerent kinds, viz. 45 and 55, 

 we obtain two heptacrons, viz. 3444555 and 4444455. 

 The octacron 44444466 has but one kind of tetrapleural 

 summit, viz. 4646, and the pairs of opposite summits of 

 the basic quadrangle being of the same kind 46, we obtain 

 from it only the heptacron 4444455. 



It may be remarked that for the five heptacrons respec- 

 tively the values of the sum y^ + hy^ + ^y^ + • • • are 



1+3 + 2, i+l + 2 + 2> 3 + 2 + 2 + 1 + Ij 1 + 1 + 1 + 1+1, 1+i+l, 



giving for 2(2/1 + 4^2+3^3+ * • •) ^^^ value 14, as it should 

 do. 



