be cut from any doubly reversible .r-edron having k triangles about 

 its (# l)-gonal base. 



Whenever k or I in the function ii(x, k, I) is not integer, the func- 

 tion, by a geometrical necessity, is to be considered =0. 



U(x,k,l) = ii(x,k,l), 



ll\2x + 1, 2k, l)={ii(2x + 1, 2k, l)ii(x + 1, k, 1)}, 



Il 3 (3x+l, 3k, l)={ii(3x + 1, 3*. l)-ii(x + 1, k, 



FF(2jr + 1 , 2k, 1) = ii(x + 1 , k, 1), 



1*1* (3x + 1 , 3k, 1) = O + 1 , k, 1) ; 



RR(2* + 1, 2k, l)-ii(x + 1, k, 1), 

 RR(2.r + 1, 2fc + 1, l)=ii(x, k, K'~ 2)) 



tf, 2k, 1) = ii(x, k, 1) + iix, k, (/- 1 ) ; 



1 , 2k, l)={ii(2x+ 1, 2k, l)ii(x+ 1, k, /)} 

 lR(2x+l,2k + l, l)={ii(2x+ 1,2k +1,1) -ii(x, k, |(/- 

 IR(2.r, 2k, l)={ii(2x, 2k, l)-ii(x, k, l)-ii(x, k, i(/- 



R 2 R 2 (4o?+ 1, 4k, l)=ii(x+ 1, k, 1), 

 PR 2 (4tf+ 1, 4k, l)={ii(2x+l, 2k, l)-ii(x+ l,k,tf) 

 RR 2 (4^ + 1, 4k, l)=ii(2x+l, 2k, l)-ii(x+ l,k, #), 

 IR 2 (4^r+ 1,4k, I)=^[ii(4x+ 1, 4k, l) + 2ii(x+ 1, k, {I) 



R 3 R 3 (6tf+ 1, 6k, l) = ii(x+ 1, k, 1), R 3 R 3 (7, 3, 3) = 1, 

 PR 3 ( 6x + 1 , 6k, 1) = |{ ii(2x +l,2k, 1) - ii(x + 1 , 



l,6k,l) = ii(3x+l,3k,l)-ii(x+l,k,il), RR 3 (7,3,1) = 2, 

 1, 6k, l)=${ii(6x+l, 6k, l) + 3ii(x + l, k, 1) 

 ii(2x+ 1, '2k, l)3ii(3x+ 1, 3*, /)} , 

 IR 3 (7, 3, 2) = IR 3 (7, 3, 1) = IR 3 (7, 3, 0) = 1 ; 



I n R m O+l,A, x k)=0. 



By the aid of the above, together with the following, equations, 

 the (,r + A+/).edra having k + 1 triangular faces, an (x + k + l 1)- 

 gonal base and triedral summits, are successively found. 



I (^r, k', l') + l\x, k>) . II 2 (x, k', P) 

 + l s (x, k') . II 3 (x, k', /') + R(*. k') . IR(*, k', I') 

 + W(x, k') . IR 2 (>, V, /') + R 3 (*, k') . IR 3 <>, k', /')} ; &c.&c. 



taken for all values of k 1 -\-l' = 



