460 Royal Society : — 



By such a reductiou of a ^j-edral (/-acron P to P', of P' to P", &c., 

 P can be sliowu to be (/eiierabJe from a certain pyramid n ; by which 

 it is meant that 11 is the higliest-ranked pyramid to which P can by 

 snch reduction be reduced. 



Hereby it is evideut that the problem of enumeration of the .r-edra 

 is brought down to this: to defennine how many {y-\-n\)-edra are 

 generahle from the x-ed red pyramid. 



The autopolars so generable are first considered, as the heteropolars 

 are obtained by combination and selection of those operations with 

 which the theory of the autopolars makes us acquainted. 



Autopolarity is of three kinds, nodal, enodal, and utral. 



Every even-based pyramid is nodally autopolar ; i. e. it cannot 

 but have two nodal summits. For example, the 5-edral and 

 7-edral pyramids have the signatures of their faces and summits 

 thus arranged, — 



1234 123456 



4321 65432 1* 



the upper line showing the triangles, and the lower the triaces about 

 the base, which as well as its pole the vertex, is signed zero. The two 

 triaces in the triangle 5 are 3 and 2 ; the two triangles in the triace 

 1 are 6 and 1 in the 7-edron, and 4 and 1 in the 5-edron. The nodal 

 summits and faces are 3 and 1 in the 5-edron, and 4 and 1 in the 

 7-edron. No other mode of autopolar signature is possible in these. 



Every odd-based pyramid is utrally autopolar. The 6-edral and 

 8-edral pyramids may receive either of the signatures following : — 



12345 1234567 



54321 7654321 



12345 1234567 



45123 567123 4* 



the first of which lines exhibits nodal faces and summits 31 and 41, 

 while in the second every triangle is opposite its polar triace, and no 

 face or summit is nodal. 



No pyramid is enodally* autopolar, i. e. capable of only enodal 

 signature. If we draw a 7-gon whose summits are 1234567, and 

 then the dotted lines 73 and 75, and next taking three points in it, 

 complete the 5-gon 34089, and join 93, 92, 81, 87, 06, 05, 04, we 

 can sign the faces thus : — 



045 = 1, 506 = 2, 6087 = 3, 781=4, 1892=5, 293=6, 39804 = 7, 

 2371 = 0, 3754 = 8, 567=9. The type now represents an enodally 

 autopolar 1 0-edrou, in which no pair of gamics meet each other, or 

 can by smy autopolar arrangement be made to meet. The 18 edges 

 of the solid are well represented thus, the odd places in a quadruplet 

 showing summits, and the even, faces : — 



1520 2630 3748 4158 5269 6379 7410 0783 8795 



0251 0362 8473 8514 9625 9736 0147 3870 5978* 



The gamic pairs stand together, and no quadruplet exhibits fewer 

 than four numbers. A nodally autopolar must always be, and a 



