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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 3 1 and 4 1 , 

 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-edron, in which no pair of gamics meet each other, or 

 can by any 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 

 utrally autopolar may always be so signed, that two pairs of gamics 

 shall exhibit in each quadruplet a duad of the form act. In the 

 above type it is observable that every duad, as 15, occurs four times. 

 The same thing is to be seen in every autopolar type of edges. 



If we make use of the closed 10-gon 1239804567, as directed in 

 a paper "On the Representation of Polyedra," in the 146th volume 

 of the Transactions of the Royal Society, a paradigm of this 1 0-edron 

 can be written out, exhibiting to the eye all the faces, summits, 

 angles, and edges of the figure. 



The problems following are next proposed and solved. 



To find the number of autopolar (r + 2)-edra generable from the 

 (r-f \}-edral pyramid. 



