220 



5. The asymmetric poly edr a are the most numerous of all. 

 The data required for the enumeration of the P-edra Q-acra are 

 the following : 



A. A table of the hexarchaxine, tetrarchaxine, and triarchaxine 

 P-edra Q-acra and Q-edra P-acra, both zoned and zoneless, with all 

 the zonal signatures, and an account of the terminations of their prin- 

 cipal axes, and with the zonoid signatures. 



The zonoid signature in a zoneless or mixed symmetry gives the 

 number of zoneless polar faces, summits, and edges. 



B. A table of j anal poles, containing an exact account of every 

 polar feature terminating a janal axis, with the repetition if it be 

 zoneless, and with the traces and zones if zoned. 



C. A table of poles, containing an exact account of every polar 

 feature terminating either a janal or a heteroid axis, with zonal 

 signatures and traces when they exist, or with the zoneless repe- 

 tition. 



D. A table of faces and summits having a diagonal trace of a 

 single zone, with the zonal signature. 



E. A table of objanal monozone faces and summits, with their 

 zonal and zonoid signatures. 



F. A table of contrajanal anaxine pairs of edges, with the faces 

 A B intersecting in them. 



G. A table of all edges (A B) with their intersecting faces, the 

 polar edges with their zones, the monozone zonal and epizonal edges 

 each with its zone. An epizonal edge is cut by a zone. 



These data being first obtained for the P-edra Q-acra and for the 

 Q-edra P-acra, the numbers of the following solids and faces and 

 summits are exactly known for both P-edra Q-acra and Q-edra 

 P-acra ; and we suppose that all the inferior polyedra are registered 

 in like manner : 



a. The m-zoned monarchaxines with their zonal signatures, and 

 their principal poles ; 



b. The zoned triaxines with their zonal signatures ; 



c. The m-zoned homozones with their zonal and zonoid signatures, 

 and their exact principal poles ; 



d. The zoneless monarchaxine janals with their repetition and zo- 

 noid signatures, and their principal poles ; 



e. The zoneless triaxines ; 



