224 



3. A plane submargin of the subject is never charged. 



4. Every charge adds a primary effaceable in a mixed reticulation, 

 and may add any number of secondary effaceables, if a solid charge. 



5. If the penesolid 402, which consists only of two marginal tri- 

 angles, is employed to form a compartment of the new signature 

 (which is the compartment (2) r , or (2) r times repeated in the r-ple 

 marginal signature), the resulting reticulation is closed to further 

 marginal increments, and can only be crowned. 



The zonal and zonoid signatures of reticulations give an account of 

 all zoned and polar effaceables, so that after coronation of a reticula- 

 tion we can always enumerate the results of effacement, with their 

 symmetry and signatures. Any effaceable can be effaced independ- 

 ently of others. 



Asymmetric mixed reticulations are never constructed, unless such 

 as can be crowned by a line ; we thus escape the construction of the 

 most numerous class of mixed reticulations. And asymmetric corona- 

 tions, or asymmetric results of effacement, are never registered, unless 

 the coronation be by a line, or the effacements be about & perfect edge. 



An edge is a 2-aee summit which has always e(^:0) effaced 

 effaceables that can be restored in one way only. When they are 

 restored we have a perfect 2-ace, which is to be constructed and 

 registered, whether symmetrical or not. 



The removal of an edge lays bare in all cases either a polyedron or 

 a penesolid, which may be either plane, full, or mixed. 



A penesolid is a reticulation which can be crowned with a line, 

 thus becoming a polyedron. 



The order of the marginal features is not regarded in the marginal 

 signature of a reticulation, as we keep no exact account of the faces 

 about the crowning p-ace ; but the marginal signature of a penesolid 

 is always so written that the exact faces of the crowning line shall be 

 known by inspection of the signature. A penesolid may have any 

 number of effaceables which are registered with the marginal and 

 zonal or zonoid signatures. The results of effacement about the 

 crowning edges are the data C G, and these are exactly given by our 

 methods. 



The marginal signatures of penesolids are of the form 

 [2a2b], [226], [34c], [3aA<?], 



