144 TERMINOLOGY. . 184. 



+ rlE (0 Fig. 21., -r^(0 Fig. 22., 

 4 4 



+ 1 IE (*'') Fig. 23., 1 IE (*'") Fig. 24. 

 4 4 



These four solids yield six binary aggregates : 



1. + r IE. r IE, which is = 1 IE Fig. 30. ; 



4 4 2iii 



2. + r IE. + 1 IE, which is = + Fig. 25. ; 



4 4 2i 



3. + r IE. _ 1 IE, which is = IE Fig. 28. ; 



44 2 ii 



4. rlE. +1 IE, which is = + Fig. 27. ; 



44 2 ii 



5. -rlE. _ 1 IE, which is = -IE Fig. 26. ; 



4 4 2i 



6. + 1 IE. 1 IE, which is = r IE Fig. 29. 



44' 2iii 



Of these, 1 and 6 are pentagonal-icositetrahedrons, 1 is 

 the left, and 6 the right one ; 2 and 5 are tetrahedral 

 trigonal-icositetrahedrons, 2 is in the normal, and 5 in the 

 inverse position ; and 3 and 4 are trigrammic tetragonal- 

 icositetrahedrons, of which 4 is in the normal, and 3 in 

 the inverse position. Every two homogeneous forms of 

 these six reproduce by combination the tetracontaoctahe- 

 dron itself. 



The halves and fourths belong to the second degree of 

 regularity. 



The preceding methods of resolving the original forms 

 of several axes yield all those forms which have been de- 

 scribed above (. 57. 77)> and which could not be ob- 

 tained by immediate derivation. Thus, resolution com- 

 pletes what by derivation would have remained imperfect ; 

 and we are entitled to consider as complete the number 

 of simple forms of several axes. 



The method of resolving simple forms, is not confined 

 to those which possess several axes, in as much as it may 

 also be applied to pyramids of every description, and even 



