GEOMETRICAL DEDUCTION OF SEMIKEGULAR ETC. 23 



38. The foregoing investigation leads to the following conclusion 

 as to the nets of fourdiinensional space. 



If the edges are the subject there are only vertex gaps. 



If the faces are the subject there are edge and vertex gaps. 



If the limiting bodies are the subject there are face, edge, and 

 vertex gaps. 



If the constituents are the subject there are body, face, edge, 

 and vertex gaps. 



The vertex gaps are filled by polytopes determined by their 

 vertices. Their limiting bodies are regular or semiregular polyhedra. 



The edge gaps are filled by fourdimensional prisms determined 

 by edges parallel to their axes. Their bases are either regular or 

 semiregular polyhedra and their other limiting bodies are prisms. 



The face gaps are filled by prismotopes determined by parallel 

 positions of a face and are limited by two groups of prisms. 



The body gaps are filled by fourdimensional prisms determined 

 by two parallel positions of a regular or semiregular polyhedron. 



Contraction applied to the nets. 



39. One or two examples will suffice to shew the application 

 of this process to the nets. 



If in the net e { N(0,T) (Fig. 18) (A. 24) the CO corresponding 

 to the vertices of the original octaheclra be made the subject of 

 contraction, the tO are reduced to CO, the tT to 0, while the 

 CO remain unchanged. Thus ce^N^O,! 1 ) denotes a net composed 

 of and CO (A. 18). 



40. In the net e 2 NC 2!i: (Fig. 19) the polytopes filling the vertex 

 gap (#) may be made the subject of contraction, when the folio wing- 

 changes take place. The polytope a. remains unchanged except in 

 position; the prism /3 is reduced to a tetrahedron common to two 

 of the polytopes a; the CO of 7T remain unchanged while the RCO 

 are reduced to cubes. Thus the net of three constituents is re- 

 duced to one of two constituents, one limited by SCO andlGT 7 , 

 the other by 24 C and 24 CO. 



Tables. 



41. The chief results of this memoir are tabulated in the 

 Tables I and II. 



Table I gives the 48 polytopes of expansion (the regular polytopes 

 included) and the 42 polytopes of contraction. The first set has 



