I 381 ) 

 munication for the six series of parallel sections of the eightcell we 



indicate the results of the intersection <>f a single sixteen cell Gig 

 in two different manners. A first plate will give t he projections of 

 the limiting elements of the sixteencell on the diameter normal to 

 the intersecting spaces, which will enable us to deduce the sections 

 from it tabularly ; a second plate will give the sections themselves 

 in parallel perspective, included in the sections with the polarly 



circumscribed G-V ' or the bodily circumscribed C\ . Moreover a third 

 plate will contain two groups of diagrams, the first of which will 

 elucidate the manner of deduction of the projections given on plate I, 

 whilst the second is concerned with the space-tillinus obtained by 

 the intersection of the net (G' 16 ). In order to facilitate the survey 

 of these space-fillings we deviate from the way followed in the 

 second communication and treat together the more or less regular 

 space-fillings presenting themselves here, instead of joining each of 

 them separately to the corresponding generating three series of in- 

 tersecting spaces. 



We now first consider the four diagrams of the first group of 

 plate III dominating the deduction of the projections of plate I. In 



tig. 1 we once more show how the inclined cell Clë " , indicated 



by its vertices only, is inscribed in the cell Gs~. If we indicate by 

 A, B, C, D the vertices of one of the sixteen limiting bodies, by 

 A', B' , C' , D' the opposite ones, the sixteen limiting tetrahedra are 



ABCD 



ABCD 

 ABCD 

 A B CD 

 ABCD 



ABCD 

 A'B CD 

 A'BCD' 

 ABCD 

 ABCD 

 ABCD' 



ABCD' 

 A'B CD' 

 A'B' CD' 

 ABCD 



A' BCD 



Of these five groups of 1,4,6,4,1 tetrahedra those of the first, 

 the third, and the fifth groups are inscribed in the eight limiting 

 cubes of the eightcell, whilst the four vertices of each of the tetra- 

 hedra of the second and the fourth group always split up witli 

 reference to two opposite limiting cubes of the eightcell into one 

 vertex and three vertices. 



