( 286 ) 



By this value the place of N on pp' is perfectly determined ; 

 however in fig. 5 we may — and, if d t has been determined by 

 Pj and J/ a , we must - - assume for S not the point on the right 

 of M x corresponding- to this ratio but the point on the left. 



As a second example we consider the group IX, of plate VI 6 to 

 which — according to the second memoir - corresponds a series 

 of planes normal to the plane of tig. 4 parallel to up, 111 (pj 11 being 

 the midpoint of sv). We draw through a and q the lines sS" and 

 qS' parallel to t'p, 111 and determine now the ratio of pS' to />/>' 

 by means of similar triangles as follows. These triangles give 



sv _ w'sr __ P s" _j,s" _ pic __ pic' 



qw' w'w" ps ic'u w"u \<pi 

 So we have 



S'ir '2'jtr' pw' 2s S 



pp' qu pp' i/-\-s d 



and finally 



P S'_pw--SV_ bJ -- \ sjd-s± = (e-l)(S-e) __ 



pj'~ pp' ~d\ ' d + *J d <l±s)~ 2(«+-l)" 



In this way is obtained the complete system of the twelve different 

 ratios ;. given in the following table, where, when I differs from 

 i, the value smaller than \ always appears. For all the groups in 

 any horizontal row X has the value indicated in the last column 

 but one. In the last column are given the numbers of centimeters 

 corresponding to these ratios, when the length of the side of the 

 starpentagon (fig. 5) is 20 centimeters. Finally the last column but 

 two indicates the direction of the trace of the intersecting planes 

 normal to the plane of lig. 4, by means of which the values of X 

 have been calculated. (See table p. 287). 



For the sake of clearness the values of ;. with the side (20 

 centimeters) of the starpentagon of fig. 5 as unit have beer, indicated 

 separately in fig. 6. By transferring this scale division in fig. 5 to 

 each of the sides P t P,, etc. we are enabled to draw immediately 

 each of the traces / in question with accuracy. 



12. By means of the preceding the polygon of intersection of the 

 polyhedron situated in any assigned diagonal plane can be constructed. 

 To this end we indicate in tig. 7, which is a repetition of fig. 4, 

 for the twelve different cases of the table by the numbers 1, 2, . . . , 

 12 the traces of the intersecting planes passing through the edge in 

 q normal to the plane of the diagram, and show how we can 

 obtain all the measures necessary for the construction of these 



