204; TERMINOLOGY. . 155. 



also = n 1 = (. 152. vi.) ; and consequently r + n m 

 = Pr. 



The faces of the horizontal prism Pr + n lv () appear as 

 rhombs if combined only with P and Pr. This horizontal 

 prism therefore is = Pr 1 (. 152. vii.). 



The same horizontal prism produces parallel edges of 

 combination with P + n (6), in the place of its more ob- 

 tuse terminal edges. Hence Pr 1 and P + n are co- 

 ordinate forms, n is = 1, and P + n = P 1. 



The horizontal prism Pr belongs to P. But at the same 

 time it also belongs to, or produces parallel edges of com- 

 bination with d, or that scalene four-sided pyramid of a 

 dissimilar section with P, whose double representative sign 

 has been expressed, either by (P + n n )' n or by (Pr 4- n n ) m . 



If we suppose the corresponding finite diagonals of the 

 horizontal prism, and the mentioned pyramid to be equal ; 

 the axes of the two forms must necessarily be also equal, 

 and since Pr belongs to P, the same applies to this funda- 

 mental pyramid ; and hence we infer that the ratio of the 

 said diagonal in the secondary pyramid to its axis is the 

 same, which takes place in the analogous lines of the fun- 

 damental form itself. 



Suppose in the pyramid, which is to be determined, the 

 ratio of the three perpendicular lines, equal to a' : b' : c'. 

 (. 53. 6.) ; we have 



a' : b' == a : b. 



The horizontal prism Pr 1 belongs to P 1, but, on 

 account of the parallel edges of combination, also to the 

 pyramid d. If we proceed in comparing the axis and the 

 diagonals as above, we find the ratio of 

 a' : cf = | a : c = a : 2. c ; 

 and therefore the ratio of all the three lines 

 a' : b' : c' = a : b : 2. c. 



If now we compare the co-efficients of this ratio with 

 the general co-efficients for (P + n) m (. 94.) ; that is to 

 say 



1:1:2, 

 with 2". m : 1 : m ; 



