. 114. OF THE CONNEXION OF FORMS. 117 



Draw now the line BG, parallel to the axis ; we have 

 MG = QB = 1 ; and FG = 1 MF. 



In the similar triangles FBG and F3M, we have 

 GB : GF = M3L : MF, 



: 1 _ MF 1 : MF, 

 6 2 



and consequently 



3 m + 1 



from which it appears that the angles of the transverse 

 section are dependent solely upon m, and that consequent- 

 ly they are the same in all pyramids derived according to 

 the same m, whatever may be the fundamental rhombohe- 

 dron. 



If the section does not intersect the lateral edges of the 

 pyramid, its figure is an unequiangular, but equilateral 

 hexagon. The angle at the obtuse edge is as above ; but 

 the angle at the acute edge is likewise dependent upon m. 

 For let CPF' be in the plane of the section ; the lines CP 

 and PF' will determine its figure. But CP is = 1 ; and 



PF = 3 m \ as it follows from the similarity of the 

 3 m + 1 



triangles Q3B and PF'. Now 



CP : PF = 3 m + 1 : 3 m 1, 



a ratio solely dependent upon m. Hence all scalene six- 

 sided pyramids, derived according to the same m, have 

 their sections through the terminal edges similar to each 

 other. 



. 114. SERIES OF SCALENE SIX-SIDED PYRAMIDS. 



From every member of the series . 110., several 

 scalene six-sided pyramids may be derived. The 

 axes of those which are derived according to one 

 and the same m, and consequently the pyramids 

 themselves, produce a series which proceeds agree- 



