. 110. OF THE CONNEXION OF FORMS. Hi 



AX : A'X = C'X : BX = 2 : 1. 



Hence the axis of the derived rhombohedron is equal to 

 half the axis of the fundamental one, if the horizontal pro- 

 jections are equal. 



In order to find the ratio between the axis of the more 

 acute rhombohedron and that of the fundamental form, let 

 AB'XC' be the principal section of the latter, the side of 

 the horizontal projection being C'P. In this case ACXB 

 will be the principal section of the derived rhombohedron, 

 the side of its horizontal projection = BQ, and its axis equal 

 to that of the fundamental rhombohedron. 



From the point C' draw the line C'B'" parallel to the 

 axis, produce AB to B'", and complete the rhomboid 

 AB'X'B"'. This is the principal section of the derived 

 rhombohedron, the side of the horizontal projection being 

 B'"Q' = C'P. 



The similar triangles ACX and AB'X' give 

 AX : AX' = AC : AB' =1:2. 



The axis of the more acute rhombohedron is conse- 

 quently double the axis of the fundamental form, their ho- 

 rizontal projections being equal. 



. 1 10. SERIES OF RHOMBOHEDRONS. 



By continuing the derivation, a series of rhom- 

 bohedrons is obtained, whose axes increase and de- 

 crease as the powers of the number 2 ; their hori- 

 zontal projections being always supposed to be 

 equal. 



This series corresponds exactly to the series of scalene 

 four-sided pyramids (. 90.), in respect to the ratio be- 

 tween the axes of its members, in as much as it depends 

 upon the same fundamental number. 



Designate the fundamental form by R ; R 4- n will be 

 the general expression of an indeterminate member of the 

 series. 



Fig. 44. will assist us in giving a clearer idea of this se- 



