. 90. OF THE CONNEXION OF FORMS. 75 



powers of the number 2 ; the horizontal projections 

 always being supposed equal. 



There exist certain constant ratios in the homologous di- 

 mensions of any two scalene four-sided pyramids, thus de- 

 rived from each other ; for the sake of an easier comparison, 

 they can be expressed by constant ratios between their axes, 

 if referred to one and the same horizontal projection. 



Supposing the horizontal projection to be equal in all 

 the pyramids considered, designate the fundamental form 

 by A ; and by B, C, D ... the derived pyramids whose axes 

 are decreasing, by B', C',[D' ... those whose axes are increas- 

 ing : a fragment of the series, containing such members as 

 are nearest to the fundamental form, will be represented by 



...D, C, B, A, B', C', D'... 

 Let the axis of A be = a, the axis of B will be = |. a, 

 that of C = |. A. a = \. a, &c., that of B' = 2. a, that of 

 C' = 2. 2. a = 4. a, &c. ; hence the fragment of the series 

 given above, as expressed by the axes of its members : 



... |. a, . a, |. a, a, 2. a, 4. a, 8. a ... 

 and their ratio to each other = 



... | : i : : 1 : 2 : 4 : 8 ... 

 that is to say = 



... 2 3 :2 2 : 2 l : 2 : 2 1 : 2 2 : 2 3 ... 

 The general member of this series, or the expression of 

 the axis of an indeterminate n th member, will be = 2. a, 

 where a is the axis of the fundamental form, and 2 the 

 Law of Progression. The number 2 is the Fundamental 

 Number of the series. 



Upon laws of this kind is founded the method of Crys- 

 tallograpJiic Designation^ which is comprised under the fol- 

 lowing rules. The fundamental form is expressed by any 

 arbitrary letter. The same letter serves also for denoting 

 such derived forms as are of the same quality as the funda- 

 mental one, as is the case in the present instance, where the 

 derived and the fundamental scalene four-sided pyramids 

 possess similar bases ; but if the derived forms are of an- 

 other kind, the letter is transferred to these derived mem* 



