124 TERMINOLOGY. . 118. 



The two triangles APC, A'MG' are equal and similar to 

 each other. Therefore A'M = AP ; which if the axis of 

 the pyramid be called a', may be expressed thus : 



k- a' = *- a, 

 and consequently 



a' = |. a. 



The above-mentioned constant co-efficient (. 54. 4.) ac- 

 cordingly is = f . 



. 118. SERIES OF ISOSCELES SIX-SIDED PYRAMIDS. 



There is a series of isosceles six-sided pyramids 

 in connexion with the principal series of rhombo- 

 hedrons, with which it proceeds after the same law, 

 and is limited by infinite six-sided prisms. 



The axes of the members producing this series, are equal 

 to the axes of the rhombohedrons, multiplied by , the ho- 

 rizontal projections being always supposed equal ; so that, 

 if P + n denotes an undetermined n th member of the se- 

 ries, its axis will be = 2 n . a. In the expression for the 

 axis of any member, the common factor . a is contained ; 

 and by dividing with it, we find that the axes of the series 

 increase and decrease like the powers of the number 2 ; 

 and 2 n consequently expresses also in this series the law of 

 progression. 



The limits of this series are isosceles six-sided pyramids 

 belonging to rhombohedrons, whose axes are on one side 

 infinitely long, on the other infinitely short. It is evident 

 that an isosceles six-sided pyramid of an infinitely long axis 

 can be nothing else but a regular six-sided prism, whose 

 transverse section is equal and similar to the horizontal 

 projection of the pyramid ; and that an isosceles six-sided 

 pyramid of an infinitely short axis can be nothing else but 

 a plane figure perpendicular to the axis, and equal and si- 

 milar to the same horizontal projection. 



The regular six-sided prism, which limits the series of 

 isosceles six-sided pyramids, can be distinguished by its 



