90 TERMINOLOGY. . 96. 



mid, the base of which is similar to that of P. The same 

 result is obtained by applying at once the third method of 

 derivation (. 82.) to the intermediate form in . 92. This 

 would require to lay planes on the edges formed by the 

 intersection of the faces of ( + n) m with those of (P + n) m , 

 the inclination of these planes being such, as to make their 

 common intersection a rhoHib, similar and parallel to the 

 base of the fundamental form. 



It is now wanted to determine the ratio of the axis of 

 the derived pyramid to that of P, the horizontal projections 

 of the two pyramids being supposed equal. 



For this purpose draw the lines : AA', bisecting the edge 

 BC' in K ; MK in the plane of the base, and XA' in the 

 plane of the lower rhomb, parallel to MK. From the si- 

 milarity of the triangles AMK, AX A' follows 



XA' = 2. MK. 



Draw now the line A'a, and another K&' parallel to it ; 

 and M9L' will be half the axis of the derived pyramid, its 

 horizontal projection being reduced to BCB'C'. But from 

 the similarity of the triangles EA'X and 2TKM follows 



XA' : MK = Xa : Ma' 

 or, since 



aX = &M + MX = (m + 1) a : 



2 : 1 = (m + 1) a : 

 and consequently, 



a'M = "^1 a. 

 2 



The number __ is termed the co-efficient of the sub- 

 2 



ordinate series, and prefixed in the crystallographic sign of 

 one of its members, to the sign of the member of the 

 principal series, from which the derivation started, so that 



m ~*~ P 4- n is the designation of an indeterminate mem- 



ber of the subordinate series. 



If m 4- 1 becomes a power of the number 2, the deriva- 

 tion yields a member of the principal series itself; and if 



