. 93. OF THE CONNEXION OF FORMS. 81 



(P + n)> and (P + n) 



are the crystallographic signs for the derived pair of pyra- 

 mids. 



The axis common to both these forms is = 2 n . m. a; 2". a 

 being the axis of P + n, and a the axis of P. 



. 93. THE RATIO OF THE DIAGONALS OF THE 

 BASES IS DEPENDENT ON m. . ^ 



If m be supposed equal, the bases of all (? + n) m , 

 and, on the other hand, the bases of all (P -f n) m , 

 are equal and similar to each other. 



It is evident from the preceding paragraph, that the 

 figures of the bases, or their dimensions, are determined 

 by the situation of the points S, S', S", S'", or, which is the 

 same, by the length of the lines MS, MS', &c. Draw the 

 line AA' which bisects BC', the lateral edge of the fun- 

 damental form in H, and the lines HM and A'X, perpen- 

 dicular to the axis, it will follow that 



A'X = 2. HM = V (b* + c 2 ). 



From the similarity of the triangles 2CSM, and aA'X, 

 we obtain 



aX : XA' = 3M : MS ; or 

 (m + 1) a : V (b 2 + c 2 ) = m. a : MS. 

 Therefore 



MS=_?1_ VC> 2 + c 2 > 

 m 4- 1 



From this expression it appears, that for a given ratio 

 of b and c in the base of P, the quantity of the axis a en- 

 ters for nothing in the determination of the bases of the 

 derived forms, and consequently, that the angles of these 

 depend only upon the number m. 



If, according to this process, pyramids of dissimilar bases 

 are derived from several scalene four-sided pyramids, ac- 

 cording to a determined m ; the bases of all these derived 

 pyramids will be equal and similar to each other, in as 



vor.. r. F 



