88 TERMINOLOGY. . 95. 



viz. from Pr + n under the sign (in 1) (Pr 4- n) m '. 

 It. depends here upon the value of m, whether the deriva- 

 tion proceeds from an intermediate form belonging to the 

 principal series, or to a subordinate one (. 96.). 



Let m be = 2, and the secondary pyramid therefore 

 = (P 4- n) 2 ; we have 



(m 1) (Pr 4- n)^T == (Pr 4- n) 3 ; 

 the ratio of the perpendicular lines = 2. 2* . a : 2. b : c, 

 exactly as in (P + n) 2 . Let m be = |, it will follow, that 



(m 1) (Pr 4- n) m * = 4 (Pr 4- n) 5 = (Pr 4- n I) 5 , 

 and the ratio of the lines given above = 3 . 2 n *. a : f . b 

 : c = f . 2 n . a : |. b : c, as in (P 4- n)*. 

 In both these examples, Pr 4- n belongs to the principal 



scries. But suppose m = 4 ; (m 1) (Pr 4- n) * 



will be = 3 (Pr 4- n) 3 , and the ratio of the three lines will be 

 = |. 2 n . a : 4. b : c, of which the first member still must 

 be multiplied by 3 to make it equal to that of (P 4- n) 4 , as 

 the crystallographic sign requires. Here Pr 4- n belongs 

 to that subordinate series, whose co-efficient is the num- 

 ber 3. 



In the two first of these examples, it appears that one 

 pyramid may be transformed into another, whose number 

 of derivation is greater ; in order to obtain forms in every 

 respect more analogous to the rest of those contained within 

 the compass of forms derived from the scalene four-sided 

 pyramid. The last example shews how a pyramid de- 

 rived from a member of a subordinate series, may be trans- 

 formed into another which belongs to a member of the 

 principal one. These two kinds of permutation sufficiently 

 account for the utility in the employment of the double 

 mode of deriving and of designating forms, which, for their 

 absolute dimensions, might be considered as being identical. 



The values of m most commonly found in nature are 3, 

 4 and 5, particularly the first of them ; and there is scarcely 

 a species to be found presenting forms in connexion with 



