86 TERMINOLOGY. . 95. 



2". a : b : c, 



the ratio of the analogous lines in the same succession 

 will be 



for (Pr + n) m = m + 1 . 2 n . a : b : m _l. c : 



2 m 1 



for (Pr + n> = m "t_L 2". a :^J. b : c. 



2 m 1 



Let S'B'S, Fig- 39., be half of the base of the derived 

 pyramid (Pr + n) m ; in the ratio of the three lines 

 3M : MB' : MS 



9LM and MS are already Imown, that is to say, expressed 

 by a, c, and m ; and the only expression still wanting, is 

 that of MB' by means of a function of m and b. 



The triangle CIS is similar to the triangle MB'S; 

 therefore 



CS : C'I = MS : MB'. 

 But we hare 



C'S = MS MC = ?_ m _. c c = m _TL_ 1 . c. 



m + 1 m+1 



Therefore 



2m . 



: b = -. e 



m -f 1 m + 1 m 1 



and 



Since this is the required expression, it follows that 

 : M1B' : MS = 2" m. a : 2 m .b : 2 m - 



m 1 m+1 



or if the co-efficient of that diagonal to which the pyramid 

 belongs, is supposed = 1 , the same ratio will be expressed 



by 



= ?1. 2-a: ^Ltib : c. 

 2 m 1 



By exchanging the diagonals b and c, the ratio of the 

 three lines of (Pr 4- n) m will be obtained 



= m+1 - 2- a : b : m + *- c. 

 2 m 1 



The co-efficients of the axis and the two diagonals, as de- 

 veloped here, if substituted in the expressions for the co- 



