. 151. OF COMBINATIONS. 193 



. 151. DEVELOPEMENT OF PYRAMIDAL COMBI- 

 NATIONS. 



The following developements will shew how the 

 knowledge of binary combinations relative to the 

 pyramidal system in . 149-, is to be applied in 

 particular cases. 



Fig. 07- represents a pyramidal combination, whose un- 

 determined designation is 



P + n. P + n 1 . (P + n)". P + w. [P + oo]. 

 a b c d e 



First of all, the more obtuse of the two four-sided pyra- 

 mids, or P + n (#), is supposed to be the fundamental form. 

 The value of n will therefore be = 0, and P -f n = P. 

 This determination must precede that of the vertical prisms, 

 which, however, is now very easily effected ; the prism d, 

 or that whose intersections with P are horizontal, being 

 P + co, whilst <?, the other prism, is [P -f- co], and produces 

 intersections with the faces of P, which are parallel to the 

 terminal edges of this form (. 149. i. 4.). 



The edges of combination between P and P 4- n 1 (6), are 

 parallel among themselves, but at the same time also to 

 the perpendicular lines drawn upon the faces of the former, 

 and to the terminal edges of the latter pyramid. The 

 forms are in a diagonal position to each other ; and they 

 are therefore in the relation of P + n and P + n + 1 ; and 

 since n = 0, P + n 1 will be = P + 1, as follows immediate- 

 ly from the derivation. 



The scalene eight-sided pyramid c belongs to P ; for it is 

 in a parallel position with it, and the faces of the four-sided 

 pyramid appear as rhombs in the place of the apices of the 

 eight-sided pyramid (. 149. ii. 1.). For n 11 = 0, (P + n n ) m 

 becomes = (P) m . 



P + 1 is in a diagonal position with (P) m ; its faces, how- 



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