186 TERMINOLOGY. . 149. 



principal section, and the edges of combination consequent- 

 ly parallel to the alternating terminal edges of that isosce- 

 les four-sided pyramid, from which the scalene eight-sided 

 pyramid is derived, or to which it belongs. Examples of 

 both cases are contained in pyramidal Zircon, Vol. II. 

 Fig. 99. The scalene eight-sided pyramids (P) 3 Or), (P) 4 (?/) 

 and (P) 5 (~) are in a parallel position with P + cc (/), but 

 they are in a diagonal position with [P 4- oo] (A). 



If, therefore, the edges of combination produced by a 

 scalene eight-sided pyramid and an isosceles four-sided one, 

 in a parallel position, are found to be parallel to those 

 edges which are produced by the same eight-sided pyramid, 

 and a rectangular four-sided prism in a diagonal position ; 

 it follows that the isosceles four-sided pyramid and the 

 eight-sided pyramid must be co-ordinate forms. 



4. Let n' be = n 2, and m' = 3. The combination 

 will be P + n. (P -f n 2) 3 , and the forms in a pa- 

 rallel position. The faces of the four-sided pyramid ap- 

 pear in the place of the more acute terminal edges of the 

 eight-sided pyramid ; the edges of combination being pa- 

 rallel among themselves, to the above-mentioned terminal 

 edges of the eight-sided pyramid, and to those lines in the 

 four-sided pyramid, which, from its apices, can be drawn 

 perpendicularly to its lateral edges. Examples occur in 

 pyramidal Zircon, of (P) 3 and P 4- 2. 



Suppose A'M, Fig. 69., to be half the axis, A'C the 

 more acute terminal edge of the eight-sided pyramid. If 

 CM be made equal to half the side of the horizontal pro- 

 jection, MA 7 becomes half the axis, A'C the above-mention- 

 ed peiy.endicular Hue upon the face of the four-sided pyra- 

 mid ; and consequently, supposing MD, half the side of 

 the horizontal projection = |, we have 



n / . i n' 



MA = 2'. a = m +- 2*. a. 



2 



If, according to the supposition, m' be = 3, the values 

 of n and n' will follow thus : 



n = n' + 2, and n' = n 2. 

 But if we substitute 4, instead of m', we obtain 



