. 149. OF COMBINATIONS. 185 



ii. P + n. (P + n')"'. 



1. Let n' be = n. The forms in this case are co-ordinate, 

 and as such, in a parallel position. The faces of the four- 

 sided pyramid appear in the place of the apices of the eight- 

 sided one as rhombs, their edges of combination being 

 parallel to the terminal edges of the four-sided pyramid. 

 Ex. P (c) and (P) 3 (*), or P (c) and (P) 4 (x) in pyramidal 

 Garnet. Vol. II. Fig. 96. The situation of the edges 

 produced between the two forms, is independent of m', but 

 in the parallel position already mentioned, it equally re- 

 quires the above relation of the forms, whatever may be 

 the value of m. 



If the forms are in a diagonal position, and therefore not 

 co-ordinate ones, there exists for every sea ne eight-sided 

 pyramid a particular four-sided pyramid, whose faces 

 appear as rhombs in the place of the apices of the former. 

 Ex. P 1 (o) and (P 2) 3 (a), or P (P) and (P I) 3 (2) 

 in pyramidal Garnet. Vol. II. Fig. 96. In these combina- 

 tions the rhombic figure of the faces of the isosceles four- 

 sided pyramids depends on a certain relation of m', n' 

 and n, which is expressed in the equation : 



n n'4- 1 



m' = 2 ~ + 1. 



This equation is very useful, from two of these quantities 

 being known, to find the value of the third. 



2. Let n or n' be = co . One of the forms becomes 

 = P co, and as a face perpendicular to the axis, it pro- 

 duces horizontal edges of combination (i. 3.). 



3. Let n be = + cs. The pyramid P -f n in this case 

 appears as a rectangular four-sided prism. In either posi- 

 tion its faces assume a rhombic figure, by their intersec- 

 tion with those faces which form the lateral solid angles of 

 the pyramid. In the parallel position the angles of those 

 rhombs, and therefore the situation of the edges of combi- 

 nation are altogether dependent on m', and different in 

 different pyramids. In the diagonal position, however, 

 these angles and edges of combination do not depend on 

 m', since the angles are equal to the angles of the rhombic 



