. 149. OF COMBINATIONS. 187 



2 2 = |. 2 2 =|. 2>. 2 2 = f. 2 ; ., 



and P 4- n = | P 4- n' 4- 2, or that member of the second 

 subordinate series (. 107-), which belongs to P 4- n' 4- 2. 

 For m' 5, ths result is 



n ' * s 



= 3. 25 = _J_. 2*. 



and P + n = -^j- P 4- n' + 3, that member of the first 



subordinate seiics (. 107-), which belongs to P + n' + 3. 



5. Let n' be = n 3, m' == 4. The combination is 

 P + n. (P + n 3) 4 ; under these circumstances, the 

 forms will be in a diagonal position. The faces of the 

 four-sided pyramids appear in the more obtuse terminal 

 edges of the eight-sided ones. The edges of combination 

 are parallel to these, to the perpendicular lines upon the 

 faces of the four-sided pyramid, and among each other. 



Suppose, Fig. 70., MA' to be half the axis of the eight- 

 sided pyramid, and A'B its more obtuse terminal edge. If 

 now BM is half the side of the horizontal projection, we 

 have in MA' half the axis, and in A'B the perpendicular line 

 upon the face of the four-sided pyramid ; and consequent- 

 ly, if half the side of the horizontal projection MD is sup- 

 posed = , MA will be 



n *'-* 



= 2 2 . a = m'. 2 * . a. 

 But we have m' = 4 ; therefore 



n = n' 4- 3, and n' = n 3. 

 If m' be = 3, it will follow that 



n' 1 n'l n' + 2 



2* = 3. 2~ir = _JL_. 2 l. 2 -ir _ _JL_. 2 ; 



and thus P 4- n becomes z /a P 4- n' 4- 2, or that mem- 

 ber of the first subordinate series, which belongs to 

 P 4- n' 4- 2. 



m' = 5 makes 



n'-T n'-l n'4-3 



2 5 = 5. 2 2 = |. 2 2 . 2 2 = |. 2 2 . 

 P 4- n therefore becomes = | P 4- n' 4- 3, which is that 



