. 152. OF COMBINATIONS. 197 



of the corresponding terminal edges of (Pr 4- n) 3 . The 

 edges of combination are parallel to each other, and to the 

 above mentioned edges of the pyramids. For, if in the 

 general co-efficients of (Pr + n) ra (. 95.), m is supposed 



= 3, the ratio m + l 2. a : m _L c becomes equal 

 2 m 1 



to 2 n . a : c, which is identical with the ratio of the ana- 

 logous lines in P + n. Ex. P (P) and (Pr) 3 () in Serpen- 

 tine. Vol. II. Fig. 33. 



If m' be = 5, and n' = n 1, the same situation of the 

 edges takes place. A similar result is obtained by substi- 

 tuting 4 instead of m'. In this case, however, the pyra- 

 mid, from which (P -f- n') :n/ is derived, would not be one be- 

 longing to the same series as P 4- n, but it would be a py- 

 ramid, belonging to that subordinate series of which f is 

 the co-efficient. This becomes evident from a comparison 

 of the general co-efficients. The observed parallelism of 

 the edges in one direction alone is therefore insufficient for 

 the determination of the form, if there is not another da- 

 tum supplying this want from another side. 



v. P + n. (Pr -f n') m '. 



Let n' be = n ; m' = 3, or the combination P -f n. (Pr 4- n) 3 . 

 As the preceding case (iv.) refers to the more obtuse ter- 

 minal edges, so the present one applies to the more acute 

 ones ; which is evident from the comparison of the general 

 co-efficients of the forms concerned. The situation of the 

 edges being as described here, if m be supposed to assume 

 such values as do not make m' 4- 1 equal to a power of the 

 number 2 ; the pyramids P 4- n and P 4- n' will not be- 

 long to one and the same series. 



vi. P 4- n. Pr 4- n'. 



The pyramids and horizontal prisms considered here are 

 supposed to belong to one and the same series. 



Let n' be = n. The faces of the horizontal prism ap- 

 pear in the place of the acute terminal edges of the pyra- 

 mid, and the edges of combination are parallel among each 



