ITS MODIFICATIONS. 165 



It is however extremely improbable that this pre- 

 cise relation between the dimensions of the prism and 

 the law of decrement should ever exist. The charac- 

 ter of the modifying planes, as given above, may 

 therefore be considered to exist in all the prisms 

 belonging to this class. 



Modification a would produce a series of four- 

 sided pyramids on each summit, resting on the lateral 

 edges of the prism. And if the modifying planes 

 were so enlarged as to efface the primary planes, a 

 series of octahedrons with square bases would result. 



The planes produced by this modification incline 

 equally on M and M', but at a different angle from 

 that at which they incline on P. This character of 

 the plane , will distinguish it from plane a of the 

 cube; and its equal inclination on M and M', will 

 distinguish it from plane a of the right rectangular 

 prism. 



Class b. Solid angles replaced by two planes. 



The new figures produced by this class would be 

 eight-sided pyramids, similar to those produced by 

 rf, h, i, 7c and m of the octahedron with a square base. 



MA ,M8W1 A3(J3HKr33ft. 



.M. .Vj 



Class c. Edges of summit replaced by single 

 planes not forming equal angles generally with the 

 adjacent terminal and lateral planes. 



Produces a series of four-sided pyramids resting 

 on the lateral planes of the prism ; and by an exten- 

 sion of the modifying planes, a series of octahedrons 

 with square bases would result. 



