104} TERMINOLOGY. . 104. 



than 1 + >J 2 into another (P -f n') m/ , in which m' is great- 

 er than 1 + ^/2. It is 



(P + n') m/ = (m 1) (P + n 1) -'. 

 Thus, for instance, 2 being less than 1 -f- *J 2, the sign 

 of a pyramid (P + I) 2 may be transformed into another 



2 + 1 



(2 1) (P + 1 1)2-1 _ (P) 3j where the exponent 3 

 is greater than 1 + ^2. 



The position in which the scalene eight-sided pyramid is 

 obtained from the isosceles four-sided pyramid, supposing 

 m to be greater than 1 + ^ 2, is the parallel position, that 

 which differs from it for 45, the diagonal position. In the pa- 

 rallel position, a plane through the axis and the acute termi- 

 nal edge of the eight-sided pyramid, passes at the same time 

 through the acute terminal edge of another eight-sided py- 

 ramid, or through the perpendicular line from the apex, 

 upon the face of an isosceles four-sided one, whilst in the 

 diagonal position, the same plane passes through the obtuse 

 terminal edge of the other eight-sided, or through the ter- 

 minal edge of the four-sided pyramid. 



Pyramids expressed by the sign (P + n) 3 are frequently 

 met with in nature, as in pyramidal Garnet, in pyramidal 

 Zircon, c. ; those dependent upon other values of m oc- 

 cur more sparingly, as (P + n) 4 in pyramidal Garnet, and 

 (P + n) 6 in pyramidal Tin-ore. 



. 104. THE BASES OF THE SCALENE FOUR-SIDED 

 PYRAMIDS DEPEND UPON m. 



For one and the same m, the bases of all forms 

 contained under the sign of (P -f- n) ni 3 are equal 

 and similar to each other. 



The demonstration of this proposition follows from . 93. 

 The lines denoted in that paragraph by b and c, obtain 

 here the determined value = *J 2 ; and hence we have 



, ro 2. m 

 MS = . 



m+ 1 



