LAWS OF DECREMENT. 357 



HEXAGONAL PRISM. 



ef\ eg, but efis half an edge of either of the equi- 

 lateral triangles b i c or h i c, whence 



ef : e g : : \ m : n : : m : 2 n y 



which thus becomes the unit of comparison for simple 

 or mixed decrements on the angles of this prism. 



The laws of intermediary decrements may be deter- 

 mined by means of spherical triangles adapted in the 

 manner already described. 



Decrements on the terminal edges. 



A decrement by I row on the edge b e, fig. 356, 

 would intercept proportional parts of the edges b d, 

 b c, and consequently if the whole of b d were inter- 

 cepted by the new plane, the whole of b c, e g', and 

 e //, would be intercepted also, and d h would be the 

 edge of the new plane d h c g. And we observe that 

 the entire of the line b , which is perpendicular to 

 d h, would also be intercepted by the same plane. 

 The ratio of b a : b c may therefore be taken as the 

 unit of comparison for determining the laws of decre- 

 ment on the terminal edges of the hexagonal prism. 



But b a is perpendicular to d i 9 the base of the 

 equilateral triangle d i b ; 



whence d b : b a :: R : sin. 60 :: 1 : Sin * 60 



R 



But . . db -.be :: 1 : - 



m 



Therefore b a : b c :: m sin. 60 : n R. 



The law of decrement on the lateral edges of the 



c5 



prism, will be represented by the units contained in 

 the ratio of the edges of the defect occasioned by such 

 decrement. 



