48 TERMINOLOGY. . 56. 



cos. y - (_< 



2 [ (3 m* + 1) a" + 9] 



cos. x , __ 



2 [ (3 m 2 



^I)- + "\ . 



+ 1) a 2 + 9] / 



_ 



(3 m 2 + 1) a 2 + 9 

 cos. y = 1 + cos. x + cos. z 2. ^ [(1 + cos x) (1 cos. z)] ; 

 cos.x= 1 + cos.y+cos.z 2. A/[(! + cos.y)(l cos.z)]; 

 cos. z = (1 + cos. y + cos. x + 2. ,J [ (1 +cos.y)(l + cos.x)]). 



. 56. SCALENE EIGHT-SIDED PYRAMIDS. 



The scalene eight-sided pyramids, Fig. 12., are 

 contained under sixteen scalene triangles. 



1. The scalene eight-sided pyramids have three different 

 principal sections ; the first of these B'SC'S'BS^CS" is an 

 equilateral octagon, of alternately equal angles ; the other 

 two A'CXX'C and A'B'X/B on one side, and A'SX"S'" and 

 A'S'X'S" on the other are rhombs. 



2. The remaining sections are pyramidal and prismatic ; 

 so are likewise the axes. Of the prismatic axes, every two 

 B'B, C'C, and SS'", S'S", pass through equal solid angles. 



3. Those edges which are not terminal, are edges at the 

 base. 



4. The horizontal projection is equal and similar to the 

 base or the pyramidal principal section. 



5. Let the axis A'X' of the scalene eight-sided pyramid 

 be = m. a (where a is = AX, the axis of an isosceles four- 

 sided pyramid, the side of the horizontal projection of 

 which, SS y is = 1, and m a variable co-efficient, greater 

 than 1 + *J 2 ( 103.)) ; the acute terminal edge = y ; 

 the obtuse = x ; the edge at the base = z : we obtain 



(m + l)a 9 + 2 

 /(m* + l)a-2y 

 \m + 1) a 2 + 2/ 



