. 117. OF THE CONNEXION OF FORMS. 



above mentioned algebraic expressions, gives the cosines of 

 the edges for the members of the subordinate series. 



The position of the members of the subordinate series 

 towards each other, and towards those of the principal se- 

 ries, follows from their derivation. The limits are com- 

 mon to the principal, and to all subordinate series. 



. 117. DERIVATION OF THE ISOSCELES SIX-SIDED 

 PYRAMIDS. 



From every rhombohedron an isosceles six-sided 

 pyramid may be derived, whose axis is to the axis 

 of the rhombohedron in the ratio of 2 : 3, the ho- 

 rizontal projections of the two forms being sup- 

 posed equal.' 



The third method of derivation (. 82.) is applicable to 

 the present case. 



Let ABXC, Fig. 48., represent the principal section of 

 the given rhombohedron, and suppose a horizontal plane 

 to pass through M, the centre of its axis. In this plane 

 is situated the base of the six-sided pyramid, which is to 

 be derived. 



The terminal edge AC of the rhombohedron, being 

 produced to Z, will be changed in AZ the terminal 

 edge of the pyramid. In the same way XB is changed 

 into XH, so that, if we draw AH and XZ, AZXH will 

 be the principal section of the derived isosceles six-sided 

 pyramid, its axis being equal to that of the rhombohedron, 

 the side of its horizontal projection MZ = MH. 



Draw the lines BG and CG' perpendicularly to HZ ; 

 MG' will be = MG = PC, equal to the side of the hori. 

 zontal projection of the given rhombohedron ; and if now 

 the lines GA', G'A', GX', G'X', be drawn parallel to the 

 sides of the principal section of the pyramid, A'X' will re- 

 present the axis of this form for the side of its horizontal 

 projection MG' = PC, that is to say, equal to the horizon- 

 tal projetion of the given rhombohedrpn., 



