. 99. OF THE CONNEXION OF FORMS. 97 



2. DERIVATIONS FROM THE ISOSCELES FOUR-SIDED 

 PYRAMID. 



. 99. DERIVATION OF HOMOGENEOUS FORMS. 



From every isosceles four-sided pyramid another 

 form of the same kind may be derived, which is 

 more obtuse, and in a diagonal position to the fun- 

 damental one. 



The derivation is effected by laying tangent planes on 

 the terminal edges AB, AC, &c. of the given pyramid 

 ABCB'C'X, Fig. 45., and enlarging them till they wholly 

 include the space AFGF'G'X (. 80.). The result of this 

 operation is at once the derived form itself, because the 

 terminal edges of the fundamental pyramid are equal to 

 each other. 



In this case, also, the process may be inverted. Draw for 

 this purpose the perpendicular lines from the apices to the 

 lateral edges upon each of the faces of the given isosceles 

 four-sided pyramid, and by planes, laid through every two 

 adjacent ones of these lines, detach those parts of the form 

 which are situated towards their outside. The remaining 

 form is the same isosceles four-sided pyramid, from which, 

 after the process described above, the given more obtuse 

 pyramid has been derived. 



The terminal edge of the given pyramid coincides with 

 the perpendicular line, drawn upon the face of the derived 

 form, from the apices towards the lateral edges, as is evi- 

 dent from the circumstance that these planes touch the 

 given pjTamid in its edges. The base of the derived 

 pyramid is the square circumscribed about the base of the 

 fundamental form ; it is inscribed, if the method has been 

 applied inversely. The two pyramids, and thus every 

 two which are in the same relation, assume such a posi- 

 tion towards each other, that the diagonals of the base of 

 the more acute pyramid are parallel to the sides of the base 



