. 92. OF THE CONNEXION OF FORMS. 79 



but this form being contained under triangular faces, must 

 undergo a preliminary operation, before the process can 

 be applied. 



Let AX, Fig. 40., be the axis, BCB'C' the base of the 

 fundamental form, of which the faces BAG', C'AB', &c. are 

 contiguous to the upper, and BXC', CXB', &c. to the lower 

 apex of the pyramid. Enlarge now the planes of these 

 faces upwards and downwards, beyond the edges BC', C'B', 

 &c. ; and in these enlargements describe the triangles B A'C', 

 BX'C', &c. and C'A"B', COfB', &c. equal and similar to 

 the faces of the fundamental form. This process deter- 

 mines the situation of the points A', A", &c. X', X", &c. 

 which, being joined by straight lines, will produce rectan- 

 gular figures, similar and parallel to the base of the inter- 

 mediate form (. 90.). These rectangular figures are per- 

 pendicular to the axis of the fundamental form, which they 

 intersect in the points A and X. This mode of transform- 

 ing triangular planes in such as are rhomboidal, is the pre- 

 paration of forms mentioned above (. 81.). 



After this preparation, let the axis of the fundamental 

 form be produced on both sides to an indefinite but equal 

 length, so as to have A& = X or Ma = M ; and 

 draw straight lines from the points A, A", &c. of the lower 

 rectangle towards 21, which is the upper point, from the 

 points X', X", &c. of the upper rectangle towards , which 

 is the lower terminal point of the lengthened axis, and 

 from the angles B, C, B', C', of the base of the fundamen- 

 tal form, towards both these extremities. If planes be now 

 laid on every contiguous pair of these lines, those faces 

 which are inclined towards the upper apex, will intersect 

 those which are inclined towards the lower apex, in the 

 lines BS, SC', C'S', &c., and thus produce a form con- 

 tained under sixteen scalene triangles. 



The triangles BM9L and C'Ma are rectangular in M, 

 and the line Ma is common to both. But BM is either 

 greater or less than C'M ; therefore, also, B& will be 

 greater or less than C'JJ. Hence the two faces B2S 

 and C'3S of the derived form, contiguous to the edge 3S, 



