14-3. OF COMBINATIONS. 157 



from the parallel and the diagonal position of the forms 

 themselves, and of their horizontal projection, which dif- 

 ferences cannot be comprehended in a single formula. A 

 single equation again is sufficient in the prismatic system, 

 where there exists no such difference. 



These equations contain every possible case in respect to 

 the kind and the position of forms within one and the same 

 system, and to certain differences in the edges of combina- 

 tion, in as much as these may be produced by faces conti- 

 guous either to the same, or to different apices ; and which 

 again belong either to that side which may be conceived to 

 be turned towards the observer, or to the opposite side 

 of the forms under consideration. These differences are 

 expressed in the equations by the addition of the signs 4- 

 and .* The possibility of thus comprehending every 

 binary combination of a system in one, or, at the utmost, 

 in two expressions, is at the same time the most convincing 

 proof of the simplicity and generality of the method. 



I shall now shortly explain the use of the Line of 

 Combination, in the developement of compound forms, 

 which is here reduced to the determination of the relations 

 among the simple forms contained in the compound one, 

 that are already known as to their kind and position. 



Let ABC, A'B'C', Fig. 50., represent the faces of the 

 same forms as in Fig. 49. ; and the points G, G', F be 

 identical with those in the same figure which are marked 

 by the same letters. Combine now a third form with these, 

 whose face is A'^'C", and whose dimensions are such as 

 to have the points G and G' common to all these three 

 forms. The points G must always keep the same place, if 

 we suppose the horizontal projection to be equal. The 

 edge of combination thus produced between the new form, 

 and any one of the others, will evidently coincide with that 



* Vide Gilbert's Annalen der Physik. 1821. 8., where 

 these formulae have been published, along with examples 

 of their application. 



