100 TERMINOLOGY. . 101. 



like the powers of the square root of 2 ; the hori- 

 zontal projections of these forms being always sup- 

 posed equal. 



As in . 90. the fundamental pyramid is designated by 

 P, the more obtuse members in their succession by P 1, 

 P 2, P 3, &c. the more acute members by P + 1, 

 P + 2, P + 3, &c. This designation not only rests upon 

 the same principles as that in . 90., but is in fact exactly 

 the same. Since it is necessary to know before hand 

 whether P is an isosceles or a scalene four-sided pyramid, 

 if the forms derived from it are to be taken into consider- 

 ation ; this identity of the designation can neither in this 

 nor in any other case, admit of, or give rise to, any ambigui- 

 ty. Suppose the axis of P = a ; the series of pyramids, and 

 that of their axes, will appear as follows : 

 ... P 3, P 2, P 1, P, P + 1, P + 2, P 4- 3 ... 



V2-a, 2. a, 2 J 2. a... 



2^2 2 V 2 



The ratio of the axes is 



1 



2 V2 2 V 2 



that is to say 



The axis of an undetermined n th member, or of P + n, is 



n 



= A/ 2 n . a = 2 5 . a ; and this expression is the Law of Pro- 

 gression of the series, whose fundamental number is 



V 2 = 2. 



It is evident, that subsequent members of the series are 

 in a diagonal position, alternating members in a parallel 

 position ; and since the position of P may be taken for 

 normal, all members of an even exponent will be in a 

 parallel position, those of an odd exponent in a diagonal 

 position. 



The algebraic expressions in . 52. refer to the edges of 

 P. Those of P + n are obtained by substituting 2". a 2 in 

 the place of a 2 . 



