76 TERMINOLOGY. . 90. 



bers, along with the modification or alterations thereby ren- 

 dered necessary. Under both circumstances, the place of 

 the member in the series to which it belongs, is expressed 

 by the appropriate exponents of the fundamental form, to- 

 gether with + or , their positive or negative signs. In the 

 fundamental form, the exponent is = 0, and therefore not 

 expressly indicated. 



Let P designate the fundamental form of the above- 

 mentioned series ; the fragment of that series will be 

 ... P 3, P 2, P _ 1, P, P + 1, P + 2, P + 3 ... 



An indeterminate n th member receives the designa- 

 tion P + n, the (n + l) th , P + n + 1, where the number n 

 may be either positive or negative. 



Since the ratio of the diagonals of the base b : c (. 53.) 

 is known from the dimensions of P, and remains the same 

 in all the members ; the dimensions of any required mem- 

 ber can be found, if the ratio of its axis to that of the fun- 

 damental form be known, and this ratio is indicated by 

 the designation. One of the advantages of this designation 

 consists, therefore, in affording a distinct idea of the forms 

 themselves, speaking as it were to the eye ; at the same 

 time, it expresses the connexion existing among them, in 

 as far as they are derived from each other ; and, moreover, 

 it contains every thing required for calculating the dimen- 

 sions of any member, if those of the fundamental form, or 

 of any other member, be known. 



The expressions of the cosines of the edges, as given for 

 the scalene four-sided pyramid (. 53.), refer to those of the 

 pyramid P. If, instead of a 2 , 2 2n a 2 is substituted in these 

 formulae, they are changed into the following expressions, 

 which refer to the pyramid P + n, 



22a 2 b 2 (22n a 2 + b 2 ) c 2 

 cos. y = ^ '- ; 



22.1 a 2 1,2 + (2*1 a 2 + fc2) C 2 



cos x = 2 2n a 2 c 2 (22" a 2 + c 2 ) b 2 . 

 2 *n a 2 C 2 + (2 a a 2 + c 2 ) b 2 ' 

 cos z = b 2 c 2 (b 2 4- c 2 ) 2_a 8 

 "" b 2 c 2 + (b 2 + c 2 ) 2 2n ~a2 * 

 In order to find the cosine of any of these angles for a 



