, 95. OF THE CONNEXION OF FORMS. 87 



sines of the edges of a scalene four-sided pyramid (. 53.), 

 will produce similar expressions for the edges of the de- 

 rived pyramids, in as far as the process of derivation has 

 been applied to the intermediate form. 



The pairs of scalene four-sided pyramids thus produced, 

 according to an equal m, from all the intermediate forms, 

 which belong to the members of the series . 90., will them- 

 selves likewise form two series, which proceed according 

 to the general law of the former, and are limited by planes 

 perpendicular and parallel to the axis. The complete de- 

 signation of the two series between their limits, is 



P co ... (Pr + n) m ... (Pr + co); 



P co ... (Pr + n) ra ... (Pr + co)<". 



It has already been stated, that by these two different 

 modes of derivation, we obtain exactly the same forms. 

 This may be demonstrated, by proving how one form, whose 

 derivation from P is expressed by the sign (P + n) m , may 

 likewise be derived from the intermediate form Pr -f n', 

 under the sign (Pr + n') m/ , or that (P + n) m may be 

 = (Pr + ri') m/ . 



The ratio of the axis and the diagonals, that is to say, of 

 the three lines perpendicular to each other, is in 



(P 4- n) m = m. 2 n . a : m. b : c ; in 



!^i. 2-.a: "L+2. b : c. 

 2 m' 1 



If we suppose the co-efficients of b in the two pyramids 



to be equal, we obtain m' = ^"t-l , which being sub. 

 m 1 



stituted in the ratios for (Pr + n') m/ gives 



= m . 2'. a : m. b : c. 

 m1 



The equality of the three lines supposes therefore also 



-J!L_. 2' to be = m.2n; and consequently 2*':= (m1) 2". 

 m l 



Any scalene four-sided pyramid, which belongs to P -f n, 

 under the sign of (P 4- n) m , may therefore likewise be con. 

 sidered as deriving from the intermediate form of P + n, 



