. 94). OF THE CONNEXION OF FORMS. 



c=; and 



m 4- 1 

 SN = MB' = b. 

 Therefore 



^W(b 2 + c 2 ):b = V(b 2 +c'):(m+ 1) b ; 

 m-H 1 



and 



(m + 1) b = MB 4- MB' = MB + b, 

 from which follows 



MB = m. b. 



For (P -j- n) m , therefore, will follow that 

 aM : MB : MC' = 2". m. a : m. b : c, from which, by 

 the mere permutation of the diagonals, the ratio of the 

 analogous lines of (P 4- n) m is found to be 

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



Since 



AM : BM = aM : BM ; 

 AM : CM = M ; C'M ; 



the triangle AMB is similar to aMB, and AMC' to aM', 

 and aB parallel to AB, aC' parallel to AC'. If therefore 

 from any member of the series . 90., according to whatever 

 m, a pyramid of dissimilar base with the fundamental form 

 be derived ; the terminal edges contiguous to similarly situ- 

 ated, although unequal diagonals, b and m. b or c and m. c. 

 of the two pyramids, will always be parallel to each other. 



The number m may be so great, that m. c becomes great- 

 er than b. Nevertheless m. c remains the line corres- 

 ponding to the diagonal c, which is here supposed to be the 

 short one. The correspondence between two diagonals 

 must not therefore be judged of according to their absolute 

 length, but according to their situation. This will require 

 some attention, in order to avoid being confounded by the 

 apparently different position of such pyramids. 



If the ratios obtained just now between the axis and the 

 diagonals of (? + n) m , and (P + n) m be substituted in the 

 general formulae for the edges of the scalene four-sided 

 pyramid (. 51.); the result will be other formulae, similar 

 to those in . 90., and as general ; and they will refer to sca 



