. 97. OF THE CONNEXION OF FO11MS. 91 



m+ is a power of the number 2 greater than 1, n is in- 



creased for the exponent of this power. Any other m pro- 

 duces one member of a subordinate series from every mem- 

 ber of the principal one. In this case is divided by 



that power of the number 2, which, in the series of these 

 powers, differs least from the mentioned number ; n is in- 

 creased for the exponent of that power, or diminished if 

 that power be negative, and the quotient thus obtained is 

 now considered as the co-efficient of the member of the 

 subordinate series. 



The expressions for the cosine of the edges referring to 

 those members of the subordinate series, are developed, as 

 has been pointed out before in several similar occasions. 



The limits of the subordinate series evidently coincide 

 with those of the principal series. 



From the value of m = 3, = 4 and = 5, the co-efficients 

 of the subordinate series are found = f and = f . These 

 and their inverse | and * have already been found in na- 

 ture ; |, for instance, in prismatic Hal-baryte, in pris- 

 matic Lime-haloide, f and in prismatic Sulphur. 



. 97. HORIZONTAL PRISMS. 



To every scalene four- sided pyramid, derived 

 from P, as well as to P kself, belong two Horizon- 

 tal Prisms, one of which refers to the long, the 

 other to the short diagonal of the base of the fun- 

 damental form. 



In any scalene four-sided pyramid, we may suppose one 

 of the diagonals of the base to be increased continually, 

 while the other remains unchanged. The value of the ter- 

 minal edges changes with the increase of the diagonal. The 

 edge which is contiguous to the unchanged diagonal ap- 

 proaches to 180. That contiguous to the increasing one 

 approaches to equality with the angle of the principal 



