. 98. OF THE CONNEXION OF FORMS. 95 



The method of derivation, applied to the scalene four- 

 sided pyramid, is so general, that it will remain unaltered, 

 and yield similar forms and relations, even though the axis 

 of the fundamental form should be inclined at an angle to 

 the plane of the base. 



This Inclination of the Axis may take place, either in the 

 plane of only one of the diagonals of the rhombic base, 

 as in Fig. 41., or in the planes of both diagonals, as in 

 Fig. 42. In the first two principal sections, ACA'C' and 

 BCB'C' are rhombs, and one ABA'B' is a rhomboid; 

 in the second, only BCB'C' is a rhomb; and both the 

 others, ACA'C' and ABA'B' are rhomboids. A third case 

 is still possible, where all the three principal sections would 

 yield rhomboidal figures, upon which supposition, how- 

 ever, the diagonals CC' and BB' themselves intersect each 

 other at oblique angles in the point M. From the want 

 of sufficiently accurate observations, it is at present impos- 

 sible to decide which of the two last cases, or whether 

 perhaps both of them take place in nature, while the 

 fact of an inclination of the axis in the plane of one of 

 the diagonals has already been established by numerous ob- 

 servations. According to the principles laid down in . 87. 

 as respects the determination of fundamental forms, it will 

 be impossible to limit the number of these to four, because 

 forms whose axis is inclined, cannot be derived from others 

 whose axis is perpendicular to the base by any of the given 

 processes of derivation. Without entering here into the 

 full developement of the theory of these forms, and without 

 drawing all the necessary consequences from this import- 

 ant fact, it may be useful to mention some of the alge- 

 braic formulae, dependent upon the inclination of the axis 

 in the plane of one of the diagonals. 



Let a : b : c : d denote the ratio between the four 

 lines AP, BM, CM, and MP, in Fig. 41., the following 

 formulae will be obtained : 



cos. y = 



a 2 (b 2 + c 2 ) + c 2 (b + d) 



cos. y' a2 ( b2 c 2 )-c 2 (b 



a 2 (b 2 + c 2 ) + c 2 (b_d)- 



