429 
of them are determined from observation the angular or linear 
elements can be determined from them. 
The linear elements have hitherto been almost universally 
used as a concise means of expressing any form. Their dis- 
advantages will be explained hereafter. 
The angular elements are in reality more concise, because 
they can express the forms they represent to any degree of 
accuracy which can be derived from observation. 
They have also this great advantage, that by the use of 
angles alone they can express the relations of any form to 
another without determining the linear elements at all. 
Thus in the following table p x for any form gives the incli- 
nation of the face for which it stands to that of the adjacent face 
of the cube in any combination of these two forms. 
Faces of all the twenty-four faced trapezohedrons lie in the 
same zone C-fid-. Hence the value of for any of these faces 
gives the inclination of that face to that of the cube in that 
zone. 
For instance (fig. 43*, Plate IV.*), m 2 is the pole of a face 
of the twenty-four-faced trapezohedron, for which the value of 
p 3 = 78° 54', X 3 =ll° 19', linear elements 1, 5, 5; Z 2 is the pole 
of another twenty-four-faced trapezohedron, where jp 3 = 76° 22', 
X 3 =14° 2', linear elements 1, 4, 4. 
For m 9 ; ^ = 15° 48'. And for l 2 y ^ = 19° 28'. 
Hence^54M4'-15M8'=Om 2 ; 54° 44'- 19° 28'= 0l 2 ; and 
19° 28'— 15° 48' = m 2 Z 2 . 
Results procured by simple subtraction when the angular 
elements are used ; but only found by retranslating the 
linear indices obtained from angular observations of the 
goniometer back again into angles, by trigonometrical 
formulae. 
Again, referring to (fig. 43*, Plate IV.*), we see that G v 
U 3 , Q s , 7T 3 , h 2 , F7 2 , f v Ap P v H x all lie in the same meridional 
zone. 
The values of y> 1 for each of these forms enable us to 
determine the distances of these poles from each other in the 
zone by simple subtraction of angles. 
2 s 
VOL. II. 
