OP ARTS AND SCIENCES. 87 



rliorab, as a constant result, 60'^. The supplementary angles, 120*^, are 

 usually truncated ; but not unfretpiently (as shown on Plate III., Fig. 2) 

 we meet with perfect terminations of this kind. When these occur 

 on the same sprays as the acute angles, they uniformly appear in their 

 proper relative positions, at right angles to the supplementary termina- 

 tions, and we have frequently seen both terminations on the same 

 rhombic plate. Less frequently, we find the sixty-degree angles 

 truncated ; and, as the result of such truncation, we have observed 

 isolated hexagonal plates as perfect as those of the hexagonal iodide, 

 from which outwardly they only differed in their color. 



We have represented in Plate III., Fig. 1, a spray, presenting, for the 

 most part, quite a different set of terminations from those before fig- 

 ured, which, although they are not quite so well defined as the others, 

 nevertheless measure very constantly 82*^ and SS*^. When found on 

 the same sprays, the 98* terminations are, in general, parallel to the 

 60°, and the 82^^ to the 120°. Moreover, the 98° terminations are 

 very frequently found bevelling the constantly recurring terminations 

 of 60°, and, on the other hand, as often, the terminations of 120° bevel 

 those of 82°. 



Sometimes one of the two bevelling planes disappears, or, at least, is 

 reduced to such small dimensions as to be imperceptible on the sec- 

 tion as shown under the microscope. This kind of hemihedrism is shown, 

 in the largest termination, both on Plates II. and III. 



The angles 98° and 82°, or more accurately 98° 12' 50" and 81° 

 47'' 10", are the angles of the rhombic prism {120}, derived from the 

 prism {110} of 120° and 60°. The relation of these two prisms is 

 shown by the figure below the drawing, on Plate II. ; and it will be 

 noticed that all the termination edges in the drawing are parallel to 

 one or the other of the lines wliose relative positions are thus defined. 

 It is a very remarkable property of the prism of 120°, that the derived 

 prisms {120} and {230} have identically the same angles; the only 

 difference being that the relative positions of these angles (98° 13' 

 and 81° 47') are reversed. This is shown in the same figure as 

 before, where the pi'ism {230} is also represented, but by dotted lines. 

 The property referred to depends on the circumstance, that one-half 



I- 9 , 



and two-thirds of the tangent of 60° (^ and ~j^) are reciprocals, 



and must therefore be the tangent and cotangent of the same angle, 

 which is 40° 53' 35", or one-half of the prismatic angle 81° 47' 10" 

 named above. The same values are also equal, respectively, to the sine 

 'and cosecant of 60°, or to the cosine and secant of 30°. Not only. 



