CRYSTALLOGRAPHY. 



467 



Matliema- 



n al 

 Theory. 



CO 



Fie. 



XIV. 



1G. 



Fig;. 19. 



fig. 13. 



Hence the above proportion becomes 



"*" 2 u + 3 , 



6 k — 3 / — - 

 —7 v i "- : p z 



( 2m + 3 )(6m _3) 



4>.6« V 3«o 



Vf ^ 



Now, comparing op with o 2, we obtain 



Let gzzt/lipzZt/Si as in the tourmaline; and let 

 us suppose n—\, which indicates a decrement by two 

 ranges. We will have, on the one part, he (Fig. 16.) 

 :.ce:: a/6 : 1 ; which gives 135° 35' 4" for the inclina- 

 tion of p t o on k I o. 



On the other part, we have op. (Fig. 19,)$px,:.: 

 */27 : 1, which gives 158° 12' 4S" for the inclination of 

 o t h to r t h 



As the law of decrements varies, three of the longi- 

 tudinal edges contiguous to each summit, such as t o, 

 t r, (Fig. 1 6. ) preserve the same inclination to the axis, 

 being always parallel to the oblique diagonals of the 

 nucleus, while the three intermediate edges make great- 

 er or smaller angles with the axis, by rising up or sink- 

 ing down. There is therefore a point, when the six 

 edges, being equally inclined to the axis, become equal, 

 so that the solid assumes the form of a dodecahedron 

 composed of two right pyramids, united by their bases. 

 Let us ascertain whether this result can be produced by 

 a regular law of decrement. 



It is evident that, in this case, g n (Fig. 18.) — nl, 



or *A „■*_— T_ Vl,e- 



i«1 



This equation gives us 



n=±. That is to say, that the decrement takes place 

 by three ranges. Crystallization furnishes us with an 

 example of this decrement in the faces, which form a 

 kind of ring round the bases of a variety of corundum, 

 called ttniternaire by Flatiy, and in those faces which 

 are situated laterally in pairs in the hlnoiernary variety 

 of specular iron ore. . . 



Let us ascertain if there be a case when the dodeca- 

 hedron having its triangles, two and two, on the same 

 plane, is converted into a rhomboid. At this point, the 

 cosine p z (Fig. 18.) of the angle o z p vanishes. As- 

 suming then the analytical expression for p z, and sup- 

 pressing its denominator, we have (2 » + 3) (6 n — 3) 



V~ a' 1 g- = ; or simply, 6n — 3=0. This gives us 

 11=1, which indicates a decrement by a single range of 

 molecules. 



Let us now ascertain, in a general manner, the rela- 

 tion between the two scnikliagonals g' and p' of the 

 secondary rhomboid. 



We have on one side g n (Fig. 1 8. ) : t n : : v| g' 2 : 

 WVp': 1 — Sg' 2 '- ■ V;; 77 : Vop'~-—g%. 



On the other side, gn : in 



±fyw9p'i — 3g-, because «= | 

 g>*:8p i *—g'*:: l 4g"' 



. 2ra + 3 , 



f(9p*— 3g>)::g* 



Hence 

 12 f-*4,g*. 



Taking the products of the extremes and means, and 

 then reducing, we obtain 



From which we get this 



12p*g'*— 3g*g'*=3g*p f *. 



proportion, g' : // : : V3 g* : 



V4 p"—g": 



In calcareous spar, g — ■/?>, p-=i/2. So that in it 



g' : )/ : : \/ l 5 : ^/5. That is to say, that the horizontal 



diagonal of the secondary rhomboid;, is to the 



oblique, as the horizontal semidiagonal of the nucleus is 



to the edge of the same nucleus. 



Another property of the secondary rhomboid, which 

 we will consider here, consists in this. The plane an- 

 gles are equal to the respective inclinations of the faces 

 of the primiti/e rhomboid, and reciprocally. Farther, 

 the angles of the principal section are the same on one 

 side and the other. 



Let us resume the formulas relative to these three 

 species of angles. 



1. For the acute plane angle. 



r : Cos. : : _g°- -f p' : z±zg'=+zp*. 



2. For the smallest inclination of the faces. 



r : Cos. : : 2 p' 1 : z±zg'z+zp-. 



3. For the acute angle of the principal section. 



Mathema- 

 tical 



Theory. 



Sin. : Cos. : : Vc 



P 



But if we make g=*./3, p=\Z2, as in the primitive 

 rhomboid, and take the upper Signs in the fourth term 

 of the proportions, the first ratio becomes 5:1, the se- 

 cond 4:1, and the third 3:1. 



And if we make g=\/3, p=»/5, as in the secondary 

 rhomboid, and take the lower signs in the fourth term 

 of the proportions, the first ratio becomes 4 : 1, the se- 

 cond 5:1, and the third 3:1. So that the third an- 

 gles are the same ; and with respect to the two others- 

 they are the inverse of each other. This suggested the 

 name inverse calcareous spar, by which Hauy has de- 

 noted this variety of crystal. 



5. Decrements on. (he Inferior Angle. 



These decrements are analogous to those that take 

 place on the superior angle, both because in general 

 they produce rhomboids, and because they take place 

 both in breadth and in height. In the first case, the 

 faces produced incline towards the superior part of the 

 axis ; in the second, they incline the contrary way, or 

 towards the inferior part of the axis. We shall consi- 

 der, in the first place, the decrements in breadth. 



Let adsg (Fig. 20.) be the principal section of the 

 nucleus, p d the oblique diagonal of the secondary 

 rhomboid, and u d the inferior edge contiguous to that 

 diagonal. The measuring triangle d ho will not differ 

 from that which has been already considered in the case 

 of decrements on the inferior edges, (Fig. 12.) We 

 in the present case, d h ( Fig. 20. ):oh::2np 



Decre- 

 ments PIT 

 die inferior 

 angle. 



Plate 

 CCXX1V. 

 Fig. 20. 



havi 



Vg«==p*. The only difference will be, that the num- 

 ber of diagonals subtracted, which in the preceding 

 case was equal to the number of 'ranges subtracted, in 

 the present case, will only be'eqUal to half that number. 

 By proceeding in the same way as in the case al- 

 luded to, we shall have ap= - Vdp % — 3g*~ and 



2« + l , 



Let us now ascertain the general expression for the 

 ratio of the two semidiameters g' and p' of the secon- 

 dary rhomboid. 



Let /~ be the semiperpendicular to the axis relative 

 to this rhomboid. We shall have 



tz:p ■ 



I" 



, 2M-I-1 , ,_ 



f/9/ 4 — 3 g'\ 



