462 



CRYSTALLOGRAPHY. 



Mathema- 

 tical 

 Theory. 



Plate 

 C''XXtV. 

 F.sr. 9. 



»e-re- 

 metfts on 



the supe- 

 rior aniHe 



place, the cosine e k disappears ; so that, in that case, 



n — 1 / 3 a- /r a 

 TiJ 2 — —o.orn — 1 = 0. Hence »! = 1* 



an — 1 "* iOff?| a i ' 



This was already evident, from what was said before. 



Let us determine the two demidiagonals of this rhom- 

 boid. Let g' and p' be these two lines ; s m (Fig. 0.) 

 being the oblique diagonal of the rhomboid, in u will be 

 the perpendicular upon the axis. Then m u =z V ■£ g'\ 



| -I 



But on the other hand, muzz. — • V^g 4 = 2V^g s . 



Hence g'z=2g. 



In this case, the line in u is elevated so as to be in 

 the direction of gn. This is a a necessary consequence 

 of s m being the oblique diagonal. Hence s u—l sr, 

 and 5 m=2 s d. So that 2 p'=2\ / g i + r>' i or ftzzX^gv+p*. 

 That is to say, that the horizontal demidiagonal g' is 

 double that of the nucleus, and that the oblique demi- 

 diagonal p' is equal to the edge of the nucleus. 



This case exists in a variety of crystals, and particu- 

 larly in that variety of calcareous spar which Hauy has 

 called equiaxe. In it gz=\/3, pzz\/2. Hence g'=*/12, 

 and p'=z\/5. From these data, it is easy to determine 

 the angles, employing the formulae above explained. 



Let us suppose -that the secondary crystal is a cube, 

 and let us enquire what, in that case, ought to be the 

 ratio between the two demidiagonals of the nucleus. 

 We may make g'— 1, p'zztl: Then substituting in the 

 equations g'=2g, p'zz'/g' 1 -f-p 1 , we obtain lz=2g, or 

 g=\. _l = v / g 1 +p'-, or l = g 1 +p*=±-£p*. Hence 

 p—^/f=\\/3. And g : p : : 1 : </3. That is to say, 

 that the nucleus is an acute rhomboid, with angles of 

 60° and 120°. This would be the case with the cube 

 of fluor spai - , if, in place of the octahedron, which is 

 the real nucleus, we were to substitute the rhomboid 

 which results from the application of two regular tetra- 

 hedrons upon the two opposite faces of the octahe- 

 dron. 



2. Decrements on the Superior Angle. 



These decrements always give rhomboids for secon- 

 dary forms. Let us continue to employ Fig 9. in which 

 a o represents the oblique diagonal of one of the faces 

 of the secondary crystal, and s o the edge contiguous to 

 that diagonal ; so that, if from the point o we draw a 

 perpendicular to the axis, it will coincide with dr, since 

 the point o ought to be situated opposite the third part 

 of the axis. The angle atz, which in the preceding 

 case supplied the place of the measuring angle, becomes 

 here the real measuring angle ; and the quantity n will 

 always signify'the number of diagonals subtracted, with 

 this difference, that we must double the number to have 

 that of the ranges subtracted. 



Let us express, in a general manner, the ratio be- 

 tween the two semidiameters g' and p' of the secondary 

 rhomboid, supposing us to know g, p, and n. 



We have, in the first place, or : ar : 

 fvg/)' 2 — 3g'-. And because the expressions for mil 

 and a u remain the same as in the case of decrements 

 on the superior edges, we will have o r : a r : : m u : 



^^F" : 2 -^nr i * / 9p'*—3g> : : v'fg 7 "* -. 



</*g"- 



3n 



((2n—iy3p*+(n + iyigi-~ (2w — l) s g 2 jg'* = 

 (n -f- l) s 12g 2 p' 2 , and developing (« -f iy4?g° — (2n — l) s g*, 

 and reducing, l(2n — l) 2 3 P * + (l2n + 3)g^g'* =s 

 (m + 1)* 1-2 g 2 p' 2 . 

 Then g':p'i-y(n+iyi2g t : v / (2«—l) 5 3p 2 -f(12n + 3)g«. 



If we suppose the decrement to take place by two 

 ranges, and that the nucleus is a rhomboid, in which 

 g =x /9, pz=«/10, we will have n= 1, and the proportion 

 becomes g' : p' : : a/14-1 : t/55. This result will be 

 found in the variety of oligiste iron ore, or iron glance, 

 called hinaire by Hauy. 



Le\ us enquire whether, among all the possible se- 

 condary rhomboids, there be one similar to that which 

 results from a decrement by one range upon the supe- 

 rior edges. 



We have seen that the oblique diagonals of this last 

 rhomboid coincided with the superior edges, such as 

 a g of the nucleus. On the other side a in is one of the 

 oblique diagonals of the first rhomboid, and since they 

 are similar, we must have g a n—m a u, and conse- 

 quence the rectangular triangles ang, auih are like- 



wise similar 

 n + 1 



n 

 2 it — 1 



Vfg' 



Therefore 

 2« — 1 



vV :: V\g- : \a. Or 



a n. Or 

 n -f 1 



3 n 



1 : i. From which we set nzz2. Hence 



the decrement will take place by 4- ranges. 



If we make 11—$, which is the case when the decre- 

 ment takes place by one range, we have (taking the 

 ratio between in u and a u), 



1 V7* :: (3 n + 3) V^p • 



^.aA);/* — 3g'*; or, removing the radical signs, and 

 reducing, we have e' 2 : 12 7/2 — 4ff* : : fn-4-lV 2 p 3 : 



(2— iHv-ir) 8 



1 aking the product of the extremes and means, and 

 transposing, we get 



11 so 3n 



(2 n — 1 ) V~a z : : f V±p : Va*. Thus the ratio be- 

 tween in u and a u in the present case becomes infinite, 

 which indicates that the diagonal a itself is infinite, 

 and of course the face upon which it falls is horizontal. 

 This case occurs in calcareous spar, in the tourmaline, in 

 sulphate of iron, Sec. In them, either a second decre- 

 ment takes place, from which there result lateral faces, 

 whose intersections limit the superior face, or there re- 

 main faces parallel to those of the nucleus. 



If we suppose now decrements in height, it is easy 

 to see that the faces resulting from them will throw 

 themselves to the opposite side of that where the decre- 

 ment takes place, so that we shall still have secondary 

 rhomboids, always less and less obtuse as the height of 

 the plates increases. Let us point out the method of 

 calculating the effects of these decrements. 



Let assd fFisc. 11.) be the principal section of the Piate 

 nucleus, and azt the measuring triangle, in winch a z 

 measures a single range, that is to say, is equal to an 

 oblique semidiameter of a molecule ; and t zis equal to 

 as many lengths- of molecules as there are ranges sub- 

 tracted in height. 



If we prolong t a above a g, the line ay will coincide 

 with the oblique diagonal of the secondary rhomboid, 

 the principal section of which will be ap.sk. 



Having continued sg till it meets ap, draw yu per- 

 pendicular to the axis a s. We must, in the first place, 

 determine the ratio between uy and a 11. 



1. For u y. The similar triangles sgm,syu give 

 sg : sy : : g m : uy. But s g=2 p ; sy—sg-\-gy. But 

 we do not know the value of gy. To find it, the similar 

 triangles a t z, ayg, give us az : t z. : : gy : a g, or 



Fi s . 11. 





