CRYSTALLOGRAPHY. 



463 



Mathenu 



tical 

 Theory. 



p : «Vg*+p : : gy : V^g*+^f\ Hence gy= — • And 



2np+p 



P 



of consequence, syzz2p -J = 



So that the proportion becomes 



2np + P it- ; 2« + l ,-r 



Lastly, gmzz 



2. For aiu au=.as — ns—v'9p z — 3 g 1 — u s. But 

 we do not know the value of u s. To find it, the tri- 

 angles sing, suy, give us sg : sy : : sm : us, or 2p : 

 2n»4-p , /-- TT 2?t+I , — 



o 11 g =z ! Yd 2 . 



\ Va? 



u s. Hence 



3 n 



Therefore we have 

 »— 1 



=^-B*>" = 



3ra 



V« ! . 



Therefore we have finally 



2h+1 ,—- n — 1 /— 



(6« + 3)^fg« 



: (2« — 2) Va *. 



Let us now determine, in a general manner, the re- 

 lation between the two demidiagonals g' and p' of the 

 secondary rhomboid. 



In tlie first place, it is evident that, in the secondary 

 rhomboid, / in is the semiperpendicular upon the axis, 

 and a m the third of that axis ; and because I m and a m 

 are proportional to u y and a u, we have 

 {6n-\-3)v'^~t '• (2n — 2)V9p" — 3g i :: V'-fg' 2 : 



$V , 9p' i —3g\ 



And, reducing and suppressing the radical signs, 

 (2« + l) 2 4g 2 : (2n— 2)? (3 p' 2 —g l ) : : g' 2 : 3// 2 — g' 2 . 



Taking the product of the extremes and means, we 



get \V2n— 2)* 3p"— { 2n— 2) 2 g 2 + ('2h + 1 ) 2 4g 2 V 2 = 



(2w -f 1 ) 2 1 2g i p'' 2 j and developing the quantities (2« — 2) 2 

 and (2n-j-l) 2 , then reducing and taking the ratio of g' 

 top', we obtain 

 g' : p' : : V / (2« + l)^3g' 2 : \/(«— 1 ) 2 3/>* -f (3«* -f 6«)g 2 - 



Let ?z=l, g=l, p = Y / 3, as in the acute rhomboid 

 of 60° and 120°, we get g' -. p' : : ,y/8 : s/3, a result si- 

 milar to that to which we would obtain by supposing a 

 decrement by two ranges in breadtli upon any two op- 

 posite angles of the cubic nucleus. This result, applied 

 to the acute rhomboid, is realised in a variety of grey 

 copper ore. 



It is remarkable, that the same rhomboids which re- 

 sult from a decrement in breadth upon the superior an- 

 gle, the faces of which are turned towards the oblique 

 diagonals of the nucleus, are still susceptible of being 

 produced in consequence of a decrement in height, such 

 that their faces correspond with the edges of the nu- 

 cleus. Let us obtain a formula by means of which, 

 the law being given relative to one of these rhomboids, 

 we may know likewise that upon which the other de- 

 pends. Let n be, as usual, the number of ranges sub- 

 tracted by the decrement in breadth, and let n' denote 

 the decrement in height. In order that the two rhom- 

 boids should be similar, it is necessary that the ratio 

 between the semiperpendicular on the axis and the 

 third of that axis be equal to each other. Therefore, 



~ Vfg" 2 = —. !jPVv-3g* :: &»'+3Vf^ : 



2n'—2\ / 9p'—3g i . Or, simplifying, n+1 : In— 1 :: 

 2«'-J-l : 2h' — 2. Taking the product of the extremes 



3 



and means, we have 2 n «'-j-4 n~4i n' — 1. Hence w= 



4«'— 1 



, and n'=z 



4«-f-l 



2n'-|-4.'"' 4— 2 n 



Let n'=.i-, asm the former case. 



1—L 



Mathema- 

 tical 

 Theory. 



rior edges. 



Phtk 

 CCXXIV. 

 Pig. 7, 12. 



We will then have 

 w=4-; a decrement which has not hitherto been ob- 

 served in the mineral kingdom. Let n=2, then »'=§ 

 or an infinite quantity. From hence we learn, that in 

 such a case, the Ime a p coincides with the line a g, that 

 is to say, that the secondary rhomboid is similar to that 

 which results from a decrement by one range on the su- 

 perior edges of the nucleus. 



3. Decrements on the Inferior Edges. 



The secondary solids produced by this kind of de- Dene- 

 crement are always dodecahedrons, with scalene tri- menu 

 angular faces, one of the sides of which coincides with the in, , e 

 one of the edges b d, df,fg, &c. (Fig. 7-) of the pri- 

 mitive rhomboid. 



Let adsg (Fig. 12.) be the principal section of this 

 rhomboid, p u the axis of the secondary dodecahedron, 

 pd,du two contiguous edges of that dodecahedron. 

 Let dhobe the measuring triangle in which h o is equal 

 to the length of a molecule, and d h to as many oblique 

 diagonals of molecules as there are ranges subtracted. 

 Let n be the number of these diagonals, p' the half 

 of a single diagonal, and g' the half of the horizontal 

 diagonal. 



We will have h o-=zv'g'' i -\-p'~ i , and d h=2np': 



Let us, in the first place, determine the part a p of 

 the axis of the secondary crystal, or the quantity that 

 this axis exceeds in length the axis of the nucleus. 



Having produced g a till it meets dp, we will have 

 the similar triangles p a I, p s d which give us ds:p s .: 

 al: a p. But, 



1. d sz= \Zg".\.p-. 



2. p s—a p-{-a s=a p-{- V9 p' — 3 g' 1 . 



3. For a 1. The similar triangles dho, da I, give us 

 dh: oh:; ad: a I; or 



7§ /2 +/>" 2 



2 np' : Vg't+p'z ::2p:a I: 



And because the dimensions of the molecules are 

 proportional to those of the nucleus, we have, by sub-. 



ff 2 -f-B 2 „ gr'-Lp'S 1 



stitutm<r 5 — — — ioi- -s — '- 1 —, a 1= — yVs-Ln'-i. 

 a p- pi* n & •* 



Hence the proportion d s : p s :: a l:ap becomes 



Vg- -j-j)- : a p -f v'9p i — 3g- : : - Vg" -j-7; 2 : a p. There* 



fore a p— i/9p ! — 3'g". 



11 — 1 ' 



Having thus obtained a p, let us determine the re- 

 spective coincidences of the faces of the dodecahedron at 

 the edges contiguous to the summit. Let a s (Fig. 13.) Fig. U 

 be the nucleus, and b p d, d pf,fp q three of the faces 

 of the dodecahedron. Draw the horizontal semidiago- 

 nal de of the rhomb dfq s, then having produced pf, 

 draw d h perpendicular to it, and join the points k, e. 

 The angle d k e wil measure half the inclination of dpf 

 to fpq. 



Draw the horizontal semidiagonal fc of the rhomb 

 abdf m&fz perpendicular to dp, and join the points 

 c, z. The angle j'z c will measure half the inclination 

 of fp d to b p d, and it is easy to see that this incidence 

 will be always greater than the first. 



Let us, in the first place, find the value of d e and 

 e k. But it is evident that d ezzg. We have only there- 

 fore to find c /:. 



