460 



CRYSTALLOGRAPHY. 



Mathema. 



tical 



Theory. 



rhomboid. The cosine of its smaller plane angle is al- 

 ways a rational quantity, provided the expressions for 

 the squares of the diagonals be rational. 



2. For the inclination of the respective faces, as, for 

 example, that of a b df or dfg s. From the point m, 

 draw m i perpendicular to df, and prolonged till it meet 

 fs. The angle a m i measures the inclination wanted : 

 a i is the sine of that angle, taking a m for radius. We 

 have only to discover the ratio between a m and i m, the 

 cosine. 



We have already a m : 



De :re- 

 merits on 

 the supe- 

 rior edges. 



FJg. 0. 



J\ 



'P 2 



PUTE 



CCXXIV. 

 Fig. 7, 8. 



g 2 +V l 

 Draw a 1c (Fig. 8.) perpendicular to g s ; it ^ill be 



equal to a i (Fig. 7-) Now, ai = = 



J if(9P~*&) _ J 3 g*P*- g' _ a ( 



V 4p 2 V p 2 



IW—g 



Therefore, am: ai: 



f 4/j 2 



And 



/ 4 p 2 / 4 p 2 



(3 p 2 — g 2 \ 



If the rhomboid is acute, the proportion will be 

 a m: i m : : 2 p 2 : p 2 — g 2 . 



Thus the expression for the cosine of the angle of in- 

 cidence is rational, as well as of the plane angle of the 

 rhomboid. 



3. For the angles of the principal section, ale (Fig. 

 7.) will be the sine of the angle g, taking a g for ra- 

 dius; and from the preceding investigation, it is evident 

 that (comparing together the sine and cosine) we have 

 ale: kg::*/3g 2 p' 2 — g*:g 2 — p 2 . If the rhomboid were 

 acute, the proportion would be V3g 2 p 2 — g* : p 2 — g 2 . 



In the rhomboid which constitutes the primitive form 

 of calcareous spar, we have g=\/ 3 and p—^/2 *. If 

 we substitute these values in place of g and p in the 

 preceding proportions, we obtain, 



1. af-.fm ::5:1 ; which gives fa »n=ll° 32' 13", 

 therefore b af is 101° 32' 13". 



2. a m : i m : : 4 : 1 ; which gives for the angle ami 

 75° 31' 20". 



3. ak:kg (Fig. 8.) ::3:1; which gives for the 

 angle ags 71° 33' 54". 



Let us now determine the results of the different 

 laws of decrement of which the rhomboid is suscep- 

 tible. There are five different kinds of decrement pos- 

 sible, which give secondary forms, namely, 



1 . A decrement on the superior edges a b, af. 



2. A decrement on the superior angle a. 



3. A decrement on the inferior edges d b, df. 



4. A decrement on the lateral angles b,f. 



5. A decrement on the inferior angle d. 



We shall consider here only the secondary forms re- 

 sulting from a decrement by one range of molecules. 



1 . Decrements on the Superior Edges. 



These decrements in general produce dodecahedrons, 

 with triangular faces, three edges of which, taken al- 

 ternately, will coincide with the edges a b, af, ag, &c. 

 of the nucleus ( Fig. 7. ) and the others will be raised 

 above the oblique diagonals ad, aq, &c. The axis of 

 the secondary crystal will be the same as the axis of the 

 nucleus. 



Let ad sg (Fig. 9.) be the principal section of the 



nucleus, a m the edge of the secondary costal which 

 rises above the diagonal a d, and which must be in the 

 plane that passes through a, d, s ; let * m be the lower 

 corresponding edge, Avhich coincides with the edge sd 

 of the primitive rhomboid. 



Let a z t be the measuring triangle, which we will 

 consider here as if the decrements took place upon the 

 angle a, observing, that to one range of molecules sub- 

 tracted towards the edges a b, af (Fig. 7.) corresponds 

 the oblique diagonal of a molecule, which measures 

 the quantity that one plate of superposition passes ano- 

 ther. 



The first point is to determine the ratio between the 

 sides a z and t z of this triangle. Let a be the length 

 of a molecule, and p' its oblique semidiameter. Cal- 

 ling n the number of diameters subtracted, we have 

 a z:tz:: 1p' y.n: a; and, because the dimensions of a 

 molecule are proportional to those of the nucleus, a z : 

 tz:: 2 tip: Vg z -j-p*. 



Let us determine likewise the ratio between m u per« 

 pendicular to the axis relative to the secondary dode- 

 cahedron, and the part a u of the axis comprehended 

 between the summit and that perpendicular. 



1. Yovmu. The similar triangles m s u, dsr, give 

 ds : d r : : s m : m u. But ds = V ' g z -f- p l and dr — 

 V'jg 2 . We only want to know s m, or rather its part 

 d m, since the rest of it is known. The similar trian-i 

 gles azt, ad m, give a z 



1 , 



::2p:dm=-</g* + p* 



Ivfathema* 



tical 



Theory. 



Plate 

 CCXXIV. 



Fig. 9. 



-/g z +p z ~: 



:"4V?+? 



t z: :ad: dm, or 2np: A /g 1 -\-p i 



Therefore * m = Vg 1 -\-p 1 + 



Hence the proportion d s 



: dr 

 n + 1 



s m : m u 



Vg*+p 



* : m uzz 



, becomes Vg l + p l : V% g l : 

 n + 1 



n 



2. For a u. 



Vf£\ 



Let us find su, and subtract it from 



We have ds :rs::sm: su, or */g 2 +pi • ^. a/o, ^2__ 3^2 

 : -^t-Vg 2 -L.p 2 ; s u — ~- </9p 2 — 3g 2 . There- 



fore a u = V9f — 3g 2 — \^Y~ ) ^9/ — 3g 2 = 

 \/Qp 2 3g 2 . Hence mu:au:: -±— \/* g 2 : 



*/9p* — 3g 2 . 



3n 



2n — 1 

 3ra 



Let b' am, f am, {Fig. 10.), be two neighbouring 

 faces situated towards the superior summit of the se- 

 condary dodecahedron, and so chosen that the edges 

 a b', af, coincide with those marked with the same 

 letters in Fig. 7; in which case, the edge a m (Fig. 10.) 

 will be that which rises above the diagonal a d (Fig. 7.) 



The semidiameters of the nucleus g and p, and the 

 number n of decrements being given, let us determine 

 the incidence of b' a m (Fig. 10.) ony a m, and that of 

 b' am on the face adjacent to it, on the other side of 

 a b'. 



Let us suppose a plane b' y f perpendicular to the 

 axis ao. Draw b'o,f'o, yo, upon this plane, and 

 f i", f p, perpendicular the one to y 0, the other to a y, 



These ratios are probably not rr.iite correct. Hence some little error in the angles found, 



Fig. 1». 





