446 



CRYSTALLOGRAPHY. 



Theory. (Fig. 3.) be the face of the rhomboid (Fig. 1.) mark- 

 ed with the same letters, subdivided into the bases of 

 the molecules of which it is composed. If we suppose 

 that the angle g" undergoes a decrement by a single 

 row of molecules, the small rhomboid represented by 

 o n 2»" will be wanting; hence, it is obvious, that the 

 edge of that plate will have the direction o z, and that 

 the distance between the angle g", from which the de- 

 crement sets out, and the edge o z, will be measured by 

 the semidiagonal r g" of a molecule. If the decrement 

 takes place by two ranges, the edge of the first plate 

 of superposition will correspond with c d, and the dis- 

 tance between it and the angle g" will be measured by 

 the diagonal g" n of a molecule. From this we may 

 conclude, that, in general, in decrements on the angles, 

 the distance between one plate and the succeeding one, 

 which is the same with that of the angle from which 

 the decrements began, and the first plate of superposi- 

 tion, is equivalent to as many semidiagonals of a mole- 

 cule as there are ranges taken away ; while, in the case 

 of decrements on the edges, the distance between two 

 successive plates is equivalent to a number of diagonals 

 equal to that of the molecules taken away. 



This being understood, let us suppose a decrement 

 of two rows upon the angle g". In that case, the qua- 

 drilateral neap (Fig. 2.) being a section made on the 

 first plate of superposition, the edge of that plate in 

 which the decrement takes place will coincide with en, 

 since g"n is the same diagonal as in Fig. 3. There- 

 fore, if we draw the straight line g"eh, it will coincide 

 with the face produced by the decrement. But in this 

 case g"h is parallel to the axis S s, as may be easily de- 

 monstrated by the assistance of mathematics. Hence 

 it follows, that the secondary faces constitute the faces 

 of a prism. 



If the decrements went on more rapidly, if they 

 took place, for example, by four ranges, in which case 

 the edge of the first plate of superposition will coin- 

 cide with the line y g, then the line g"qS' indicates the 

 position of the secondary faces. We see that they rise 

 above the nucleus, and form the surface of a rhomboid 

 more acute than this nucleus. If, on the other hand, 

 the decrements took place in height, then the line ug"s', 

 which we suppose to indicate the secondary faces pro- 

 duced, would incline towards the inferior portion of the 

 axis. Hence, it is obvious, that the faces of the se- 

 condary crystal (still a rhomboid) would incline in the 

 opposite direction of the primary faces. 



The hypothesis of a decrement by two ranges in 

 height, gives, in this case, a remarkable result; the 

 secondary crystal is precisely similar to the primitive. 

 Hauy has made it probable that such secondary crys- 

 tals exist both in quartz and tourmaline. 



Let us pass to the superior angle S, and let us sup- 

 pose at first a single range of molecules taken away. 

 If from t, the centre of the equal diagonal Sp, we 

 draw t x parallel and oblique to p a, this line will coin- 

 cide with the edge of the first plate of superposition, 

 since the distance between the angle S and the edge is 

 equal to a semidiagonal of a molecule. Hence the line 

 Sx h will coincide with the secondary face produced by 

 the decrements, which is obviously perpendicular to 

 the axis. 



A more rapid decrement, as by two ranges in breadth, 

 would produce faces inclined as the line S ai is ; that 

 is to say, that the secondary crystal would be a rhom- 

 boid, inclined as the nucleus, and more obtuse. If the 

 decrement takes place in height, then the secondary 



faces produced, one of which corresponds with the line Theory. 

 KS m, will incline to the other side of the axis ; hence '-' ■ y - .J 

 the secondary rhomboid will have a position the reverse 

 of the primary. 



These observations and examples we conceive suffi- 

 cient to make the nature of the decrements on the 

 angles obvious to every reader who takes the trouble to 

 consider the subject. But there is still other kinds of 

 decrement which remain to be explained. 



3. Mixed Decrements. 



This name is applied to those decrements in which Mixed de- 

 the number of ranges taken away in breadth and height elements, 

 give ratios, the two terms of which surpass* unity. As, 

 for example, decrements by two ranges of molecules 

 in breadth, and three in height, or by three ranges in 

 breadth and two in height, &c. It is easy to see that 

 the theory may be with facility reduced to that of 

 decrements, in which there is only one row of mole- 

 cules taken away in one of the two directions. 



4. Intermediate Decrements. 



We have seen, that in the case of a decrement by Interne- 

 one row of molecules round the same solid angle, the diate de- 

 three faces produced are always in the same plain, and crement5 - 

 that, in that case, it is only necessary to consider the 

 effect of the decrement with respect to one of the plain 

 angles which concur to the formation of the solid angle, 

 conceiving this effect to be prolonged over the neigh- 

 bouring faces. In that case, the decrements on these 

 last faces are considered as subsidiary, to favour the 

 action of the principal decrement. 



In general, whenever the solid angle of a primitive 

 crystal undergoes decrements which tend to produce a 

 face in its place, whatever the law may be to which we 

 reduce the production of that face, there are always 

 auxiliary decrements, the concurrence of which is ne- 

 cessary, in order that the new face may be of the requi- 

 site magnitude. Now, when the decrement which we 

 consider in preference takes place, by two ranges of 

 molecules, or by a greater number, the auxiliary de- 

 crements in continuity with it follow a peculiar law, 

 which it is necessary to explain. 



Let A A' (Fig. 4.) be a parallelopiped of any kind Piate 

 which undergoes a decrement by two ranges on the CCXXNI, 

 angle EOI of its base AEOI. It is obvious that the F 'S- 4 - 

 edges of the plates of superposition will have the di- 

 rections be, r s, parallel to the diagonal EI, and so 

 situated that there will be upon the sides OE, OI 

 two rows of molecules comprehended between the 

 angle O and the line b c, and likewise between b c 

 and rs. But, as has been already said, the plates 

 applied upon the adjacent faces IOA'K, EOA'H un- 

 dergo likewise auxiliary decrements, which continue 

 the effect of the decrement upon the angle EOI. But 

 such, in this case, are the effects of these decrements, 

 that the edges of the plates applied upon IOA'K have 

 the directions eg, st ; and those of the plates applied 

 upon EOA'H the directions b g, rt. For since the 

 lower edge of the first plate applied upon AEOIcoincides 

 with be, and the height of this plate corresponds to 

 that of a single molecule, a little attention will satisfy 

 us that the plane beg, which on one part coincides like- 

 wise with be, and on the other separates from the base 

 AEOI, by a quantity measured by Og the height of a 

 single molecule, is necessarily parallel to the face pro- 

 duced by the decrement. The same holds with the 

 plane rts. From this it follows, that if we suppress 



