igo 



NATURE 



IJan. 7, 1873 



point are perpendicular, and one of these bisects the 

 angle formed by the remaining two lines, the sines of the 

 angles taken in the proper order are in the harmonic 

 ratio. Another point illustrated was that a sheaf of four 

 lines presents the same anharmonic ratios of their sines as 

 does a sheaf of four lines severally perpendicular to them. 

 Reverting to the subject of the traces of the faces of a 

 zone on their own zone plane, it was now seen that we 

 can discuss the subject of relations of any four planes in 

 the zone by considering those of their nonnals the 

 angles between which are measured on a great circle of 

 the sphere. But it remains to obtain an expression that 

 shall connect these angles with the symbols of the poles 

 or faces of the zone. Such an expression obtained by 

 Prof. Miller in the first case involves a relation of the 

 simplest kind. In short, the anharmonic ratio of four 

 planes is the ratio which we obt.ain directly from the 

 determinants of the symbols for the four planes. Since, 

 however, the symbols for a zone as obtained from the 

 symbols of different pairs of faces of the zone may, and 

 generally do, differ by a common factor, it is advisable to 

 put the expression for the anharmonic ratios of four 

 tantozonal planes under the form of a convenient symbol 

 given them by V. von Lang, viz., for the four planes 

 PQRS:— 



\PSr\ . \P_S~\ = sin P Q . smPS ^ m 

 V.QRy'\~SKA sm{PR-PQ)' sm{PR-PS) n 

 where the letters on the left side of the expression stand 

 for the symbols of the planes of which the determinants 

 are to be taken. This very important expression offers 

 the means of determining one unknown symbol or one 

 unknown angle among those belonging to the four planes ; 

 another result that flows from it is the necessity for the 

 anharmonic ratios of four planes in the zone, i.e. the 

 magnitudes m and ;/, being always rational if the planes 

 belong to a ci-ystal. And this is another and more 

 general way of stating the fundamental crystallographic 

 law, that of the rationality of indices. 



Prof. IVlaskelyne next proceeded to discuss some of the 

 further results deducible from this great law. Firstly, 

 since the harmonic ratio of four planes brings those 

 planes under the requisite condition of rationality, we can 

 say of any zone in which two of the planes are perpen- 

 dicular to each other, that for any third plane of the zone 

 inchned on one of them at an angle (^, a fourth plane may 

 also exist as a possible plane of the zone, also inclined on 

 the first plane at the angle <^ ; and further, the professor 

 went on to state that if we ask the question what are the 

 conditions for three consecutive planes in a crystal zone to 

 include the same angle <^, we find for answer that only in 

 those cases is this possible where cos. ^ is rational, and 

 that this is only so where (/> possesses one of the values 

 90°, 60°, 45°, and 30°. 



After a review of the results thus' far obtained, the pro- 

 fessor entered upon the subject of symmetry, and de- 

 fining the different varieties of geometrical symmetry ; 

 such as, firstly, the symmetry of a plane figure to a centre 

 of symmetry, to one or to several lines of symmetry, or to 

 a pivot of symmetry ; and secondly, that of a solid figure 

 to a centre of symmetry, to one or to several planes of 

 symmetry, and to one or to several axes of symmetry : 

 he defined certain terms which would be found useful 

 in the discussion of the symmetry of crystals. Thus, 

 a plane figure was cntliy-symmetrically divided by a 

 single line of symmetry or ortho-symnictrically dW\it£.6. by 

 two lines of symmetry perpendicular to each other ; while 

 an axis of, for instance, hexagonal symmetry became one 

 of di-hexagonal symmetry, where each repeated element 

 of form is itself doubled, as by reflection, on a plane of 

 symmetry. 



In applying the principles of geometrical symmetry to 

 crystals, it was shown that the best and simplest method 

 was that of dealing with the distribution of their poles on 

 the sphere of projccticn. 



The condition requisite lor a single plane of symmetry 

 to exist upon a crystal was then shown to bo that this 

 plane should be at once a zone plane and a possible face 

 of the crystal. On the other hand, for a crystal to be 

 symmetrical to a centre, no particular condition was re- 

 quisite, since the direction and not the requisite position 

 of a crystal plane has been seen to be the important point 

 regarding it, while again every plane passing through the 

 origin may be represented by the symbol of either of its 

 poles indifferently. Now, an axial system as previously 

 defined involves five variable quantities ; namely, the 

 three angles between the axes : 



^, the angle 1 ' Z 



t), the angle 2: A' 



f, the angle X 1 ' 

 and the two ratios involved in the parameters, namely, 



- and -. 



/> b 



Hence, for a crystal to be centro-symmetrical, all these 

 five quantities may vary from one substance to another. 

 If, however, the crystal system be divided symmetrically 

 by a plane, two of these axial elements are absorbed in 

 satisfying the two requisite conditions of that plane being 

 at once a cr)'stal face and a zone-plane. 



A crystal system that is simply centro-symmetrical pre- 

 sents the kind of symmetry characteristic of what is called 

 the Anorthic system of crystallography ; a crystal that 

 obeys the principle of symmetry to a single plane belongs 

 to the Oblique or Clinorhombic system. 

 ( To be continued^ 



TWO REMARKABLE STONE IMPLEMENTS 

 FROM THE UNITED STATES 



THE similarity of stone implements, both modern 

 and prehistoric, that obtains throughout the world, 

 has been commented upon so frequently as scarcely to 

 need further illustration. Within a few days, however, 

 I have found two forms of arrow and javelin points that 

 are so unusual in their shapes, and otherwise of interest, 



Fic. I.— (Natural size.) 



that I believe drawings of the two, and a brief note con- 

 cerning them, will be welcomed by arch.-cologists. 



Fig. I represents a "flame-shaped" arrow-point, as 

 this shape has been well called by Mr. E. B. Tylor 

 {I'idc "Anahuac," by E. B. Tylor, p. 96, Fig. i). Although 

 I have collected fully ten thousand specimens of " Indian 

 relics" from the immediate neighbourhood of Trenton, 

 New Jersey, U.S.A., of which a very large proportion 

 were spear and arrow heads, I have not been able before 

 to duplicate this form, or to find any unmistakable trace 

 of it in the bushels of fragments that here cover the 

 ground in some places. This aiTow-head, accompanied 

 by the javehn (Fig. 2) and several of the leaf-shaped 



