If the formulæ obtained by these analyses may be considered as expressing the real constitution of this mineral, it would appear that it consists of a mixture of neutral silicate of alumina with a variable amount of neutral hydrate; the former mineral is not, I believe, known in an isolated state, but the latter as Gibbsite frequently occurs: were we acquainted with the former also, we might imagine allophane to be the transition, or a series of transition stages between it and Gibbsite; this is well shown by the collected formulæ :? {A1203. 3SiO3.

What relation the specimens formerly examined by various analysts bear to those now described, cannot, unfortunately, be ascertained, since in one case only can I find a sample which had been dried at 100° C., and that still contains an amount of water quite incompatible with any formula analogous to those preceding.

I will subjoin the analyses containing the maximum and mìnimum amounts of water which have been hitherto obtained*, for the sake of comparison with my per-centage numbers :

It is scarcely conceivable that this very great loss of water should be peculiar to the Charlton allophane; but it yet seems singular that no chemist but Guillemin should have made the observation that such a loss occurred, and that in his case it should have been comparatively so small: obviously in these specimens, at least, the water in the native mineral exists in two very different states of combination; whether the entire amount is necessary or superfluous to its constitution as allophane has yet to be shown. My present object is to draw attention to the fact, that, regarding the water expelled at 100° C. as uncombined, or simply adherent, formulæ may be assigned to the mineral which do not militate against the generally received views of the constitution of saline compounds. This, however, cannot be urged in defence of such formulæ as have hitherto been given, those of Dana for instance,

A12 02, SiO3+5HO, the dried specimen of Kersten, are scarcely in accordance with existing ideas, and certainly do not tend to elucidate the confessedly obscure question of the constitution of the silicates.

XLVIII. On the Application of Elementary Geometry to Crystallography. By W. H. MILLER, M.A., F.R.S., Professor of Mineralogy in the University of Cambridge*.

1. THE instruments of analysis hitherto employed in investigating the general geometrical properties of crystals, have usually been either analytical geometry of three dimensions, or spherical trigonometry. Lately, however, Professor Sella of Turin has deduced the symbol of a zone containing two known faces, the symbol of a face common to two known zones, and

* Communicated by the Author.

the equation of condition that a face may belong to a zone, and has shown that the faces of a crystal may be referred to the axes of any three zones as crystallographic axes, with the aid of elementary geometry only (Nuovo Cimento, vol. iv.). The following investigation shows, that not merely the propositions established by Professor Sella, in which it has not been considered requisite to adhere closely to the steps of his demonstrations, but all the more important geometrical properties of crystals admit of being easily and concisely proved by the methods of ordinary elementary geometry. The relation between the segments formed by the mutual intersection of four straight lines is frequently used. Though well known, a proof of it is given in order to save the trouble of reference.

2. Let A, B, C, D, E, F be the points of intersection of four straight lines, as shown in the annexed figures. Let AH, parallel to BC, meet DF in H. Then

AF.BD=FB. AH, and CE. AH=DC.EA.

Therefore AF.BD.CE=FB. DC. EA.

=

b

a

3. Let XOX', YOY', ZOZ' be any three straight lines given in position, all passing through a given point O; a, b, c any three lines given in magnitude; h, k,l any three positive or negative whole numbers, one or two of which may be zero. Take OH= measured along OX or OX', according as h is positive or negative; OK: along OY or OY', according as k is positive or negative; OL= along OZ or OZ', according as l is positive or negative. Let the symbol hkl denote any plane parallel to HKL, including HKL itself, on the same side of the point O. It is now proposed to investigate some of the properties of the system of planes obtained by giving to h, k, l different numerical values. 4. Let the point O be called the origin of the system of planes;

k

the straight lines XOX', YOY', ZOZ' its axes; the three lines a, b, c, or any three lines in the same ratio, its parameters; and let h, k, l, or any three whole numbers in the same ratio, and having the same signs, be called the indices of any plane parallel to HKL. Then

When one of the indices h, k or 7 becomes O, the corresponding point H, K or L will be indefinitely distant, and the plane hkl will be parallel to the corresponding axis. When two of the indices become 0, the plane hkl will be parallel to the two corresponding axes.

When a numerical index is negative, or a literal index is taken negatively, the negative sign will be placed over it.

The planes hkl, hkī are obviously parallel, and on opposite sides of the point O.

5. The line in which any two planes of the system intersect will be called an edge.

Let O be the origin; OX, OY, OZ the axes; a, b, c the parameters of a system of planes. Let OB=6; and let the planes hkl, pqr, passing through B, inter

sect in the edge BM, meeting L the plane ZOX in M, and let them meet OZ and OX in L, R and H, P. Then

AOH=OBOL,

2OP=OB=OR.

α

R

Fig. 3.

E

1.OL=kc, h. OH=ka, r. OR=qc, p. OP=qa,

lr. LR=(kr-lq)c, hp. PH=(hq-kp)a.

But by (2),

HM.LR.OP=ML.RO.PH.

Therefore if

u=kr―lq, v=lp-hr, w=hq-kp,

wl.ML=uh. HM, wl. LH=-vk. HM, uh.LH=—vk.ML. Draw MD parallel to OZ meeting OX in D.

LH.OD=ML. OH, and LH. MD=MH.OL.

Hence v. OD=-ua, and v.MD=— WC. Take DE equal and parallel to BO, EF equal and parallel to MD. Then

v.DE=—v.OB=-vb. BM is obviously parallel to OF, the diagonal of a parallelopiped the edges of which are respectively parallel to the axes OX, OY, OZ, and proportional to -ua, -vb, -we, or to ua, vb, wc.

Any line parallel to BM will be denoted by the symbol uvw.

The whole numbers u, v, w will be called its indices.

6. In OY take OB-b, and let hkl, pqr be the symbols of any two edges BM, BS passing through B, and meeting the plane ZOX in M, S. Let MS meet OZ in W, and OX in U. Draw MD, SG parallel to OZ. Then, by (5),

=

Fig. 4.

W

B

M

k. OD=-ha, k. DM -lc, q. OG=-pa, q. GS=-rc. Hence

(kr-lq)OU=(lp-hr)a, (hq-kp)OW=(lp-hr)c.

Therefore

where

OU=OV=OW,

u=kr-lq, v=lp-hr, w-hq-kp.

Since u, v, w are integers, the plane UBW is one of the planes of the system. Hence a plane of the system may always exist parallel to the edges formed by the intersections of any two pairs of planes of the system.

7. When uvw is the symbol of a plane parallel to the edges hkl, pqr, the indices u, v, w are derived from h, k, l, p, q, r exactly in the same manner as the indices u, v, w are derived from h, k, l, p, q, r, when uvw is the symbol of the edge in which the planes hkl, pqr intersect.

If the two symbols followed by the first and second indices of each, be written one underneath the other, and three letters X in the intervals between every four indices beginning with the second, it will be seen that u product of indices joined by the thick stroke of the first X- product of indices joined by the thin stroke; and that v and w are formed in the same manner from the products of the indices joined by the thick and thin strokes of the second and third letters X respectively.

h k l h k

XXX u kr-lq, v=lp-hr, w-hq-kp.

p q r p q

By this rule numerical values of u, v, w may be formed at