once from those of h, k, l, p, q, r without the trouble of substituting the latter in the expressions for u, v, w. 8. The planes of a system that intersect in parallel lines are said to constitute a zone. A line through the origin parallel to the edge formed by any two planes of a zone is called the axis of the zone. A zone and its axis are denoted by the symbol of the edge formed by any two planes belonging to the zone. Let uvw be the symbol of a plane parallel to the edges hkl, pqr. Then u=kr-lq, v=lp-hr, w-hq-kp. Therefore, multiplying the first equation by h, the second by k, and the third by 1, and adding, we get hu+kv+lw=0, which expresses the condition that the zone hkl may contain the plane uvw. Any three whole numbers which, when substituted for u, v, w, satisfy the above equation, are the indices of a plane in the zone hkl; and any three whole numbers which, when substituted for h, k, l, satisfy the same equation, are the indices of a zone containing the plane uvw. 9. Let the plane uvw meet the axes of the system of planes in U, V, W, and the zone-axis efg in P. Draw VP meeting WU in M, and WP meeting UV in N. The indices of the edge VM will be ev, gw+eu, -gy, and the indices of the edge WN will be ew, fw, eu+fv. The edges VM, WN are in the plane UVW. Therefore by (5), = W Fig. 5. M eu. WU (eu + gw) WM, and fv. VN-eu. NU. But by (2), PV.NU.WM=MP.VN. UW. Therefore fv. PV (eu+gw)MP. Hence = fv. MV(eu+fv+ gw)MP. In like manner, if the plane mno passing through V, meet OP in D, and OM in G, fn. GV = (em+fn+go)GD. But by (2), OD.GV.MP=OP.GD. MV. Hence v(em+fn+go)OD=n(eu+fv+gw)OP. So also, if the zone-axis hkl meet the plane uvw in Q, and the plane mno in E, v(hm+kn+lọ) OE=n(hu+kv+lw)OQ; and if the zone-axis pqr meet the plane uvw in R, and the plane mno in F, v(pm+qn+ro)OF=n(pu+qv+rw)OR. = Hence, if a' (em + fn + go)OD, b'= (hm + kn+lo)OE, c'=(pm+qn+ro) OF, we shall have Since u', v', w' are whole numbers, it follows that the planes may be referred to the zone-axes efg, hkl, pqr as axes of the system; and that the symbol of the face uvw, when referred to these axes, becomes u'v'w'. 10. It appears from accurate measurements of the mutual inclinations of the faces of a crystal, and from calculations founded on those measurements, that, if through any point within a crystal planes be drawn parallel to the faces of the crystal, and any three intersections of these planes, not in one plane, be taken for axes, the positions of the faces of the crystal will be found to obey the law which has been assumed for the system of planes which has been the subject of the preceding investigation. Hence all the conclusions arrived at respecting such a system of planes, hold equally for the faces by which a crystal is bounded. 11. Let P, Q, R, S be four faces in one zone, of which P, Rpass through the origin O, and Q, S pass through any point K in the axis OY. Let hkl, uvw be the symbols of the faces Q, S which are projected into KF, KR respectively; efg, pqr the sym. bols of any zone-axes in the faces P, R which are projected into OP, OR respectively. OD, OP are the projections of the portions of the zone-axis efg intercepted between O P Fig. 6. R and the faces hkl, úvw; and OF, OR are the projections of the portions of the zone-axis pqr intercepted between Ŏ and the faces hkl, uvw. Therefore by (9), Hence, if v(eh+fk+gl)OD=k(eu+fv+gw)OP, OF.OP=i.OD.OR. This equation remains true when K is no longer a point in the axis OY, and O is no longer the origin. Let PQ, PR, PS denote the angles which the faces P, Q, R, S make with the face P, supposed to be all measured in the same direction. Then 12. PS is given in terms of PQ, PR, and the indices of P, Q, R, S by equations easily deducible from the above. These are and i(cot PS-cot PR) cot PQ-cot PR = isin PQtan@=sin(PR—PQ), tan(PS—¿PR)=tan¦PRtan("—0). v The indices of S may be obtained by eliminating and พ น between either of the last equations and uu+vv+ww=0, where uvw is the symbol of the edge formed by the intersection of any two of the three faces P, Q, R. 13. Sin (PR-PQ) sin P = i sin (PR - PS) sin PQ is readily transformed into cos (2PR-PQRS)=(1−i) cos (PQFRS) + i cos (PQ±RS), the upper or lower sign being taken according as PR is greater or less than PS. This equation gives PR when the angle PQ has been observed in one crystal, and the angle RS in another; or when, as not unfrequently happens, the crystal consists of two individuals in positions not accurately parallel, one of which has the faces P, Q, and the other the faces R, S. Fig. 7. 14. Let efg, pqr be the symbols of any two edges OP, OR; KF, KR any two edges in the face POR; hkl, uvw the symbols of any two faces containing the edges KF, KR respectively. Then, since the edges efg, pqr meet the face hkl in D, F, and the face uvw in P, R, OF.OP=i.OD.OR. K U F R Draw DU parallel to KR, meeting OR in U, and FV parallel to KR, meeting OP in V. Then OF=i. OU, OV=i. OD. Draw OQ, OS parallel to KF, KR respectively. Then sin (POQ-sin POR) sin POS i sin (POS-POR) sin POQ. = 15. POS is given in terms of POQ, POR, and the indices of the four edges by or i(cot POS-cot POR) = cot POQ-cot POR; i sin POQ tan 0= sin (POR-POQ), tan (POS-POR)= tan POR tan( These expressions are of use in calculating the angles between the faces of crystals, and also in constructing models, and drawing figures of crystals. XLIX. Remarks on Foam and Hail. By Professor TYNDALL. ON N Monday, the 13th of April, I walked from Ventnor to Freshwater Gate along the southern coast of the Isle of Wight. The day was stormy, and the sea consequently high. Owing to the retardation of their inferior portions by friction, the waves, as they approached the shore, became steeper, the crests advanced, and finally fell over like cataracts. Sometimes three or four such waves were observed following each other and charging with great impetuosity up the beach, bounding over isolated rocks which, when the wave receded, stood six or eight feet above the water. Sometimes two waves intersected each other, and at the point of junction, as both fell forward, a dark cavernous space, somewhat resembling those observed at the ends of glaciers, existed for an instant. In the next moment the air within it was whipped into foam. The production of a great quantity of foam was indeed the consequence of the commotion. Large masses of it, which shook like elastic jelly in the wind, were collected wherever it could find shelter, from which pieces were sometimes detached and blown like gigantic snow-flakes inland. The remark I have to make here has reference to the temperature of this foam. On passing a nook where a quantity of it had collected, I dipped my hand into it and found it bloodwarm; its taste, moreover, was intensely bitter. On examining the water of the sea from which the foam had been produced, I found it very cold. In fact it was a raw sleety day, and the temperature of the sea-water was what might be expected in such weather. The warmth of the foam appeared to be due to the air enclosed by the waves when they fell over, and in the compression and heating of which a portion of the vis viva of the falling mass was expended. The air thus entrapped has, on account of its warmth, an increased capacity for aqueous vapour : this, and the evaporation from the extended surfaces in contact with the free atmosphere, concentrate the solution of which the films are composed, and is probably the cause of the increased bitterness. I am tempted to make a remark here upon another subject. While reading, a short time since, an able and elaborate summary of the opinions at present entertained regarding the formation, shape and constitution of hail, I found one circumstance omitted which must have some influence, and may have a very sensible one. While crossing the Hoch Joch in the Tyrol last year, when Dr. Frankland and myself were assailed by a fierce hailshower, the thought occurred to me that the conditions in front and behind a hailstone, during its motion, must in some cases affect its shape and constitution. The theory of meteorolites generally received at the present day is, that they are opake bodies raised to a state of incandescence by their motion through the atmosphere. Whether this theory be correct or not, the resistance of the atmosphere is a true cause, and a generation of heat equivalent to the vis viva lost through this resistance is inevitable. When a hailstone moves through the air, it has a condensation in front and a rarefaction at its back; a possible melting temperature before it, with a possible freezing temperature behind it, which may sensibly influence both the shape and constitution of the hailstone. In the shower to which I have referred, the predominant shape of the hailstones was conical, and this shape is also noticed by Dr. Hooker in his journal. It seems a fair matter for inquiry how far this shape, and the stratified appearance sometimes exhibited by hailstones, may be dependent on the condensation and rarefaction to which I have alluded. Both the circumstances referred to may have been already noticed; but I am not aware that they have, and this will perhaps excuse me if I should be calling attention to things already sufficiently known and appreciated. As regards the foam, the facts recorded appear to have some bearing upon the determination of the mechanical equivalent of heat by the churning up of water. But the skilful experimenter to whose industry we owe this determination has, I doubt not, excluded every such remote possibility of error. Royal Institution, April 1857. Phil. Mag. S. 4. Vol. 13. No. 87. May 1857. 2 B |