Page images
PDF
EPUB
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

If we substitute for e, f, g; h, k, 1; p, q, r their values in terms of the indices of the poles D, E, F, in which the zonecircles EF, FD, DE intersect, the resulting expressions for u',v', w' resemble those obtained by Frankenheim (Crelle's Journal für Mathematik, 1832, vol. viii. p. 181), but are not identical with them; for though the notation for faces is the same in both, the axes are different.

To find u, v, w in terms of u', v', w'.

Let efg, hkl, pqr be the symbols of the poles D, E, F, in which the zone-circles efg, hkl, pqr mutually intersect. The symbols of the faces 100, 010, 001, when referred to the new axes, become ehp, fkq, glr respectively; and the indices of the zones containing every two of them will be

kr-lq, qg-rf, fl-gk;

lp-hr, re-pg, gh-el;

hq-kp, pf-qe, ek-fh.

But these are e, h, p ; f, k, q ; g, l, r respectively. Therefore

n=eu thi tp,

v=fu' + kv'+qw',

(11)

w=gu' + lv' + rw'.

To find the symbol of the zone uvw, when referred to the axes of the zones efg, hkl, pqr as crystallographic axes.

Let efg, hkl, pqr be the symbols of the faces common to every two of the zones, not contained in the zones efg, hkl, pqr respectively; u'v'w' the symbol of the zone uvw when referred to the new axes. The zone uvw contains the faces Owv, wОū, vũ0.

Their symbols, when referred to the new axes, are

fw-gv kw-lv qw- rv,

ew-gu hw-lu

[blocks in formation]

pw-ru,

-fu hv-ku pv-qu.

The symbol of the zone containing the second and third of these becomes

(hv-ku) (pw-ru) — (pv—qu) (hw—lu),

(pv-qu) (ew-gu) — (ev— fu) (pw—ru),

(ev-fu) (hw-lu) — (hv— ku) (ew—gu).

Hence, multiplying out, and rejecting the common factor u,
u' = (kr—lq)u+ (lp—hr)v+ (hg−kp)w,
v'=(qg―rf)u+ (re−pg)v+ (pf−qe)w,
w'= (fl−gk)u+ (gh—el)v+ (ek−fh)w.

Or

u'=eu+fv+gw,

v=hu+kv+lw,

w'=pu+qv+ rw.

To find u, v, w in terms of u', v', w'.

(12)

Let the faces 100, 010, 001 be referred to the axes of the zones efg, hkl, pqr as crystallographic axes. Then (10) their symbols become ehp, fkq, glr. Therefore (12),

u=en+hy+pw,
v=fu' +kv'+qw',

w=gu'+lv' + rw'..

(13)

Having given the symbols of five poles, and the positions of four of them, to find the position of the fifth.

Let efg, hkl, pqr, mno, uvw be the symbols of the poles D, E, F, G, P (fig. 4). Let D, E, F, G be given in position. Therefore the segments into which any of the angles EDF, FED, DFE are divided by the arcs GD, GE, GF, are known. Let X, Y, Z be the poles of EF, FD, DE; and let m'n'o', u'v'w' be the symbols of G, P when referred to the axes of the zone-circles EF, FD, DE as crystallographic axes. Then

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[blocks in formation]

Hence, two of the angles EDF, FED, DFE, and the segments into which they are divided by PD, PE, PF, being known, the position of P is determined.

Having given the symbols and positions of four poles, and the distances of any two of them from a fifth pole, to find its symbol. Retaining the construction and notation of the preceding article, let the symbols and positions of D, E, F, G, and the distances PD, PE be given, to find the symbol of P. Since PD, PE are known, the segments into which any of the angles EDF, FED, DFE are divided by PD, PE, PF may be computed. The ratios of u', v', w' may then be found from any two of the three equations

[blocks in formation]

The indices u, v, w are then given in terms of u', v', w' by (13).

XVI. On two new Forms of the Stereoscope, intended for the purpose of uniting Large Binocular Photographic Pictures, by JAMES ELLIOT, Teacher of Mathematics, Edinburgh; and on a third Form of the same Instrument, for the same purpose, by GEORGE WATERSTON, Stationer, Edinburgh*.

IT may be rash to announce as new any form of an instrument

in exhausting the varieties of which every effort of ingenuity has already been employed, and on which, it may be said at present, literally as well as metaphorically, the eyes of the nation are fixed, so that we cannot tell what a day may produce, or what it may already have produced. I therefore advance the claim with all due hesitation, and with the distinct proviso that novelty is asserted only as far as our own knowledge goes, and the knowledge of others to whom the instruments have been shown, and only so far as I shall further limit the claim in the following description.

To commence that limitation, then, there is no novelty in the mere fact of uniting large binocular pictures. Professor Wheatstone's instrument does that; but for such pictures it is liable to several objectionst. In the first place, according to his method, the pictures must face each other; and when large, they must be placed a long way apart, so that they cannot easily be attached to the instrument; consequently you must either put up supports for both, or hang them on opposite walls of the same room, at the risk of finding these at an unsuitable distance, unequally illuminated, and subject to other inconveniences. Secondly, there is some trouble in placing them exactly parallel. Thirdly, if the instrument is not placed precisely equidistant from the two, the one will appear larger than the other, and they will refuse to unite. Fourthly, there is a considerable loss of light in reflexion. Fifthly, a double image is formed by the two sides of the glass, the separation being rendered more conspicuous from the oblique angle at which the mirrors are placed.

It is not, however, as rivals to Professor Wheatstone's instrument that these are introduced by us, but simply as furnishing other modes of attaining the same result; and, in fact, they have been brought out almost accidentally, and without any preconceived design at all. It will therefore probably be best to introduce the description of them by giving an account of their origin.

A few weeks ago, Mr. Waterston happened to have in his * Communicated by Mr. Elliot; having been read to the Royal Scottish Society of Arts, Jan. 12, 1856.

+ I mention these objections rather from the statements of others than from my own knowledge, never having used Professor Wheatstone's stereoscope, and having only once seen it.

possession two beautiful binocular photographic landscapes, by Mr. Wilson of Aberdeen. They were not small pictures, such as are usually made for the stereoscope, but were each somewhere about 12 or 13 inches long and 10 broad. Mr. Waterston had been in the habit of uniting stereoscopic views by the eye alone, without the use of an instrument. He tried the large landscapes in the same way and succeeded in uniting them, but with that picture which is usually the right-hand one placed on the left, and the left-hand one on the right. Having tried to unite the pictures in the same way, but without success, I immediately undertook to produce an instrument by which the purpose might be effected, and found the attempt successful. In fact I had thought of the same instrument before for uniting small binocular pictures crosswise. That instrument is simple enough. It consists of a wooden frame, like a box open at one end; the two sides (represented by A B and CD in fig. 1) fold together with hinges at A and C, while the front or closed end, A C, has two openings, E and E', at the distance of the two eyes

[blocks in formation]

from each other. At first these openings were hollow truncated cones, with a small aperture at the narrower end (as in the figure), and fixed to slides by means of which they could be adjusted to the exact distance of the eyes. The object of having the apertures so small, was to deprive each eye of its power of detecting distance by means of its own focal adjustment; a power which, it is well known, the eye possesses. I had another pair of slides, however, made with large apertures, and found that they answered nearly as well, and were much more easily used.

The sides are folded in, so far that the right-hand side entirely hides the right picture, P', from the right eye, and the left-hand side hides the left picture, P, from the left eye; while they are made of such a length as to leave each picture wholly seen, and no more, by the alternate eye. The extremities, B and D, of the sides are then distant about two inches from each other for pictures of the size mentioned. The height of the aperture is also at the same time contracted by a piece of cardboard, or by two

« PreviousContinue »