Part 49

NOTE 169, pp. 109, 192, 273. _A rhombohedron_ is a solid contained by six plane surfaces, as in fig. 63, the opposite planes being equal and similar rhombs parallel to one another; but all the planes are not necessarily equal or similar, nor are its angles right angles. In carbonate of lime the angle C A B is 105°·55, and the angle B or C is 75°·05.

NOTE 170, p. 109. _Sublimation._ Bodies raised into vapour which is again condensed into a solid state.

NOTE 171, p. 112. _Platinum._ The heaviest of metals; its colour is between that of silver and lead.

NOTE 172, p. 113. The surface of a column of water, or spirit of wine, in a capillary tube, is hollow; and that of a column of quicksilver is convex, or rounded, as in fig. 41.

NOTE 173, p. 113. _Inverse ratio, &c._ The elevation of the liquid is greater in proportion as the internal diameter of the tube is less.

NOTE 174, p. 114. In fig. 41 the line c d shows the direction of the resulting force in the two cases.

NOTE 175, p. 115. When two plates of glass are brought near to one another in water, the liquid rises between them; and, if the plates touch each other at one of their upright edges, the outline of the water will become an hyperbola.

NOTE 176, p. 115. Let A Aʹ, fig. 42, be two plates, both of which are wet, and B Bʹ two that are dry. When partly immersed in a liquid, its surface will be curved close to them, but will be of its usual level for the rest of the distance. At such a distance they will neither attract nor repel one another. But, as soon as they are brought near enough to have the whole of the liquid surface between them curved, as in a aʹ, b bʹ, they will rush together. If one be wet and another dry, as C Cʹ, they will repel one another at a certain distance; but, as soon as they are brought very near, they will rush together, as in the former cases.

NOTE 177, p. 123. In a paper on the atmospheric changes that produce rain and wind, by Thomas Hopkins, Esq., in the Geographical Journal, it is shown that, when vapour is condensed and falls in rain, a partial vacuum is formed, and that heavier air presses in as a current of wind. Thus the vacuum arising from the great precipitation at the tropics causes the polar winds to descend from the upper regions of the atmosphere and blow along the surface to the equator as trade winds to supply the place of the hot currents that are continually raising them into the higher regions. This circumstance removes the only difficulty in Lieutenant Maury’s theory of the winds.

NOTE 178, p. 134. _Latent or absorbed heat._ There is a certain quantity of heat in all bodies, which cannot be detected by the thermometer, but which may become sensible by compression.

NOTE 179, p. 137. _Reflected waves._ A series of waves of light, sound, or water, diverge in all directions from their origin I, fig. 43, as from a centre. When they meet with an obstacle S S, they strike against it, and are reflected or turned back by it in the same form as if they had proceeded from the centre C, at an equal distance on the other side of the surface S S.

NOTE 180, p. 138. _Elliptical shell._ If fig. 6 be a section of an elliptical shell, then all sounds coming from the focus S to different points on the surface, as m, are reflected back to F, because the angle T m S is equal to t m F. In a spherical hollow shell, a sound diverging from the centre is reflected back to the centre again.

NOTE 181, p. 142. Fig. 44 represents musical strings in vibration; the straight lines are the strings when at rest. The first figure of the four would give the fundamental note, as, for example, the low C. The second and third figures would give the first and second harmonics; that is, the octave and the 12th above C, n n n being the points at rest; the fourth figure shows the real motion when compounded of all three.

NOTE 182, p. 143. Fig. 45 represents sections of an open and of a shut pipe, and of a pipe open at one end. When sounded, the air spontaneously divides itself into segments. It remains at rest in the divisions or nodes n nʹ, &c., but vibrates between them in the direction of the arrow-heads. The undulations of the whole column of air give the fundamental note, while the vibrations of the divisions give the harmonics.

NOTE 183, p. 144. Fig. 1, plate 1, shows the vibrating surface when the sand divides it into squares, and fig. 2 represents the same when the nodal lines divide it into triangles. The portions marked a a are in different states of vibration from those marked b b.

NOTE 184, p. 145. Plates 1 and 2 contain a few of Chladni’s figures. The white lines are the forms assumed by the sand, from different modes of vibration, corresponding to musical notes of different degrees of pitch. Plate 3 contains six of Chladni’s circular figures.

NOTE 185, p. 145. Mr. Wheatstone’s principle is, that when vibrations producing the forms of figs. 1 and 2, plate 3, are united in the same surface, they make the sand assume the form of fig. 3. In the same manner, the vibrations which would separately cause the sand to take the forms of figs. 4 and 5, would make it assume the form in fig. 6 when united. The figure 9 results from the modes of vibration of 7 and 8 combined. The parts marked a a are in different states of vibration from those marked b b. Figs. 1, 2, and 3, plate 4, represent forms which the sand takes in consequence of simple modes of vibration; 4 and 5 are those arising from two combined modes of vibration; and the last six figures arise from four superimposed simple modes of vibration. These complicated figures are determined by computation independent of experiment.

NOTE 186, p. 146. The long cross-lines of fig. 46 show the two systems of nodal lines given by M. Savart’s laminæ.

NOTE 187, p. 146. The short lines on fig. 46 show the positions of the nodal lines on the other sides of the same laminæ.

NOTE 188, p. 146. Fig. 47 gives the nodal lines on a cylinder, with the paper rings that mark the quiescent points.

NOTE 189, pp. 138, 153, 156. _Reflection and Refraction._ Let P C p, fig. 48, be perpendicular to a surface of glass or water A B. When a ray of light, passing through the air, falls on this surface in any direction I C, part of it is reflected in the direction C S, and the other part is bent at C, and passes through the glass or water in the direction C R. I C is called the incident ray, and I C P the angle of incidence; C S is the reflected ray, and P C S the angle of reflection; C R is the refracted ray, and p C R the angle of refraction. The plane passing through S C and I C is the plane of reflection, and the plane passing through I C and C R is the plane of refraction. In ordinary cases, C I, C S, C R, are all in the same plane. We see the surface by means of the reflected light, which would otherwise be invisible. Whatever the reflecting surface may be, and however obliquely the light may fall upon it, the angle of reflection is always equal to the angle of incidence. Thus I C, Iʹ C, being rays incident on the surface at C, they will be reflected into C S, C Sʹ, so that the angle S C P will be equal to the angle I C P, and Sʹ C P equal to Iʹ C P. That is by no means the case with the refracted rays. The incident rays I C, Iʹ C, are bent at C towards the perpendicular, in the direction C R, C Rʹ; and the law of refraction is such, that the sine of the angle of incidence has a constant ratio to the sine of the angle of refraction; that is to say, the number expressing the length of I m, the sine of I C P, divided by the number expressing the length of R n, the sine of R C p, is the same for all the rays of light that can fall upon the surface of any one substance, and is called its index of refraction. Though the index of refraction be the same for any one substance, it is not the same for all substances. For water it is 1·336; for crown-glass it is 1·535; for flint-glass, 1·6; for diamond, 2·487; and for chromate of lead it is 3, which substance has a higher refractive power than any other known. Light falling perpendicularly on a surface passes through it without being refracted. If the light be now supposed to pass from a dense into a rare medium, as from glass or water into air, then R C, Rʹ C, become the incident rays; and in this case the refracted rays, C I, C Iʹ, are bent from the perpendicular instead of towards it. When the incidence is very oblique, as r C, the light never passes into the air at all, but it is _totally_ reflected in the direction C rʹ, so that the angle p C r is equal to p C rʹ; that frequently happens at the second surface of glass. When a ray I C falls from air upon a piece of glass A B, it is in general refracted at each surface. At C it is bent towards the perpendicular, and at R from it, and the ray emerges parallel to I C; but, when the ray is very oblique to the second surface, it is totally reflected. An object seen by total reflection is nearly as vivid as when seen by direct vision, because no part of the light is refracted. When light falls upon a plate of crown-glass, at an angle of 4° 32ʹ counted from the surface, the glass reflects 4 times more light than it transmits. At an angle of 7° 1ʹ the reflected light is double of the transmitted; at an angle of 11° 8ʹ the light reflected is equal to that transmitted; at 17° 17ʹ the reflected is equal to 1/2 the transmitted light; at 26° 38ʹ it is equal to 1/4, the variation, according to Arago, being as the square of the cosine.

NOTE 189, p. 154. _Atmospheric refraction._ Let a b, a b, &c., fig. 49, be strata, or extremely thin layers, of the atmosphere, which increase in density towards m n, the surface of the earth. A ray coming from a star meeting the surface of the atmosphere at S would be refracted at the surface of each layer, and would consequently move in the curved line S v v v A; and as an object is seen in the direction of the ray that meets the eye, the star, which really is in the direction A S, would seem to a person at A to be in s. So that refraction, which always acts in a vertical direction, raises objects above their true place. For that reason, a body at Sʹ, below the horizon H A O, would be raised, and would be seen in sʹ. The sun is frequently visible by refraction after he is set, or before he is risen. There is no refraction in the zenith at Z. It increases all the way to the horizon, where it is greatest, the variation being proportional to the tangent of the angles Z A S, Z A Sʹ, the distances of the bodies S Sʹ from the zenith. The more obliquely the rays fall, the greater the refraction.

NOTE 190, p. 154. _Bradley’s method of ascertaining the amount of refraction._ Let Z, fig. 50, be the zenith or point immediately above an observer at A; let H O be his horizon, and P the pole of the equinoctial A Q. Hence P A Q is a right angle. A star as near to the pole as s would appear to revolve about it, in consequence of the rotation of the earth. At noon, for example, it would be at s above the pole, and at midnight it would be in sʹ below it. The sum of the true zenith distances, Z A s, Z A sʹ, is equal to twice the angle Z A P. Again, S and Sʹ being the sun at his greatest distances from the equinoctial A Q when in the solstices, the sum of his true zenith distances, Z A S, Z A Sʹ, is equal to twice the angle Z A Q. Consequently, the four true zenith distances, when added together, are equal to twice the right angle Q A P; that is, they are equal to 180°. But the observed or apparent zenith distances are less than the true on account of refraction; therefore the sum of the four apparent zenith distances is less than 180° by the whole amount of the four refractions.

NOTE 191, p. 155. _Terrestrial refraction._ Let C, fig. 51, be the centre of the earth, A an observer at its surface, A H his horizon, and B some distant point, as the top of a hill. Let the arc B A be the path of a ray coming from B to A; E B, E A, tangents to its extremities; and A G, B F, perpendicular to C B. However high the hill B may be, it is nothing when compared with C A, the radius of the earth; consequently, A B differs so little from A D that the angles A E B and A C B are supplementary to one another; that is, the two taken together are equal to 180°. A C B is called the horizontal angle. Now B A H is the real height of B, and E A H its apparent height; hence refraction raises the object B, by the angle E A B, above its real place. Again, the real depression of A, when viewed from B, is F B A, whereas its apparent depression is F B E, so E B A is due to refraction. The angle F B A is equal to the sum of the angles B A H and A C B; that is, the true elevation is equal to the true depression and the horizontal angle. But the true elevation is equal to the apparent elevation diminished by the refraction; and the true depression is equal to the apparent depression increased by refraction. Hence twice the refraction is equal to the horizontal angle augmented by the difference between the apparent elevation and the apparent depression.

NOTE 192, p. 155. Fig. 52 represents the phenomenon in question. S P is the real ship, with its inverted and direct images seen in the air. Were there no refraction, the rays would come from the ship S P to the eye E in the direction of the straight lines; but, on account of the variable density of the inferior strata of the atmosphere, the rays are bent in the curved lines P c E, P d E, S m E, S n E. Since an object is seen in the direction of the tangent to that point of the ray which meets the eye, the point P of the real ship is seen at p and pʹ, and the point S seems to be in s and sʹ; and, as all the other points are transferred in the same manner, direct and inverted images of the ship are formed in the air above it.

NOTE 193, p. 156. Fig. 53 represents the section of a poker, with the refraction produced by the hot air surrounding it.

NOTE 194, p. 156. _The solar spectrum._ A ray from the sun at S, fig. 54, admitted into a dark room, through a small round hole H in a window-shutter, proceeds in a straight line to a screen D, on which it forms a bright circular spot of white light, of nearly the same diameter with the hole H. But when the refracting angle B A C of a glass prism is interposed, so that the sunbeam falls on A C the first surface of the prism, and emerges from the second surface A B at equal angles, it causes the rays to deviate from the straight path S D, and bends them to the screen M N, where they form a coloured image V R of the sun, of the same breadth with the diameter of the hole H, but much longer. The space V R consists of seven colours—violet, indigo, blue, green, yellow, orange, and red. The violet and red, being the most and least refrangible rays, are at the extremities, and the green occupy the middle part at G. The angle D g G is called the mean _deviation_, and the spreading of the coloured rays over the angle V g R the _dispersion_. The deviation and dispersion vary with the refracting angle B A C of the prism, and with the substance of which it is made.

NOTE 195, pp. 159, 164. Under the same circumstances, and where the refracting angles of the two prisms are equal, the angles D g G and V g R, fig. 54, are greater for flint-glass than for crown-glass. But, as they vary with the angle of the prism, it is only necessary to augment the refracting angle of the crown-glass prism by a certain quantity, to produce nearly the same deviation and dispersion with the flint-glass prism. Hence, when the two prisms are placed with their refracting angles in opposite directions, as in fig. 54, they nearly neutralize each other’s effects, and refract a beam of light without resolving it into its elementary coloured rays. Sir David Brewster has come to the conclusion that there may be refraction without colour by means of two prisms, or two lenses, when properly adjusted, even though they be made of the same kind of glass.

NOTE 196, p. 165. The object glass of the achromatic telescope consists of a convex lens A B, fig. 55, of crown-glass placed on the outside, towards the object, and of a concave-convex lens C D of flint-glass, placed towards the eye. The focal length of a lens is the distance of its centre from the point in which the rays converge, as F, fig. 60. If, then, the lenses A B and C D be so constructed that their focal lengths are in the same proportion as their dispersive powers, they will refract rays of light without colour.

NOTE 197, p. 165. If the mean refracting angle of the prism D g G, fig. 54, were the same for all substances, then the difference D g V - D g R would be the dispersion. But the angle of the prism being the same, all these angles are different in each substance, so that in order to obtain the dispersion of any substance the angle D g V - D g R must be divided by the angle D g G or its excess above unity, to which the mean refraction is always proportional. According to Mr. Fraunhofer the refraction of the extreme violet and red rays in crown-glass is 1·5466 and 1·5258; so D g V - D g R = 1·5466 - 1·5258 = ·0208, and half the sum of the excess of each above unity is = ·5362; consequently

(D g V - D g R)/D g G = ·0208/·5362 = 0·03879; for diamond

(D g V - D g R)/D g G = (2·467 - 2·411)/1·439 = 0·0389;

so that the dispersive power of diamond is a little less than that of crown-glass; hence the splendid refracted colours which distinguish diamond from every other precious stone are not owing to its high dispersive power, but to its great mean refraction.—SIR DAVID BREWSTER.

NOTE 198, p. 168. When a sunbeam, after having passed through a coloured glass V Vʹ, fig. 56, enters a dark room by two small slits O Oʹ in a card, or piece of tin, they produce alternate bright and black bands on a screen S Sʹ at a little distance. When either one or other of the slits O or Oʹ is stopped, the dark bands vanish, and the screen is illuminated by a uniform light, proving that the dark bands are produced by the interference of the two sets of rays. Again, let H m, fig. 57, be a beam of white light passing through a hole at H, made with a fine needle in a piece of lead or a card, and received on a screen S Sʹ. When a hair, or a small slip of card h hʹ, about the 30th of an inch in breadth, is held in the beam, the rays bend round on each side of it, and, arriving at the screen in different states of vibration, interfere and form a series of coloured fringes on each side of a central white band m. When a piece of card is interposed at C, so as to intercept the light which passes on one side of the hair, the coloured fringes vanish. When homogeneous light is used, the fringes are broadest in red, and become narrower for each colour of the spectrum progressively to the violet, which gives the narrowest and most crowded fringes. These very elegant experiments are due to Dr. Thomas Young.

NOTE 199, pp. 171, 200. Fig. 58 shows Newton’s rings, of which there are seven, formed by screwing two lenses of glass together. Provided the incident light be white, they always succeed each other in the following order:—

1st ring, or 1st order of colours: Black, very faint blue, brilliant white, yellow, orange, red.

2nd ring: Dark purple, or rather violet, blue, a very imperfect yellow green, vivid yellow, crimson red.

3rd ring: Purple, blue, rich grass green, fine yellow, pink, crimson.

4th ring: Dull blueish green, pale yellowish pink, red.

5th ring: Pale blueish green, white, pink.

6th ring: Pale blue green, pale pink.

7th ring: Very pale blueish green, very pale pink.

After the seventh order the colours become too faint to be distinguished. The rings decrease in breadth, and the colours become more crowded together, as they recede from the centre. When the light is homogeneous, the rings are broadest in the red, and decrease in breadth with every successive colour of the spectrum to the violet.

NOTE 200, p. 172. The absolute thickness of the film of air between the glasses is found as follows:—Let A F B C, fig. 59, be the section of a lens lying on a plane surface or plate of glass P Pʹ, seen edgewise, and let E C be the diameter of the sphere of which the lens is a segment. If A B be the diameter of any one of Newton’s rings, and B D parallel to C E, then B D or C F is the thickness of the air producing it. E C is a known quantity; and when A B, the diameter, is measured with compasses, B D or F C can be computed. Newton found that the length of B D, corresponding to the darkest part of the first ring, is the 98,000th part of an inch when the rays fall perpendicularly on the lens, and from this he deduced the thickness corresponding to each colour in the system of rings. By passing each colour of the solar spectrum in succession over the lenses, Newton also determined the thickness of the film of air corresponding to each colour, from the breadth of the rings, which are always of the same colour with the homogeneous light.

NOTE 201, p. 174. The focal length or distance of a lens is the distance from its centre to the point F, fig. 60, in which the refracted rays meet. Let L Lʹ be a lens of very short focal distance fixed in the window-shutter of a dark room. A sunbeam S L Lʹ passing through the lens will be brought to a focus in F, whence it will diverge in lines F C, F D, and will form a circular image of light on the opposite wall. Suppose a sheet of lead, having a small pin-hole pierced through it, to be placed in this beam; when the pin-hole is viewed from behind with a lens at E, it is surrounded with a series of coloured rings, which vary in appearance with the relative positions of the pin-hole and eye with regard to the point F. When the hole is the 30th of an inch in diameter and at the distance of 6-1/2 feet from F, when viewed at the distance of 24 inches, there are seven rings of the following colours:—

1st order: White, pale yellow, yellow, orange, dull red.

2nd order: Violet, blue, whitish, greenish yellow, fine yellow, orange red.

3rd order: Purple, indigo blue, greenish blue, brilliant green, yellow green, red.

4th order: Blueish green, blueish white, red.

5th order: Dull green, faint blueish white, faint red.

6th order: Very faint green, very faint red.

7th order: A trace of green and red.

NOTE 202, p. 175. Let L Lʹ, fig. 61, be the section of a lens placed in a window-shutter, through which a very small beam of light S L Lʹ passes into a dark room, and comes to a focus in F. If the edge of a knife K N be held in the beam, the rays bend away from it in hyperbolic curves K r, K rʹ, &c., instead of coming directly to the screen in the straight line K E, which is the boundary of the shadow. As these bending rays arrive at the screen in different states of undulation, they interfere, and form a series of coloured fringes, r rʹ, &c., along the edge of the shadow K E S N of the knife. The fringes vary in breadth with the relative distances of the knife-edge and screen from F.

NOTE 203, p. 177. Fig. 43 represents the phenomena in question, where S S is the surface, and I the centre of incident waves. The reflected waves are the dark lines returning towards I, which are the same as if they had originated in C on the other side of the surface.