I.ATORY THEORY OF LIGHT. 



UNDULATORY THEORY OF LI<illT. 



From the expression fur L we >ee that the mean illumination, or tin' 



value of (X +f'')~J l-dp t T larg *1 of I 1 ' +X'. > 2 *> which 



would I* the uniform illumination, if the streams mixed without 

 interfering. ThU is a particular example of general princi|>le from 

 which inference* have already been drawn [ ABSOIII-TKI.N], that when 

 two or more distinct streams of light interfere, no light is dtttraytd, by 

 interference, which merely causes a different distribution of the 

 illumination. 



If ft be the breadth of a fringe, which may be measured by a micro- 

 meter, = * ( ** * 6) ,whence A= *-. The distances a, 6, to the 



d a' + 6 



fanner of which a' may be deemed equal, are not small, and can be 

 measured without difficulty. The distance d was measured by Fresnel 

 by plying a screen with a small round hole at a known distance from 

 the mirrors, so that a slender beam of light from each of the imaged 

 I, T passed through the hole, and measuring by a micrometer the 

 distance between the centres of the beams at a known distance on the 

 other side of the hole whence d is obtained from the measured 

 distance by similar triangles. This is one method of measuring the 

 length of a wave of light. 



The excessive smaUness of A, indicated by this or any similar pheno- 

 menon of interference, leads to a complete explanation of one of the 

 oldest difficulties belonging to the undulatory theory, the existence of 

 rays and shadows. Conceive a broad beam of light to fall jierpen- 

 dicularlv on a screen containing a moderately small aperture, and let 

 us examine the disturbance produced at a point M, situated at a con- 

 siderable distance on the other side of the screen. For greater 

 simplicity we shall supposo the incident beam to come from a very 

 distant point, so that the incident waves may be regarded as plane. 

 I'.y Muygens's principle each : the front of a wave, as it 



arrives at the plane of the aperture, may be considered as the centre of 

 an elementary disturbance which diverges into the space behind the 

 screen, and in due time reaches M. But the disturbance at M will In- 

 by no means proportional to the size of the aperture, since the various 

 secondary waves which at a given instant reach si, and which arise 

 from incident waves which reach in succession the plane of the aperture, 

 are in a condition to interfere. First, suppose M situated at some 

 distance outside the geometrical projection of the aperture. Make M 

 the centre of a number of spheres with radii increasing by JA, :n\i\ of 

 which as many are drawn an cut the nurture. These spheres will cut 

 the aperture into numerous narrow slips, of the form of portions of 

 circular nnmili, having for their common centre the projection N of 

 the point H on the plane of the aperture. It will be readily seen that 

 the aggregate effect of the secondary wares starting from the various 

 elements of one slip will be as nearly aa possible neutralised by that of 

 the waves coming from the next slip. For the squares of the radii of 

 the annuli increase in arithmetical progression, and therefore the areas 

 of consecutive slips are equal, except as to the trifling difference 

 the change in the angle subtended at N. Neglecting for the moment 

 this small change, we readily see that to each element of the first slip 

 corresponds an equal element of the second, at a distance from :.< 

 different from that of the former element by (A, and therefore the 

 secondary waves belonging to these two elements will, as nearly as 

 possible, neutralise each other's effect. Hence the joint effect of two 

 consecutive slips as compared with that of either of them is a 

 quantity of the order A, that is the ratio of A to the other quantities 

 involved, such as the difference of distance from U of opposite sides of 

 the aperture. But as the uunilier of slips is a large quantity of the 

 order V, it might be supposed that the total effect was comparable 

 with that of one slip. This however is not the case. For the < 

 any slip taken along with half the effects of the two adjacent slips is a 

 small quantity of the order A : , and the sum of all such, v. h. n we group 

 the slips so as to take every alternate slip along with half the effect of 

 its two neighbours, is only a small quantity of the order A, unless it 

 be in consequence f tin- . .iiit of compensation at the beginning and 

 end of the series. But at the two ends, that is at the parts of the 

 aperture nearest to and farthest from the point N, the length of the 

 sli| dwindles down to zero. The peculiar case in which a part of the 

 boundary of the aperture is exactly circular, having it* centi. 

 which attaches itself to the theory of the bright point in the centre of 

 the shadow of a circular disc, is supposed to be excluded from con- 

 sideration. We infer therefor* that the disturbance at a point M, 

 situated as above described, is insensible. 



- II|I]H)M X to lie at some distun geometrical pro. 



jection of the boundary of the aperture. Imagine a scries of 

 drawn as before around M, beginning with that which touches the 

 plane of the aperture in the point . The aj i now be cut 



npas before, only that now for some dutince round x the annuli will 

 be complete, after which they will become incomplete, and will finally 

 dwindle away to nothing. The neutralisation will in this case take 

 place just as before, except as regards half the effect of the first 

 annulmi, or central circle of the system, and the disturbance will 

 therefore be sensibly- the same as if the screen were removed. 



If the point M be situated near the geometrical projection of the 

 boundary of the aperture, whether inridc IT outside, . r if (he aper- 

 ture bo so small that a few only of the spheres above mentioned cut 



it, the determination of the disturbance at M become* a more difficult 

 problem, and belongs to diffraction. The explanation of the exist- 

 ence of rays and shadows when the light is divergent, or the inter- 

 cepting screen is not perpendicular to its course, U nearly the MOM 

 a.- baforc, 



We come now to the colours of thin plates, one of the first phe- 

 nomena to the explanation of which the principle of interference was 

 applied. As, however, the whole subject has been referred : 

 present article, we shall commence with a description of the phao 

 inenon itself, and of its laws, as discovered by Newton. The r 

 of thin transparent lamina had previously been studied to a certain 

 extent, by Boyle and by Hooke, and the Utter of these philosophers 

 produced the phenomena in the instructive form in which their laws 

 have since been studied, namely, by placing two object-glasses in 

 contact. It was in this form chiefly that they were studied by Newton, 

 and from him the coloured rings formed by such glasses have been 

 called Newton's rings. 



In order to observe these rings conveniently, Newton placed two 

 convex lenses of long foci (14 and 50 feet) in contact with each other 

 at their vertices, keeping them together by means of three clamps at 

 intervals on their circumferences, so that there was between them a 

 very thin plate of air, concave on its upper and lower rides. On bring- 

 ing the jir of lenses to an open window, and receiving the rays of 

 Uirlit from the sky by reflection from them, there were observed, the 

 plates being gently pressed together, seven series of coloured rinxs or 

 bands about a block spot in the centre : beyond the seventh band the 

 colours could scarcely be distinguished. The diameters of the bands 

 being measured, where the colour in each was the brightest, Newton 

 found that those diameters were proportional to the square roots of 

 the series of odd numbers 1, 3, 5, 7, 4c. ; at the places where the 

 colours were the least bright, the diameters were found to be propor- 

 tional to the square roots of the series of even numbers 3, 4, 6, 

 The radii of curvature of the lenses being known, Newton com- 

 puted the thicknesses of the plate of air at the circumferences' in 

 which the colours of the bands had the greatest and least degrees of 

 brightness ; and he found (' Optices,' lib. ii. I that, at the mostlum 

 part of the ring nearest to the centra, the thickness wag equal to 

 aAan inch ; the thickness at the most obscure part of that ring was 

 equal to irixa inch. Hence, from the law above mention, 

 thicknesses of the air at. the most and K-ast luminous parts of the suc- 

 ceeding rings may be obtained ; those thicknesses being considered as 

 proportional to the squares of the seuiidiameters of the rings. 



If lenses whose surfaces have different curvatures arc employed, it is 

 always found that like tints are produced in the . circumfeieuces of 

 circles at places where the intervals between the surfaces are equal, 

 the eye being similarly situated, or a line supposed to be drawn to it 

 ii "ii i t ho centre making equal angles with a piano passing through all 

 the rings ; and this circumstance serves to show that the tin! - <. 

 wholly on the distances between the lenses. If the angle made by the 

 line drawn to the eye be diminished, the diameters of the ring- w ill 

 be increased, the tints nmiaining the same. Newton found that for 

 moderate obliquities the same ring was formed where the di- 

 betweeu the lenses, or the thickness of the interposed plato of air, 

 varied as the secant of the angle of incidence on ti. EMM of 



the lens, which in estimating this angle may be deemed .1 plate i>, 

 by parallel surfaces. To include great obliquities he has given an 

 empirical rule not sensibly differing from the simple rule of the I 

 except at great obliquities. It has sin l.een found l>y 



careful measures of the diameters of the rings at great obliquities, that 

 the thickness where a given ring is formed is regulated by the simple 

 law of the secant, and not by the more complicated formula given by 

 Newton. 



When the rings are formed by homogeneous light they are found to 

 be more numerous than when the light is mixed, indeed they arc almost 

 countless when the homogeneity is sufficiently ] :f< . t ; il 

 of the same colour as the light, and ore separated from one another >>y 

 narrow spaces which are quite black. The diameters of the rings in 

 the corresponding bands, at the places where the colours are the 

 brightest, are different when the bonds are formed by homogeneous 

 lights of different colours, being least when the light U violet, and 

 greatest when red; and Newton computed, ti-m the measured 

 diameters of the rings of different colours, the intervals hetnem 

 the lenses at the places where the brightest parts of tin 

 from the centra are formed : these distances are found to be 



inch for extreme red rays, and ,;,'.., ineh for extreme violet 

 rays, which it may be observed are half the lengths of an nndi. 

 for those kinds of light. 



The order and the dimensions of the coloured ritijs are the mine, 

 win t her air occupy the space between the lenses or whether the 

 lie iii the exhausted receiver of on air-pump : but when a 



. of greater refractive power than air. is interposed 

 between them, the tints are fainter and the diain, t r*of theiii, 

 less; or -m dler distances between the lenses are 

 produce the same tint* : it in ascertained that, with di.;'. 

 these distances, in the cose of a perpendicular incident ray, in 



d I., the : ridicea. 



'orrif ]mding rings ere obserTed by Newton in thin 



plates surrounded by media less dense than the platen : thus a 1- 



