525 



DIFFERENTIAL EQUATIONS. 



DIFFRACTION OF LIGHT. 



628 



article just cited for remarks on this phrase) which call dx, the diffe- 

 rential of r. Then the resulting differential of y is 



dy = (x + dx) * + (x + dx) (a? + x) 

 = (2z + 1) dx -i- (dx) 



Now (<fo) 3 is rejected as being an infinitely small part of the pre- 

 ceding term, so that (2x + 1) dx is the differential of i? + x. The 

 expression 2x + 1 (which is our preceding result) is here the coefficient 

 of the differential dx, and when a less objectionable method of obtaining 

 it came into use, still retained the name of differential coefficient. It 

 were exceedingly to be wished that some shorter term could be 

 agreed on for the expression of a result which is so often required to 

 be named. 



In the method of Leibnitz the differential coefficient is actually dy 



dil 

 divided by dx, and it is still expressed by -7- in the more modern 



methods. But this notation must now be supposed to be obtained as 

 follows : Let the change of x into x + Ax be accompanied by that of y 



Ay 

 into y + Ay ; then expresses the algebraical ratio of the change of 



dy 

 y to the change of x. In making the convention that -5^- is to stand 



Ay 

 for the limit of , obtained from the supposition that Ax diminishes 



without limit, we consider the first symbol, not as that of an alge- 

 braical fraction, but as one whose whole has a meaning, the parts 

 having r.one. This is the case with all simple symbols, as distin- 

 guished from compound symbols : thus in the figure V, standing for 

 five, the two sides of the letter have no independent meaning ; while 

 in 4 + 7, each of the three symbols, 4, + , and 7, has meaning contri- 

 buting to that of the whole, and at the same time independent of it. 



dy 

 Thus though we consider j- as pointing out that it is y which has 



been differentiated, and with respect to x, we do not, in this symbol, 

 assign to dy and dx any independent share of meaning. 



The advantage of the notation in question is the connection which it 

 preserve* between the practical use of the method of Leibnitz, and 

 the theoretical accuracy of that of limits. For instance, it is strictly 

 true, according to our conventions, that 



d,i 

 when y = X* +x, ^ = 2x + 1. 



but, giving dy and dx separate meaning, it is not true that dy = (2x + 

 1) dx. Yet the latter equation, though never true, is of this kind, 

 that it may be made as near to truth as we please, if dx may be made 

 as small aa we please. For the reason why it cannot therefore be said 

 to be true in the case where dx = 0, see FRACTIONS, VANISHING. 



The differential coefficient of a differential coefficient is called the 

 f:r.''nd differential coefficient ; a repetition of the process gives the 

 third differential coefficient : and so on. These, if the original method 

 of notation were proceeded with, should be represented by 



dy 



dx 

 but a more convenient notation 



'.Ac. 



dx 



Is derived as follows : Let the sub- 

 stitution of x + Ax for x take place any number of times in succession, 

 giving 



<t> (x) , <t> (x + Ax) , <t> (x + 2 A x) , $ (x + 3 Ax), Ac. 



Form the successive differences of the first term [DIFFERENCE] Qx or 

 y : then the following theorem can be proved. The th difference of 

 y, or A*y, divided by the th power of Ax, or (A*)*, is an expression of 

 which the limit, made by diminishing Ax without limit, gives the same 

 result as arises from differentiating y, n times in succession. We may 

 therefore (subject to the remarks already made on the first coefficient) 



A'y d* y 

 represent the limit of r^r, by r^-. > so that the function y and its 



successive differential coefficients are denoted as follows : 



^JL d *y rf 3 y d * ;/ 

 y ' dx ' dx' ' dx* ' dx* , Ac. 



It is usual to omit the brackets in the denominators. 



DIFFERENTIAL EQUATIONS. [EQUATIONS, DIFFERENTIAL.] 



DIFFERENTIAL THERMOMETER. [THEHMOMKTER, DIFFE- 

 RENTIAL] 



DIFFRACTION OF LIGHT. The peculiar modifications which 

 light undergoes when it passes by the edge of an opaque body are 

 clamed a* phenomena of the diffraction or inflexion of light. 



When a pencil of solar light U admitted into a darkened room 

 through a very small hole in a card, or is collected in a point by means 

 of a convex lens of short focus, and then diverges from that point, if 

 a small opaque plate of any outline be interposed in the course of the 

 ray, the shadow of this object received on a parallel screen behind will 

 be encompassed by a series uf coloured bands or fringes of a similar 



outline with the body, except at its angular points ; the order of the 

 colours in each fringe, reckoning from the outside towards the shadow, 

 or inside, is, as in the prismatic spectrum, from red to blue, but the 

 intermediate colours are less distinctly isolated, partaking of a mixture 

 of the extreme tints. 



The actual shadow, or dark space within the innermost fringe, is 

 also larger than the geometrical shadow which would have been cast 

 if the rays had passed exactly by the edge of the body in straight tines 

 and been received on the same screen, and the illumination instead of 

 being cut off sharply, fades continuously into the darkness of the 

 shadow, extending indeed to a small amount even within the geo- 

 metrical shadow. The space by which the actual shadow surpasses 

 the geometrical in breadth, is independent of the form or the matter 

 of the interposed body, and likewise, except in the case of a very 

 slender body, of its linear dimensions. The dilatation of the shadow 

 will therefore be most striking in the case of narrow bodies such as 

 pins, hairs, Ac. 



When the interposed body is very narrow but of sensible width, 

 streaks alternately brighter and darker will be found viithin the shadow, 

 and a white line along the middle, when the body is of a long and 

 slender rectangular form. 



All these phenomena are seen more easily and with greater brilliancy 

 if the light, instead of being received on a screen, be admitted directly 

 into the eye armed with a lens or eye-piece. 



If the incident light were homogeneous, such aa pure red, blue, &c., 

 as found in the spectrum, the colour of the fringes would of course be 

 the same as that of the incident light, and the fringes would consist of 

 simple alternations of light and shade. It is found that the scale of the 

 fringes varies with the colour, the fringes being broadest for the least 

 refrangible colours. Hence when the incident light is white the 

 fringes are variously coloured, in consequence of the encroachment of 

 the bright and dark fringes belonging to the various simple coloured 

 lights of which, as we know, white light is compounded. 



Sir I. Newton in his optics relates several modes in which he varied 

 the experiments on the inflexion of light; and, in his queries, he 

 suggests an explanation that light may be subject to the action of 

 forces sensible only at very small distances from the surfaces of bodies ; 

 that under their influence the rays in their course near the edge of the 

 body, instead of passing straightforward, describe very sinuous paths, 

 with many contrary flexures, and thus they are found some turned 

 inwards towards the shadow, others outwards, forming coloured bands ; 

 the curve surface, which is the locus of the intersections of all those 

 diffracted rays, forming the envelope of the visible shadow. Little 

 advance has been made beyond Newton's conjectures towards explain- 

 ing, according to this doctrine, the curious phenomena of diffraction ; 

 and the explanation, incomplete as it is, has been shown by Freanel to 

 be subject to serious difficulties. 



Very different has been the result with the theory which makes 

 light to consist of undulations. This theory was first applied to the 

 explanation of the phenomena by Dr. Young, who employed the prin- 

 ciple of interference, the discovery of which' we owe to him, and which 

 he so successfully applied to the explanation of the colours of thin 

 plates. The bending of light around an opaque body was assimilated 

 to the bending of waves around an obstacle, and the internal fringes 

 in the shadow of a narrow body to the interference of the light so bent 

 round the opposite edges. That these fringes are incontestably pro- 

 duced by interference, he proved by cutting off the light passing either 

 edge by means of an opaque screen placed either before or behind the 

 narrow body, when the internal fringes immediately disappeared. The 

 external fringes were attributed to the interference of the direct light 

 with the light reflected from the edge of the diffracting body (' Phil. 

 Trans.' for 1802, p. 12). Several of the observed laws of the phe- 

 nomena were shown to be explicable on this hypothesis. 



This explanation was in the first instance adopted by Fresnel ; but 

 in pursuing his researches he met with phenomena which opened his 

 eyes to the insufficiency of the explanation, and at last led him to 

 perceive that the various observed appearances flowed naturally from 

 two grand principles the principle of interference, and the principle 

 of Huygens which themselves are but particular applications of the 

 general dynamical principle of the co-existence of small motions. In 

 his celebrated memoir on diffraction (' Mem. de I'Academie,' 1821, 

 torn. v. p. 339), he has followed out this theory into its mathematical 

 details, and compared the results with observations both qualitatively 

 and quantitatively, by a series of most careful measures ; and the agree- 

 ment is complete. The theory furnishes not a single disposable con- 

 stant whereby a seeming accordance between theory and experiment 

 might have been brought about, for the length of a wave of light, the 

 only unknown constant involved, admits of being determined inde- 

 pendently of diffraction by the interference of two streams of regularly 

 reflected or refracted light. 



About the same time, and independently, Fraunhofer was engaged 

 in a series of researches on diffraction of a somewhat different class, 

 namely, those produced when a luminous point or slit is viewed in 

 focus through a telescope, and the object-glass is covered by a screen 

 pierced with one or more apertures. The reader may easily observe 

 phenomena of this kind by viewing a bright point, such as the image 

 of the sun in the bulb of a thermometer, and covering the eye with a 

 piece of fine gauze, or of tin foil or black paper pierced with one, two 



