Integrating it is

-cλ

2πr cos

2π λ

COS (vt-r+ cos 0.≈):

and taking this from -b to + b, the disturbance

at G is

25. We shall now proceed to compare the values of this expression for different values of 0.

Case 1. Suppose much greater than b.

(This will

generally be the case with sound, as for all audible sounds A varies from a few inches to several feet.)

will be a small arc, and will not differ much

from its sine putting the arc for the sine, the maximum of vibration becomes

which is the same for all values of 0.

Case 2. Suppose much smaller than b. (This will generally be the case with light).

For the part nearly opposite to the entering wave, cos ✪ is very small, and

In other parts it is to be observed that the disturbance is O when

Hence there is a succession of points in which there is absolute darkness. Of the intermediate parts, the brightest will be found (nearly) by making

the intensity of light at the brightest part of one of the bright portions is to that of the part nearly opposite to the entering

wave, as

be so small as for light (for instance if λ= 0,00002 and

ratio will be extremely small when cos has any sensible value, and we may say without perceptible error that except nearly opposite to the entering wave, there will be complete darkness all round.

26. If in the investigation we had included the terms depending on 2, we should merely have had a very small

addition to

2п

(vt − r + cos 0.x),

λ

and this addition would not sensibly have altered in value

2π

while ·(vt-r+cos 0.x) increased by 2π.

λ

It will easily be seen that in Case 1 this would have produced no effect, and in Case 2 the maxima of brightness and the absolute darkness would only have been shifted a little way; their number, relative position, and the intensity of light, remaining sensibly the same*. And if for ZG we had put its more accurate value r - cos 0., the terms added to the expression would not have been sensible.

27. The conclusion in Case 2 may also be obtained thus. In fig. 6 divide AB into a number of equal parts Aa, ab, bc, &c. such that the distance of A from G is less

λ

λ

than that of a by : that of a less than that of b by -;

and so on.

2

The waves from corresponding parts of Aa and ab are, at starting, in the same phase. Consequently when they reach G, the wave from a part of Aa is in advance

λ

of that from the corresponding part of ab by, or they are in opposite phases, and therefore, by (15), they destroy each other. Thus every part of Aa destroys a corresponding part of ab, and therefore the whole effect of Ab is 0. Similarly the whole effect of bd is 0; &c. Thus if the number of parts be even, there is no vibration produced at G: if it be odd, there is only the vibration produced by the last of the small parts. But for the position nearly in front of the wave, all the parts are nearly at the same

*We shall hereafter consider cases in which these terms are sensible.

distance, and the vibration is produced by the added effects of all the small waves coming from every part of AB. If Case 1 be considered in the same way, it appears that the paths of the waves from different parts of the opening differ so little that all when they fall on G may be considered to be in the same phase, and therefore every part of the semicircle is in the same state of vibration.

28. The conclusions at which we have arrived are very important as removing the original objection to the undulatory theory of light. It was objected that if it were produced by an undulation similar to that producing sound, it ought to spread in the same manner as sound: that if light coming from a bright point entered a room by a small hole, it ought (instead of going on straight to illuminate a spot on the opposite side) to spread through the room in the same manner as a sound coming in the same direction and through the same hole. The answer appears in the results of the last investigation: the length of the waves of air is much greater than the aperture, that of the waves of light much less and the same investigation which shews that in the former case the sound ought to spread equally in all directions, shews that in the latter the light ought to be insensible except nearly in front of the hole. We have reason to think that when sound passes through a very large aperture, or when it is reflected from a large surface (which amounts nearly to the same thing) it is hardly sensible except in front of the opening, or in the direction of reflection.

29. Our conclusion with regard to light is also important as removing one source of doubt in several succeeding investigations. In our ignorance of the law of intensity of the vibrations propagated from a center in different directions, we have supposed the intensity equal in all directions: and yet with this supposition we have found that when the aperture is much larger than A, there is no sensible illumination except nearly in the direction in which the wave was

MOTION OF A WAVE IS PERPENDICULAR TO ITS FRONT. 277

going before it reached the aperture. The same would be true if the intensity diminished according to some function of the angle made with the original direction of the wave. Since then the illumination is (as far as the senses will be able to judge) nothing, except the obliquity is small, whatever be the function, we may assume that function of any form most convenient, provided that it does not alter rapidly in the neighbourhood of the original direction, and does not increase considerably as the angle of obliquity increases.

30. From the result of this investigation it appears also that the motion of every small part of the wave is perpendicular to the front of the wave. For in fig. 5 that part of the wave which passes through the orifice AB illuminates only that part of the semi-circle which is defined by drawing a straight line perpendicular to the front: and in the same manner if we had covered AB and opened another orifice, we should have found that the only illumination was on the part determined by drawing a straight line through the new orifice perpendicular to the front. In this we see the origin of the idea of rays of light. The reader is particularly requested to observe that this theorem is proved only by the demonstration of the proposition above, and depends entirely on this assumption, that the waves of light move with the same velocity in all directions. We shall hereafter speak of cases in which the motion of the wave is not perpendicular to its front.

It will readily be seen that the whole of this applies as well to the motion of the small parts of a wave whose front is not plane.

PROP. 9. To explain the reflection of light on the undulatory theory.

31. We shall again refer to the motion of sound for an analogical illustration of this point. In fig. 7 let ABCD be the front of a wave (which for simplicity we suppose plane, every part moving in parallel directions) advancing