423 The composition of vibrations of the same period is precisely analogous, as was pointed out by Fresnel, to the composition of forces, or indeed of any other two-dimensional vector quantities. The magnitude of the force corresponds to the amplitude of the vi bration, and the inclination of the force corresponds to the phase. A group of forces, of equal intensity, represented by lines drawn from the centre to the angular points of a regular polygon, consti tute a system in equilibrium. Consequently, a system of vibrations of equal amplitude and of phases symmetrically distributed round the period has a zero resultant. According to the phase-relation, determined by (a -a ), the am plitude of the resultant may vary from (A- A ) to (A + A ). If A and A are equal, the minimum resultant is zero, showing that two equal trains of waves may neutralize one another. This happens when the phases are opposite, or differ by half a (complete) period, and the effect is usually spoken of as the interference of light. From a purely dynamical point of view the word is not very ap propriate, the vibrations being simply superposed with as little inter ference as can be imagined. 3. Intensity. The intensity of light of given wave-length must depend upon the amplitude, but the precise nature of the relation is not at once apparent. We are not able to appreciate by simple inspection the relative intensities of two unequal lights ; and, when we say, for example, that one candle is twice as bright as another, we mean that two of the latter burning independently would give us the same light as one of the former. This may be regarded as the definition ; and then experiment may be appealed to to prove that the intensity of light from a given source varies inversely as the square of the distance. But our conviction of the truth of the law is perhaps founded quite as much upon the idea that something not liable to loss is radiated outwards, and is distributed in succession over the surfaces of spheres concentric with the source, whose areas are as the squares of the radii. The something can only be energy; and thus we are led to regard the rate at which energy is propagated across a given area parallel to the waves as the measure of inten sity ; and this is proportional, not to the first power, but to the square of the amplitude. Practical photometry is usually founded upon the law of inverse squares (LIGHT, vol. xiv. p. 583) ; and it should be remembered that the method involves essentially the use of a diffusing screen, the illumination of which, seen in a certain direction, is assumed to be independent of the precise direction in which the light falls upon it; for the distance of a candle, for example, cannot be altered without introducing at the same time a change in the apparent magnitude, and therefore in the incidence of some part at any rate of the light. With this objection is connected another which is often of greater importance, the necessary enfeeblement of the light by the process of diffusion. And, if to maintain the brilliancy we substitute regular reflectors for diffusing screens, the method breaks down altogether by the apparent illumination becoming independent of the distance of the source of light. The use of a revolving disk with transparent and opaque sectors in order to control the brightnesses proposed by Fox Talbot, 1 may often be recommended in scientific photometry, when a great loss of light is inadmissible. The law that, when the frequency of inter- mittence is sufficient to give a steady appearance, the brightness is proportional to the angular magnitude of the open sectors appears to be well established. 4. Resultant of a Large Number of Vibrations of Arbitrary Phase. We have seen that the resultaiit of two vibrations of equal amplitude is wholly dependent upon their phase-relation, and it is of interest to inquire what we are to expect from the composition of a large number (n) of equal vibrations of amplitude unity, and of arbitrary phases. The intensity of the resultant will of course depend upon the precise manner in which the phases are distri buted, and may vary from ?i 2 to zero. But is there a definite intensity which becomes more and more probable as n is increased without limit ? The nature of the question here raised is well illustrated by the special case in which the possible phases are restricted to two opposite phases. We may then conveniently discard the idea of phase, and regard the amplitudes as at random positive or negative. If all the signs are the same, the intensity is n 2 ; if, on the other hand, there are as many positive as negative, the result is zero. But, although the intensity may range from to n-, the smaller values are much more probable than the greater. The simplest part of the problem relates to what is called in the theory of probabilities the "expectation" of intensity, that is, the mean intensity to be expected after a great number of trials, in each of which the phases are taken at random. The chance that all the vibrations are positive is 2-", and thus the expectation of intensity corresponding to this contingency is 2-".?t 2 . In like manner the 1 Phil. May., v. p. 331, 1834. expectation corresponding to the number of positive vibrations being (n- 1) is 2- n .n.(n- 2)*, and so on. The whole expectation of intensity is thus , 1-2<3 v-6) + ...}.... (1). Now the sum of the (u + 1) terms of this series is simply n, as may be proved by comparison of coefficients of x~ in the equivalent forms 1 O * * * L . A The expectation of intensity is therefore n, and this whether n be great or small. The same conclusion holds good when the phases are unrestricted. From (4), 2, if A = 1, P 2 = ?i+22cos(a 2 -a 1 ) (2), where under the sign of summation are to be included the cosines of the fyi(n - 1) differences of phase. When the phases are arbitrary, this sum is as likely to be positive as negative, and thus the mean value of P 2 is n. The reader must be on his guard here against a fallacy which has misled some high authorities. We have not proved that when n is large there is any tendency for a single combination to give the intensity equal to n, but the quite different proposition that in a large number of trials, in each of which the phases are rearranged arbitrarily, the mean intensity will tend more and more to the value n. It is true that even in a single combination there is no reason why any of the cosines in (2) should be positive rather than negative, and from this we may infer that when n is increased the sum of the terms tends to vanish in comparison with the number of terms. But, the number of terms being of the order u 2 , we can infer nothing as to the value of the sum of the series in com parison with n. Indeed it is not true that the intensity in a single combination approximates to n, when n is large. It can be proved 2 that the probability of a resultant intermediate in amplitude between r and ? + dr is -.. c -r-/- r d r (3). (4)> or, which is the same thing, the probability of an amplitude greater than r is The probability of an amplitude less than r is thus / e~ r2 n rdr=l -c~ r ~ n ,-r-tn (5). 05 0488 80 5506 10 0952 1-00 6321 20 1813 1-50 7768 40 3296 2-00 8647 60 4512 3-00 9502 The accompanying table gives the probabilities of intensities less than the fractions of n named in the first column. For example, the probability of intensity less than n is 6321. It will be seen that, however great n may be, there is a fair chance of considerable relative fluctuations of intensity in consecutive combinations. The mean intensity, expressed by 2 f ** _ 2/ ?l is, as we have already seen, equal to ?i. It is with this mean intensity only that we are concerned in ordinary photometry. A source of light, such as a candle or even a soda flame, may be regarded as composed of a very large number of luminous centres disposed throughout a very sensible space ; and, even though it be true that the intensity at a particular point of a screen illuminated by it and at a particular moment of time is a matter of chance, further processes of averaging must be gone through before anything is arrived at of which our senses could ordinarily take cognizance. In the smallest interval of time during which the eye could be impressed, there would be opportunity for any number of rearrangements of phase, due either to motions of the particles or to irregularities in their modes of vibration. And even if we supposed that each luminous centre was fixed, and emitted perfectly regular vibrations, the manner of composition and consequent intensity would vary rapidly from point to point of the screen, and in ordinary cases the mean illumination over the smallest appreciable area would correspond to a thorough averaging of the phase-relationships. In this way the idea of the intensity of a luminous source, independently of any questions of phase, is seen to be justified, and we may properly say that two caudles are twice as bright as one.
"- Phil. Mag., Aug. 1880.