566 ON THE LIGHT EMITTED FROM A [436
The general formula is that the probability of an amplitude between r and r + dr is
-e-'^rdr^-e-Wdr, ........................... (1)
11 n ^ '
if 7 denote the intensity*.
As regards the " expectation " of intensity merely, the question is very simple. If 0, ff, 6" .,. be the n individual phases, the expectation is
.'o Jo Jo lir Air 2>Tr Effecting the integration with respect to 0, we have
and when we continue the process over all the n phases we get finally
Expectation of Intensity = n. ' The same result follows of course from (1). The " expectation " is
But if we are not to expect any particular intensity when a large number of vibrations of unit amplitude and arbitrary phase are combined, what precisely is the significance to be attached to this result ? As has already been suggested, Ave must look to what is likely to happen when we have to do with a large number m of independent trials, in each of which the n phases are redistributed at random. By (1) the chance of the separate intensities Jj, /2, ... Im lying between /:4- d!1} J2 + dlz, etc. is
and we may inquire what is altogether the chance of the sum of intensities, represented by J, lying between. J and J+ dJ. Over the range concerned the factor e~J/n may be treated as constant, and so the question is reduced to finding the value of
under the condition that Ja + J2 + ... lies between / and /+ dJ. This isf
(m — 1)!
so that the chance of II + J2'+ ... lying between «/ and J + dJ is
* An interesting example of variable intensity when phases are at random is afforded by the observations of De Haas (Amsterdam Proceedings, Vol. xx. p. 1278 (1918)) on the granular structure of the field when a coronals formed from homogeneous light. The results of various combinations are exhibited to the eye simultaneously.
t See for example Todhunter's Int. Gale. § 272.ory of Sound, 2nd ed. § 42 a, and for a more complete theory K. Pearson's Math. Contributions to the Theory of Evolution, xv, Dulau, London. £0) = 0, viz. %z cos £0=0, 5 '135, 8-417, 11-620, etc. W. P. S.]