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,
P2 = n + 2Σ cos (α2 − α1)
(2),
where under the sign of summation are to be included the cosines of the ½ n(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 P2 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 n2, we can infer nothing as to the value of the sum of the series in comparison 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 (Phil. Mag., 1880, 10, p. 73; 1899, 47. p. 246) that the probability of a resultant intermediate in amplitude between r and r + dr is
| 2 | e−r2/n r dr |
| n |
(3).
The probability of an amplitude less than r is thus