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In their discussion, Laplace and Airy regard both n and 5 as infinite. Here it is proposed to make s infinite, so as to attain a continuous distribution of the points, but without limitation upon the value of n, which may be any integer. If, as before, cr denote the sum of the distances,
When s is very great, sin sd alternates with great rapidity, so that the integral comes to depend upon that part of the range where 6 is very small. We may then replace sm$0 by \6, and taking ty = 0s, we find
.n_N cos — - — nr-cfyr ........................ (11)
as the equivalent of (10) when s becomes infinite. This is the probability which attaches to a single integral value of I, or to a change da-, where da- = a/ 2s. Thus the probability that a- lies between cr and cr 4- da- may be written
which is the required result for a continuous distribution and is applicable to any value of n. In our former notation,
in which, however, cr now represents the sum of the distances from the centre of the line, instead of from one end of it.
If n=I, (13) reduces to
sin (1 + 2o-/a) ^ + sin (1 - 2o-/a) ^ , ,
which is unity when a lies between + ^a, but otherwise vanishes.
Again, if n = 2, we find that c/>/ (o-) = + I/a, if cr lies between + a, and otherwise vanishes, and so on.
More generally, the sequence formula may be deduced from (13), but to obtain it in the original form (6), where the distances are measured from the end of the line, we must write a-—^na for cr in (13). Then we have
7 , . .
cos -J~ (a- — *na) . — — -1- d-frdo- a, o-a a v 2 7 ^'l y ' '
[' 2^/ i N 7 2ilr/ n + 1
cos — - (cr — Jwa) do- = a cos — - cr -- — -7<r-a a a V 2
so that (6) is verified.s expanded and arranged according to cosines of multiples of 6. By summing the series and application of Fourier's theorem this term is found to be