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1919]         ON  THE EESULTANT  OF  A  NUMBEE  OF  UNIT VIBEATIONS          631
The sequence formula (6) serves well enough for the derivation of the facility curves appropriate to moderate values of n, but it does not lend itself readily to examination of the passage towards the final form when n is great. This purpose is better attained by an adaptation of a remarkable method due to Laplace*, and employed by him and by Airyf for the derivation of the usual exponential formula for the facility of error. Here again it will be the sum of the distances of the points, now reckoned from the middle of the line, that we consider in the first instance.
The distances, instead of being continuously distributed, are supposed to be limited to definite values, all equally probable,
-sb, (-s + l)!), (-s + 2)6, ...-&, 0, b, 26,,
where 2s& = a, and ultimately s will be made infinite. The question is — What is the chance that the sum of the distances of n points shall be equal to Ib, where I is a positive or negative integer ? On examination it appears that the combination follows the same laws "as the addition of indices in the successive multiplications of the polynomial
Q— is6   i   Q—i (8—1)0   i   g— i(8— 2)9   i            I   gi(i~ 2)0   i   gi(s— 1)0 j_ gisQ
by itself, supposing the operation repeated n— I times. And therefore the number of combinations required will be the coefficient of eile (which is also the same as the coefficient of e~ile) in the expansion of
\Q— isd   i   Q—i(8—i}0   i     _     i   gi{is-i)0   i   gisd\n»
" The number of combinations required is therefore the same as the term independent of 0 in the expansion of
| (eil<> + g-iM) je-™0 + g-i(«-l)fl +..'.+ e*(s-i)0 -1- fpO}nf
or the same as the term independent of 0," when
cos W [I + 2 cos 6 + 2 cos 20 + ... + 2 cos s6}n
is 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
7/1                                                                                /Q,
cos 16 \ - —r — r^ — — \ dff ...................... (9)
1       sin 40        J                                       v '
This is the number of combinations which gives rise to a sum equal to /, and win order to obtain the probability of I it must be divided by the whole number of combinations equally probable, that is (2s •+ l)w. What we have to consider is accordingly the value of
7T (2*+1)^0
cos 60-^----=^4—rz—     r  dv.............(10)
v    '
* See Todhunter's History of the Theory of Probability, p. 521.
t Theory of Errors of Observations, Macmillan, 1861, p. 8.   In a comparison of the present notation with that of Laplace and Airy, the symbols n and s will be seen to be interchanged.sary to consider the integral obtained by "the next following operation" (s =p + f - %ri). It would seem then that the above considerations are sufficient to justify the differentiation by which 0,t is obtained (jp = 0, s- -f), and a fortiori that for <fe (# = 0, s=-2), etc. W..IY.S.]elative index is then much restricted, the