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Full text of "Scientific Papers - Vi"

[Hill I          KAN'UoM   1'UiiHTX   IX   o.NK,  TWO,  OR, T1JKKE   DIMENSIONS               GO!)
Finally we obtain

\   '• itir,'
<•
as tin- probability when it is large of t.hc resultant amplitude ;/•. 11 is l,o be ivmemheivd thai ./• is limited in a, scries oi' discrete values will i a common difference i iptal to "J, ami that, mtr ajijiniximalioii has proceeded upon t.hu -«t!ppM%ui(.n ih.il ,/• is not, of higher urder than ^'n*
If the i-»'mpMiieut, aiuplil tides or si fetches he /, in place of unity, we have merely i" \vrile ./ / in place of./-.
The sjieeia! \alue e.f tiie .series ( 1 f>) i.s realixed only when n is very ^reaL U»tf jt ail'Mjil-* a 1-luser appniximation to (he (rue value than mig'hl he t;xpe(.:l,e<l \\h«-ii // is I'jily m<iderate, 1 have calrulaled the case of n.~- 10, both directly from tin- »-\ael expression (10) un<l irmn I,he series (IT)) for sill the admissible \iilu«-s ol ./•.
TAI'.M-: I.
//     10.
(HI)
•ot;ui
•IHI'JV
•tHKIllM
The \alues {or ,r «* 0 mid twice those beloiijkfinjtf to higher values of ,•*• '.liuuld tufui unity. ThuMe nbuve frujn (10) give I'OOOOI and those from (!/">) s^'jve IIHHHJH. Il \\ili be MIM-M ihat except, in the extreme ruse of i/.'. 10, the avjM-»-meuf between the tun forinuhe in Very ehiMc, Bnt»even for much higher \ahie-* uf it, the uettiul calfulutinn in .nimpler from the exact formula (10).
When / i** very small, while /i is very great., we may be able for some purjHi.ses to disreguni the discoiUimioUH eharneter ol' the jjrobubility as a fuiieh'tii nj'./•. re-placing the issolatud points by a continuous repreHentat.ive enr\'-. The diilereijee between the- alwri.swt- of coii.set'utive i.solated pointH JH i!/, ;->'» lhat. ii'(/,r be a large multiple of/, we may take
zmr
the approximate e > between x and .r
H. VI.
f the probability that the resultant amplitud
,1'J       ±2,       ±4,       ±6, ... ±n; and when n is odd, the (n + 1) values