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628 ON THE RESULTANT OF A NUMBER OF UNIT VIBRATIONS [442 If the whole number of the points be n exactly, the sum of the £'s in the denominator of (2) must vanish exactly; but if we assume this beforehand, the various |'s are not independent, as is required by the rules of the Theory of Errors. We may evade the difficulty by supposing the value of £ on any part to be the result of an independent distribution of n points over the whole length. The total of the £'s is then not necessarily zero, but if we select those cases in which the total is zero, or nearly enough zero, the original requirement is fulfilled. In point of fact no selection is required, inasmuch as the probable error of the sum of |f's is A/(2s + l) times the probable error of each and therefore proportional to V(2s +1). */(Znb/d), or V(2w), so that no error of which there is a finite probability is comparable with n. We may accordingly take (2) in the simplified form and the (modulus)2 for the composite error x is given by ^lv>=^(l2 + 22 + 32+...+s2)-Mod2 £ ?i2 x P For our purpose the sum of the series may be identified with / szds, or s3/3, J o or if we prefer it, (2,s + l)3/24, that is a3/24&3, and thus Mod2 for x=a~/Qn, ...........................(4) s, as well as n, being regarded as infinitely great. The probability of an error between x and x -+• dx in the position of the centre of gravity of the n points is accordingly //(vn\ / i l — fill *•* I ft 2 7 / / K \ / [ ____ 1 /) \3lbw Jitr /TO"//"/ I T Y V (TT) ' ' ...........................W showing in what manner the probability of a finite x becomes infinitely small as n increases without limit. The method hitherto employed requires that the total number (n) of points be very great. It is of interest also to inquire what are the various probabilities when n is small or moderate. In dealing with this problem it seems more convenient to reckon the distances from one end of the line a, and to calculate in the first instance the chances for the sum (cr) of the distances. We take <f>n (cr) da/a to represent the chance that for n points this sum lies between a- and cr + da-, and we commence with a sequence formula connecting <f>n+1 with </>„. If for the moment we suppose <£n known and consider the inclusion of an additional point, we see that faWdtr/a.........................(6) a By means of (6) the various functions may be built up in order.. In these circumstances it would appear that-_(38) is itself valid, and that it is unnecessary 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