On the Resultant of a Number of Unit Vibrations. 499 



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 f on any part to be the result of an independent 

 distribution of n points over the whole length. The total of 

 the f's is then not necessarily zero, but if we select those 

 cases in which n 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 £'s 

 is ^(25 + 1) times the probable error of each and therefore 

 proportional to ^/{2s + V) . v /(2»6/a), or ;y/(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 



»« |{fi-f-i+'«(6-fi 1 )+ .... +<£-?_,)}; (3) 



•a nd the (modulus} 2 for the composite error x is given by 

 Mod^_2^ 

 Mod 2 f~ n> (i + Z +6 +...+*■). 



For our purpose the sum of tee series may be identified with 



{ S s 2 ds, or s 3 /3, or if we prefer it, (2s + l) 3 /24, that is 



Jo 



a z \2tf>\ and thus 



Mod 2 for x = a 2 /6n, (4) 



.s, as well as ??, 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 



vA?) 



■6iix*/a* 



dx/a, (5) 



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 (<r) of the distances. We take (f> n (a)daja to repre- 

 sent the chance that for n points this sum lies between a 

 and a + da, and we commence with a sequence formula 

 connecting c/> n+1 with <j> n . If for the moment we suppose 



2M2 



