■ [ 498 . ] 



XLYI. On the Resultant of a Number of Unit Vibrations y 

 whose Phases are at Random over a Range not limited to 

 an Integral Number of Periods. By Lord Rayleigh, 



0J£. % P.R.S* 



A NUMBER (n) of points is distributed at random on a 

 straight line of: length a. When n is very great, the 

 centre o£ gravity of the points tends to coincidence with the 

 middle point of the line, which is taken as origin of coordi- 

 nates. What is the probability that the error of position, 

 that is its deviation from the origin, lies between x and 

 x -f dx ? 



Divide the length a into a large odd number (25+1) of 

 parts, each equal to b. The number of points to be expected 

 on each b is nhja. This expectation would be fulfilled in the 

 mean of a large number of independent trials, but in a 

 single trial it is subject to error. If the actual number be 

 nb/a + %, the chance that f lies between f and f -f d£ is by 

 Bernoulli's theorem 



d£ —aP-Hnb 



</(2irnb/a) 



e—**'™, (1) 



in which it is assumed that while b\a is very small, nb/a is 

 nevertheless very great f. In the language of the Theory 

 of Errors, the modulus, proportional to " probable error," 

 is y/(2nbla). 



The points which fall on any small part b may be treated 

 as acting at the middle of the part. For instance, those 

 which fall on the part which includes the origin are supposed 

 to act at the origin and so make no contribution to the sum 

 of the moments; while on other parts the moment is pro- 

 portional to the distance between the middle of the part and 

 the origin. Thus if 



be the values of the various £'s, the coordinate x of the 

 centre of gravity is given by 



_ Mg 1 -g_ 1 ) + 2fttf,-E-^+---- + *Mf.-Q ' „. 



If the whole number of the points be n exactly, the sum 

 of the f's in the denominator of (2) must vanish exactly ; 



* Communicated by the Author. 



t Compare Phil. Mag. vol. xlvii. p. 246 (1899). 



