442.
ON THE RESULTANT OF A NUMBER OF UNIT VIBRATIONS, WHOSE PHASES ARE AT RANDOM OVER A RANGE NOT LIMITED TO AN INTEGRAL NUMBER OF PERIODS.
[Philosophical Magazine, Vol. xxxvu. pp. 498515, 1919.]
A NUMBER (n) of points is distributed at random on a straight line of length a. When n is very great, the centre of gravity of the points tends to coincidence with the middle point of the line, which is taken as origin of coordinates. What is the probability that the error of position, that is its deviation from the origin, lies between x and x + dx ?
Divide the length a into a large odd number (2s +1) of parts, each equal to 1. The number of points to be expected on each b is nb/a. 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 nbja + £, the chance that £ lies between £ and £ + dj; is by Bernoulli's theorem
in which it is assumed that while b/a is very small, nb/a is nevertheless very great*. In the language of the Theory of Errors, the modulus, proportional to " probable error," is \f(2nb/a).
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 proportional to the distance between the middle of the part and the origin. Thus if
fc_ fc_ £_, a ...£_, £, fc, ... £
be the values of the various £'s, the coordinate x of the centre of gravity is given by
x = ^----------- 2 ~2-----------z----------..........(2)
Ğl _L C _L C _|__|_C x/
* Compare Phil. Mag. Vol. XLVH. p. 246 (1899); Scientific Papers, Vol. iv. p. 370.
402at random over the area of a circle of radius L. Wo start from (ĞU }, now taking- the form