508 Lord Rayleigh on the Resultant 



To return to the theoretical question, if we suppose the 

 circular arc to be very small, we see that the probability of 

 various phases of the resultant, within the narrow limits 

 imposed, follows the laws determined for the centre of 

 gravity of points distributed at random along a straight line. 

 In this case the amplitude of the resultant is n to a high 

 degree of approximation, n being the number of unit 

 components. 



But when the circul ir arc (a) is so large that sin a deviates 

 appreciably from a, the question is materially altered. We 

 may, however, frame an argument on the lines followed in 

 equations (1) and (2). Thus with a replacing a and /3 

 replacing 6, we have for the resultant whose amplitude is R 

 and phase (reckoned from the middle) (B), 



R sin ©== sin 04ft- f _0 + sin 2/3 . (ft-ftj 



+ .... + sin s/3.(ft-ftj. .... (20) 

 Rcos©= cos/3 . (ft + ftj + ...,+ cos s/3. (ft + ?_ s ). ( 21 ) 



Here (*) is a small angle, whose probability is under con- 

 sideration, but R is in general large and may then be 

 reckoned as if the distribution were uniform. Thus 



R- --( J,a cos/3^/3=(2?V«)sinJa. . . (22) 

 u Jo 

 and 



e= ray« {si " ^^ + • • • • + sin *& " f -» )} - (23) 



By the rules of the Theory of Errors, we have 

 Mod 2 "© a 2 



In (24) Mod 2 f = 2npj<x, as before, and the series of (sin) 2 

 may be replaced by 



Thus , . . , , ■ . . ; 



Mo.P0= a 4^=-^- • • • • (25) 



If a is small, this reduces to a 2 /6n, as in (4). If a = 7r, 

 that is if the distribution be over a semicircle, w T e get 7r 2 /4n. 

 If we make a==27r in (25). the result is indeterminate, 

 since although sinja = 0, n is infinite. There is a like 

 indeterminateness when a is any multiple of 27r, and this 

 was to be expected. When the arc of distribution consists 



