502 



Lord Rayleigh on the Resultant 



A plot of the curves for these cases is given in fig. 1. 

 The ordinate represents 0(<r) and the abscissa represents a 

 itself with a taken as unity x so that the area of each curve 

 is unity. 



Fur. 1. 



In order to pass from these curves in which a is the sum 

 of the distances from one end to the representative curves 

 for the mean distance, which must lie between and a, we 

 have merely to reduce the scale of the abscissas in the ratio 

 n : 1, and to increase the scale of the ordinates in the same 

 ratio, so that the area is preserved. For instance, when n = 4, 

 the middle ordinate will be increased from 2/3 to 8/3. 



The sequence formula (6) serves well enough for the 

 derivation of the facility curves appropriate to moderate 

 values of », but it does not lend itself readily to examination 

 of the passage towards the final form wlien n is great- This 

 purpose is better attained by an adaptation of a remarkable 

 method due to Laplace *, and employed by him and by 

 Airy f for the derivation of the usual exponential formula 

 for the facility of error. Here again it will be the sum of 

 the distances of the points, now reckoned from the middle 

 of the line, that we consider in the first instance. 



The distances, instead of being continuously distributed 

 are supposed to be limited to definite values, all equally 

 probable, 



- s { )y (_, + i)6, (_ 5 + 2)/> ? .. ..-b,0,b, 2b,.... sb, 



* See Todhunter's 'History of the Theory of Probability,' p. 521. 



+ 'Theory of Error of Observation,' Macrpillan., 1861, p. 8. In a 

 comparison of the present notation with that of Laplace and Airy, the 

 symbols n and s will be seen to be interchanged. 



