THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[FIFTH SERIES.] 



AUGUST 1880. 



XII. On the Resultant of a large Number of Vibrations of the 

 same Pitch and of arbitrary Phase. By Lord Rayleigh, 

 F.R.S., Professor of Experimental Physics in the University 

 of Cambridge* . 



YERDETf, in an investigation upon this subject, has 

 arrived at the conclusion that the resultant of n vibra- 

 tions of unit amplitude and arbitrary phase approaches the 

 definite value Vn when n is very great. It can be shownj, 

 however, that this conclusion is inaccurate, and that the result- 

 ant tends to no definite value, however great the number of 

 components may be. 



But there is a modified form of the question, which admits 

 of a definite answer, and was perhaps vaguely before Verdet's 

 mind. If we inquire what is the average intensity in a great 

 number of cases, or, in the language of the theory of probabi- 

 lities, what is the expectation of intensity in a single case of 

 composition, we shall find that the result is that assigned by 

 Verdet, namely n. 



A simple but instructive variation of the problem may be 

 obtained by supposing the possible phases limited to tivo oppo- 

 site phases, in which case it is convenient to discard the idea 

 of phase altogether, and to regard the amplitudes as at ran- 

 dom positive or negative. If all the signs are the same, the 

 resultant intensity is n 2 ; if, on the other hand, there are as 

 many positive as negative, the result is zero. But although 

 the intensity may range from to ft 2 , the smaller values are 

 much more probable than the greater ; and to calculate the ex- 



* Communicated by the Author. 



t Legons cl Optique physique, t. i. p. 297. 



\ Math. Soc. Proc. May 1871. 



Phil. Mag. S. 5. Vol. 10. No. 60. Aug. 1880. G 



