430 Lord Rayleigh on the Light emitted fi 



rom a 



number (n) of unit vibrations of arbitrary phases. It is 

 known that the "expectation" of intensity in any direction is 

 n times that due to a single centre, or (as we may say) is 

 equal to n. The word " expectation " is here used in the 

 technical sense to represent the mean of a large number of 

 independent trials, or combinations, in each of which the 

 phases are redistributed at random. It is important to 

 remember that it is infinitely improbable that the expectation 

 will be confirmed in a single trial, however large n may be. 

 Thus in a single combination of many vibrations of arbitrary 

 phase there is about an even chance that the intensity will 

 be less than 'In. The general formula is that the probability 

 of an amplitude between r and r + dr is 



n n 



if I denote the intensity *. 



As regards the " expectation " of intensity merely, the 



question is very simple. If 6, 6\ 0" be the n individual 



phases, the expectation is 



rt2v f*2ir r2ir 

 Jo JO i 



dd d6 f dO" r/ n a , v 



•o- o— ^— •••• [ (COS 0+COS0' + ....> 



+ (sin0 + sin<9' + .,..) 2 ]. 



Effecting the integration with respect to 0, we have 



r 2 -f- 9 - dO' d0" ri , nl 

 J J ..•.^^-....[l+(cos^ + cosr + ....) 2 



-f-(sin<9' + sin6>" + ....) 2 ]; 



and when we continue the process over all the n phases we 

 get finally 



Expectation of Intensity = n. 



The same result follows of course from (1). The "ex- 

 pectation " is 



j: 



e-^ n l.dl/n = n (2) 



But if we are not to expect any particular intensity when 

 a large number of vibrations of unit amplitude and arbitrary 



* An interesting example of variable intensity when phases are at 

 random is afforded by the observations of L)e Haas (Amsterdam Pro- 

 ceedings, vol. xx. p. 1278 (15)18)) on the granular structura of the held 

 when a corona is formed from homogeneous light. The results of various 

 combinations are exhibited to the eye simultaneous! v. 



