10' 



-m 



of a Number of Unit Vibrations. 509 



of entire revolutions, the phase of the resultant is arbitrary. 

 But if the arc differs, even a little, from an integral number 

 of revolutions, there is a definite phase favoured for die 

 resultant, and Mod 2 ® diminishes as n increases. 



The case where the arc consists of entire revolutions is 

 exceptional also as regards the amplitude, or intensity, of 

 the resultant. As we know, in that case no definite value 

 is approached, however great n may be, and the expectation 

 of intensity is n. But if there be a fractional part of a 

 reA'olution outstanding the intensity does tend to a definite 

 value, that namely which corresponds to a uniform distri- 

 bution over the arc, and this value is proportional to the 

 square of n. 



We may go further and calculate what exactly is the 

 expectation of intensity. We have to evaluate 



d0 d0 ad r , n . n , . nn N9 

 . . . T (cos 6+ cos 6 + cos + . . .) 2 



a ol a. 



+ (sin<9+sin6' / +sin<9" + ...) 2 ] 



Wd0'd0"...[n + 2 cos (0-0') 



+ 2 cos {0-0") +....+ 2 cos (0'-0") +-.'..], (26) 



the integration being in each case from — \a to + -^a. 

 Taking first the integration with respect to 0, we have 



1 \ + *° d0[n + 2 cos (0-0') + 2 cos (0-0") + .... 



+ 2cos(0'-0")+...] 

 = 4a" 1 sin i*{cos 0' + cos 0" + . . . } + n + 2 cos (0' — 0") + 



On continuing the integration the first part yields finally 

 8(n — l)a~ 2 sin 2 \ol ; 



while the remaining parts give the original terms over 

 again with omission of those containing 0. Thus 



Expectation of intensity 



= ?i + 3a- 2 sin 2 i«{/i-l + tt — 2 + ?2 — 34- +1} 



= n+4n(n-l)*- 2 sin 2 iu (27) 



If a = 0, this becomes ?i 2 , as was to be expected. If oc = 2tt^ 

 or any multiple of 'lir, the expectation is n, as we knew. 

 In general, when a becomes great, so as to include many 

 complete revolutions, the importance of the n 2 part decreases. 

 In (27) n may have any integral value. 



