506 Lord Rayleigh on the Resultant 



should be observed tliat in virtue of the exponential factor 

 only moderate values of nx 2 /a 2 need consideration. 



As a check upon (19) we may verify that it becomes 

 unity when integrated with respect to x between and <x> . 

 Starting from 



v/©f---' 



dx = 1. 



and differentiating with respect to u, we get 



~Tn I x 2 e ux 'dx = 1 , 

 and differentiating again 





Using these integrals in (19) with a = l, ?* = 6?i, the 

 required verification follows. 



The above verification suggests a remark which may have 

 a somewhat wide application. In many cases we can foresee 

 that a facility function will have a form such as Ae ~ ux *dx y 

 and then, since 



a( V (,t ^=l, 

 Jo 



it follows that A = 2\Z(w/7r). According to this law, the 

 expectation of x is zero, but the expectation of x 2 is finite, 

 if we know this latter expectation, we may use the know- 

 ledge to determine u. For 



ex 



pectation of o? 2 = i2 v / '{v/tt) \ x 2 e~ UJ '\lx = l/2u. 



We may take an example from the problem, just con- 

 sidered, of the position of the centre of gravity of points 

 distributed along a line. If x lf a? 2 , . . . x n be the coordinates 

 of these points reckoned from the middle and x that of the 

 centre of gravity, 



t . . dx x dx« . . . dx n Cri 4-Xo 4- ... + x n Y 



Mear 



%*-m 



the integrations being in each case from —\a to +-Ja. 

 Taking first the integration with respect to x n , we find that 



a 2 

 Mean x 2 = —j-; 2 + the corresponding expression with x n 



12n omitted. 



