132 Lord Rayleigh on Aberration 



screen, represented by the factor cos m(vt~x). Thus, when 

 2 = 0, there is no effect when x = 0, or a multiple of 2ir ; but 

 when x is an odd multiple of tt, there is a reversal of sign, 

 equivalent to a change of phase of half a period. And the 

 places where these particular effects occur travel along the 

 screen with a velocity v which is supposed to be small 

 relatively to that of light. In the absence of the screen the 

 luminous vibration is represented by 



<f> = cos (nt — kz), (1) 



or at the place of the screen, where z — 0, by 



(p = cos nt simply. 



In accordance with the suppositions already made, the 

 vibration just behind the screen will be 



cj) = cos m(vt — x) . cos nt 

 = -J cos {(n + mv)t — mx} + -|cos {{n — mv)t + mx} ; (2) 



and the question is to find what form <f> will take at a finite 

 distance z behind the screen. 



It is not difficult to see that for this purpose we have only 

 to introduce terms proportional to z into the arguments of 

 the cosines. Thus, if we write 



$ — \ cos {(n+m^-w-^J 



-f-^cos {(n — mv)t-\-mx—fjL 2 z }> (^) 



we may determine /ti,> 2 so as to satisfy in each case the 

 general differential equation of propagation, viz. 



dt 2 \dx 2 dz 2 ) W 



In (4) V is constant when the medium is non-dispersive ; 

 but in the contrary case V must be given different values, 

 say Vi and V 2 , when the coefficient of t is n + mv or n—mv 

 Thus 



(n + mv)* = Y 1 \m 2 + tf) j 



The coefficients fi l7 fx 2 being determined in accordance 

 with (5), the value of </> in (3) satisfies all the requirements 

 of the problem. It may also be written 



<t> = cos{mvt-mx~\(ji 1 --)i 2 )z} .cos {n£_i(^ 1+ ^z}, (6) 



