Interference Phenomena. 527 



/* being the height of the slit. If this is integrated between 

 infinite positive and negative values of x, the result is found 

 to be XF/h. This is the quantity by which we must mul- 

 tiply our expression if the whole energy passing through 

 the slit is required. As regards the width of the slit we 

 connect in the first place X and j3 by the relation 



\= — (sin j3 — sin a), 

 and hence 



dX = — cos S dS. 

 q 



F being the focal length of the telescope, the interval 

 separating the principal maxima of the two wave-lengths 

 X and dX of the slit will be 



Fdfi=FgdX/(2w cos j3). 

 Hence 



E =AFqn\dX/(27rh cos /3) 



= &ghl\ cos /3dX/{2iry), 



and as Xq = 27ry, 



E = Bhlcos /3dX = ±BsdX, 



where s is the effective surface of the telescope, and BdX the 

 energy per unit surface within the limits specified by dX. As 

 was shown previously for a single homogeneous vibration, the 

 energy contained in the first-order spectrum is one quarter 

 of that which would reach the focus of the telescope if the 

 grating were replaced by an ordinary reflecting surface. 



It must of course be remembered that in the last case we 

 have been dealing with a succession of two sets of waves for 

 each of which the energy per unit surface has been put equal 

 to BdX. 



Returning to the smaller retardation for which the spectrum 

 is crossed by dark and bright bands, we may conveniently 

 write the energy passing through the slit between the wave- 

 lengths X and X + dX 



E = i ^ [D + 2( ^_ D) co^D ]; 



where D stands for the retardation expressed as a length. 

 From this we obtain 



■pi T) 



-'-'(minimum) _ ■ L ' 



E(maximum) ±NX — D 



2 02 



