440 Mr EARNSHAW, ON THE DIFFRACTION OF AN 



3. Let us now examine the part of the screen which is inter- 

 mediate to two rays. The brightness of this part will be obtained by 

 writing 30° for 6 in equation (B) ; which by that means becomes 



or taking the central brightness as the unit 



. m 



(sin - 

 — 

 m 



2 



When /« = 0, Z = \; and as m increases Z diminishes ; at first 

 rather slowly, but afterwards rapidly, so that when 



m ?« -v/3 .„ 2l7r - 



- =. ^, or — ^— = '^ a/3 = ^^ nearly, 



there is perfect blackness ; some time before this, however, the light 

 will be too feeble for vision, and there the light on the screen will 

 appear to terminate unless the star or original luminous point of light 

 be very bright, for as m goes on increasing, Z never attains a value 



so great as ^. 



From this it follows that if there be a black ring surrounding 

 the central disc of light as described by Sir J. Herschel, its radius 



will be such that — ^ — = — — nearly. By reference to the above 



Table, we perceive that the six rays have, at that distance from the 

 centre, a brightness of about — , the central brightness being represented 



till. 



by unity. 



If this be considered sufficiently feeble to constitute a black ring, we 

 are at a loss to account for the prolongation of the six bright rays men- 

 tioned by Sir J. Herschel, since their intensity has been shewn to decrease 



