84 Prof. Powell on the Theory of the dark Bands 



the emergent pencil will originate from a geometrical focus, 

 which is not a single point., but a caustic^ whose convexity is 

 towards the edge of the prism. Hence, on the principles of 

 the wave-theory, it follows that the side of the pencil which 

 lies towards the edge of the prism is that which undergoes 

 less retardation, or has the shorter undulatory route; this dif- 

 ference varies slightly for the different primary rays. 



I had arrived at this conclusion by a different approximate 

 method, when in some correspondence the Astronomer Royal 

 pointed out to me the above view of the problem, as con- 

 nected with his investigations " on the light in the neigh- 

 bourhood of a caustic." (Camb. Trans., vol. vi. part 3*.) 



(13.) With respect to the interference-spectrum, we have 

 only to follow out the investigation given in Mr. Airy's tract 

 (Arts. 80, 83.) (as that gentleman has suggested to me) in the 

 following manner. 



Taking the focus as the origin and the axis of the object- 

 glass as the axis of x> let x y be the coordinates of any point 

 in the wave, a b those of a point in the focal image on the 

 same side of the axis, then the radius of the wave being c, 

 we have 



c~ = x* + y* ; 



and expanding and neglecting powers of y above the third, 

 we find 



7/ 

 X = C ^ . 



2c . 

 The distance from xy to a b will be 



9 = ^TO*-") 2 + (3/-) 2 ) 5 



here performing the various expansions, and for brevity 

 writing e 2 = (c #) 2 -f 2 , we at length obtain (going to the 

 third power of y] 



e x 



The terms involving the second power of y have the same 

 value on each side of the axis, and those depending on the 

 third power are found to be 



[* See Lond. and Edin. Phil. Mag. vol. xii. p. 452. EDIT.] 



