Some Ahhe Letters. 



235 



drop a perpendicular, and from the foot of this set oft' distances 

 1, 2, etc., in both directions, if possible, of the diameter equal 



successively to —^, " — ^, etc., where \q is equal to 



the 



wave- 

 n.s n.s' 



length of the light in air, and s the distance apart of the lines of 

 the grating. Erect from the points thus determined perpendiculars 

 intersecting the semicircle. Lines drawn through these inter- 

 sections from the centre of the circle give the directions of the 

 first, second, etc., diffracted beam. 



Proof. — Draw dotted lines parallel to the incident and first 

 diffracted rays, making the distance ec = s; then the lines e h and 

 d c, drawn at right angles to the incident and diffracted rays, 

 represent respectively an incident and the corresponding diffracted 

 wave-front, and the retardation (K), which is equal to the wave- 

 length — for the first order spectrum (and ? for the mth order) 



n n 



is equal io h c — d e. But & c = 5 . sin a, and d e = s. sin (3, so that 

 •^ = (sin a — sin j3), but sin a = 1 + c 1, and sin /3 = e 1, 



E = 



71. S 



thus finally 



and for the mth order 



= 



Xq 



n.s 



m = 



m.X-fl 

 n.s 



Q.E.D. 



Assuming with Fraunhofer that fine structure cannot be seen 

 unless the diffracted light produced by such structure is utilized 

 for producing the image ; and further, assuming with Abbe that 



