526 HENRY A. ROWLAND 



the intensity of the light by multiplying the final result by itself with 

 i in place of -f- i, because we have 



(A + iB} e~ ib (p ~ rt) x(A iff) e' b (f ~ rt] = A* + &, 



In cases where a ray of light falls on a surface where it is broken 

 up, it is not necessary to take account of the change of phase at the 

 surface but only to sum up the displacement as given above. 



In all our problems let the grating be rather small compared with 

 the distance of the screen receiving the light so that the displacements 

 need not be divided into their components before summation. 



Let the point x' , y', z' be the source of light, and at the point x, y, z 

 let it be broken up and at the same time pass from a medium of index 

 of refraction I' to one of /. Consider the disturbance at a point x", y" , 

 z" in the new medium. It will be 



where 



S = x"* + y' n + z m + a? + f + z* - 2 (xx" + yy" + zz") , 

 p 2 = a/ 2 + y'* + z '* + x * + f + z* 2 (xx' + yy' + zz') . 



Let the point x, y, z be near the origin of co-ordinates as compared 

 with x', y', z' or x" ', y" , z" and let a, /?, f and a!, fl, f be the direction 

 cosines of p and p. Then, writing 



R = I' V of* + y" + z' 2 + /Vz" 2 + y m + z"*, 



/I = la + I'a', 

 p. = 7/3 + /'/?, 



we have, for the elementary displacement, 



g ib [ R Vt AZ - ny vz + /tr j ] 



1 

 n ] 



where _ , [~ _ P_ _ _ / 



L V z" + y" + z' z + V x"' + y" 3 + z' 

 and r 2 = z 2 + y* + z\ 



This equation applies to light in any direction. In the special case 

 of parallel light, for which * = 0, falling on a plane grating with lines 

 in the direction of z, one condition will be that this expression must be 

 the same for all values of z. 



Hence v = . 



If N is the order of the spectrum and a the grating space we shall 

 see further on that we also have the condition 



