308 BELL SYSTEM TECHNICAL JOURNAL 



diffraction-grating, or an echelon, or past a straight-edge or a solid disc 

 or any small obstacle. Having made it, one proceeds to determine 

 the value of s at any point beyond the screen by integrating Kirchoff's 

 integrand all over the apertures — all over the vacant places of the 

 screen, as if in those places only the wave- front were intact, and else- 

 where it were abolished; as if the wave-front were cut into the pattern 

 of the apertures as by a template, and each of the segments thence- 

 forth propagated according to the law of wave-motion. 



This method is fairly easy to apply, at least when the contours of 

 the openings are simple geometrical figures, circles or rectangles for 

 instance. In practice it is used almost always; and its results are 

 ratified by experience. Yet it is not quite accurate.^ I am not re- 

 ferring here to the ever-present possibility that boundary-conditions 

 chosen for their mathematical simplicity may not properly describe 

 the actual conditions in the physical world. I am referring to a math- 

 ematical, that is to say, a logical, difficulty which is inevitably fatal. 

 A function which is equal to cos {nt — mx) over arbitrary patches of 

 the plane x = 0, but is always equal to zero over the remainder of the 

 plane — such a function does not conform to the wave-equation (6). 

 If in an actual case of light passing through a hole in a screen the phe- 

 nomena conform everywhere to the wave-equation, then the boundary- 

 condition (70) cannot be valid; if on the other hand the state of affairs 

 in the apertures is rightly described by (70), then the wave-equation 

 cannot be valid and Kirchhoff's theorem cannot be applied. 



There is no way out of this dilemma. One must either accept the 

 foregoing assumption frankly as an approximation, or else undertake 

 the vastly more difficult problem of solving the wave-equation itself 

 with the boundary-condition 5 = (or some other which is deemed ap- 

 propriate) for all the opaque area of the screen, and with other bound- 

 ary-conditions at infinity to settle the direction from which the light 

 is supposed to come. By this method one makes no assumption about 

 the values of 5 in the apertures; they are part of the solution. But 

 the general problem is formidable, and no less eminent a man than 

 Sommerfeld was required for the solving of even the simplest conceiv- 

 able case. Later I will mention his solution of the case in which the 

 barrier covers all that part of the plane .r = which lies to one side of 

 a straight line, the y-2cds for instance, and the remainder of the plane 

 is void. Meanwhile, since on the whole the second approximation is a 

 close one, I will adopt it to explore these effects of diffraction which 

 first invited the wave-theory of light, as they now are inviting that of 

 matter; which serve to determine wave-lengths of light, and of matter; 



* In some of the texts on optics this is not made sufficiently clear. 



