§4 THE AMERICAN MONTHLY [June 



the actual or veritable dynamical system; and therefore 

 cannot, as it will be i^hown they do not, produce the sup- 

 posed most perfect attainable image by reversal. 

 ' 4. More specifically, it seems evident that we are, 

 ab initio, debarred from considering the light from a 

 microscopic object as consisting of uniform plane waves, 

 except on the condition of plane-wave illumination of the 

 object. (Here, indeed, we have the secret of Abbe's con- 

 sistent enforcement of illumination by a small luminous 

 cone or pencil, which gives approximately such illumina- 

 tion). For what are uniform plane waves ? A wave- 

 system is normal to the surface called the wave-surface, 

 over which undulations from the same disturbance are in 

 the same phase. Hence the plane wave arises from the 

 Huygenian spherical wave, as a limiting case, in the 

 manner pointed out by all the standard authorities. 

 Thus Lord Rayleigh says : " So long as the radius 

 of curvature [of the spherical wave] is very long in com- 

 parison, each small part of a wave-surface propagates 

 itself just as an infinite plane wave coincident with the 

 tangent-plane." Bassett puts it similarly^ — "Spherical 

 waves concentric with the source are propagated through- 

 out the medium ; and if the eff'ect which these waves 

 produce at some portion of space whose greatest linear 

 dimension is small in comparison with its distance from 

 the source, be observed, the wave may be regarded as 

 approximately plane. We are thus led to study in the 

 first instance plane waves." 



The student of physical optics knows that this is so in 

 actual fact. To study plane- wave phenomena, or to verify 

 plane-wave dimentional calculations, he must remove his 

 source of light, itself relatively small, to a considerable 

 distance from his grating or other apparatus ; he must get 

 his beams of rays approximately parallel, that the normal 

 wavC'Surface may be approxi mately plane. This neces- 

 sity belongs to the nature of plane waves. 



