362 Transactions of the Society. 



calculation— the resulting positions of B and P, which are adjacent 

 bright lines both in the real image of the grating and in the hypo- 

 thetical diffraction image which is supposed to occupy the same field. 

 In the first place, therefore, it is necessary to determine the distance 

 separating a from a and a" respectively. For this purpose our 

 authors assume that the distance is measured by the sine of the 

 angle of diffraction, and is given by the expression a - a' = a - a" 

 = sin a .f, where a stands "for the angle of diffraction and /for the 

 principal focal length of the objective (pp. 233, 228, and 25). The 

 determination is quite arbitrary, for an earlier diagram by which they 

 illustrate the course of the diffracted beam shows this distance as equal 

 to the tangent, not to the sine, of this angle of diffraction. In ex- 

 planation of this discrepancy they cite the authority of Professor Abbe, 

 who has laid down the proposition that, " when an optical system is 

 completely aplanatic for one of its focal points, every ray emerging 

 from this point meets a plane drawn through the other focal point at 

 a distance from the axis, the linear magnitude of which is equal to the 

 product of the equivalent fecal length of the system and the sine of 

 the angle which this ray makes with the axis." But, curiously 

 enough, the very terms of Abbe's theorem exclude the particular case 

 to which they have applied it. It is expressly limited to a focal point 

 for which the instrument is completely aplanatic. Now, no Micro- 

 scope is made aplanatic for infinite distance. It is corrected for a 

 focal point on the stage, and therefore cannot be completely aplanatic 

 in the upper principal focal plane of the objective where the diffrac- 

 tion spectra appear. The whole calculation, therefore, is vitiated by a 

 fundamental error ; for the one thing that Professor Abbe's proposition 

 settles about the images a and a and a" is that the distance between 

 them will not be what Naegeli and Schwendener have chosen to 

 assume that it is. 



Having thus determined the position of their luminous points, they 

 next proceed to lay the foundation for a calculation of the inter- 

 ference phenomena to which they must give rise, by the following 

 extraordinary postulate. They say (p. 232) : " We now come to our 

 proper task, viz. to establish the effect which these diffraction 

 phenomena produce in the plane of the real image. This may be 

 simply done if we consider the aperture images in the upper focal 

 plane, the direct one as well as those due to interference, as so many 

 (secondary) sources of light whose rays interfere, as in Fresnel's 

 experiment with the mirror. For, since these sources of light are 

 point for point the optical images of the same primary source of light, 

 there is no difference of phase between them." " No difference of 

 phase between them " ! Surely no more extraordinary mistake was ever 

 made by a competent author. For why, assuming them to lie in a 

 true plane, should there be any correspondence of phase between them ? 

 The wave-fronts passing this point are not plane wave-fronts, but 

 spherical wave-fronts focussed on the point IB. It is clear therefore 



