232 Transactions of the Societtj. 



I add a few remarks about the mathematical side of the theory, 

 of which I have stated only the point of view in my paper. I 

 think you will quite understand the principle of my mathematical 

 deduction by considering the simplest case — one set of equidistant 

 lines — and observing the following notices : — 



(1) Be h the distance of the lines in a microscopical object ; 

 \ the wave length for one definite colour (Fig. 3) ; be Uq the angle 

 of incidence, in which a ray meets the grating; U_2, U_i, Uo> 

 U+i, U+2, the angles formed by the several diffracted rays —2, 

 — 1, 0, 4-1, +2, the direct ray (0) included. The perpendicular 

 line taken as zero-direction ; there is 



sin 0+2 — sin U + i = sin U + i — sin Uo = sin Uo — sin U_i = -^, 



o 



in which formula the case of normal incidence is included, of 

 course. If now the diffracted rays enter a microscope, the sines 

 of the angles' of any two consecutive rays with the axis of the 



microscope have the same difference = -^ . 



(2) If an objective is focnssed to the grating (Fig. 4), and if 

 this objective is perfectly* aplanatic for its focal point, any ray 

 forming an angle u with the axis below tlie objective is refracted 

 in such a way, that it will pass the upper (the back) focal-plane 

 of the lens in a linear distance from the axis 



A = / sin u 



if/ is the focal-length of the objective (by a theorem enounced by 

 me and by Mr. Helmholtz). From this theorem (2) in connection 

 with (1) is to be inferred : the linear distance of the diffraction- 

 spectra, which appear in tlie back-focal-plane of the objective, is 



always = ^-/, if corresponding points in every two consecutive 



spectra are considered — independent of the inclination of the 

 incident rays to the grating. If you go from central light to 

 oblique light, all the spectra move within the back-plane of the 

 system, without changing their relative position. 



(3) All the rays, which result by diffraction, from one incident 

 ray, have their oscillations in equal phase, if points are compared 

 on these rays which are situated in the back-focal plane, where 

 the spectra are formed as images of the illuminating object ; all 

 tliose raya therefore most interfere within the plane, where they 

 meet — that is the plane, where an image of the grating is formed 

 by the objective (the conjugate focus to the microscopic object). 



(4) (Fig. 5). If A be the linear distance of the two interfering 

 rays in the back-focal-plane, / the distance of the conjugate focus 



* "Perfectly aplanatic" means: without spherical aberration not only for 

 the one point on the axis, but for the points aside the axis too. 



