61 C Transactions of the Society. 



I shall further assume that these gratings are examined with an 

 optically perfect instrument. The latter assumption — which may 

 be considered closely fulfilled when the best of modern objectives 

 are used with about f of their total aperture — implies that all 

 " rays " from a point in the object are united in the conjugate 

 point ; or — to express it according to the undulatory theory, and 

 therefore more correctly — this assumption implies that all optical 

 paths uniting a point in the object with its conjugate point in the 

 image, are equal, and therefore that " rays," or, more correctly 

 speaking, plane wavelets, arrive at a point in the image in the 

 same phase-relation in which they left the conjugate point in the 

 object. This last simple and obvious deduction will be of the 

 greatest assistance in my inquiry. 



The explanation of the diffraction spectra themselves is given 

 in numerous books, and I shall therefore assume it as well known 

 that when a plane wave strikes a grating of straight, parallel and 

 equidistant slits, it is broken up into a number of "diffracted" 

 plane waves, and that when a is the angle between the arriving 

 wave and the grating, /3 that between a diffracted wave and the 

 grating, d the spacing of the grating, i.e. the distance from centre 

 to centre of the slits, X the wave-length of the light employed, and 

 X the " order " of any spectrum, we shall have the relation 



I. <% . \ = d (sin a + sin /3), 



the upper sign to be taken when the diffraction-spectrum lies on 

 the opposite side of the optical axis to that occupied by the direct 

 light, the lower sign when both are on the same side of the 

 optical axis ; this formula being the mathematical expression of 

 the fact that in the first spectrum the wavelets proceeding from 

 adjoining slits meet with a difference of phase equal to one wave- 

 length, and in any spectrum of higher order with a difference of 

 as many wave-lengths as the number of the spectrum indicates. 



But the knowledge embodied in this formula is not sufficient 

 for our purposes. The phase-relation between the direct light and 

 the several diffracted waves enters into our problem ; and I proceed 

 to discuss this, the first novel point of importance which I am 

 going to raise. 



Assuming a grating of extremely narrow slits, no question can 

 arise, as it is obvious that all the diffracted waves leave points in 

 the grating in phase with the arriving wave, and remain in that 

 phase-relation, as they are not exposed to any further interferences 

 — the wavelets from successive slits being too small to have any 

 difference of phase within themselves, and joining up with a 

 difference of whole wave-lengths, and therefore in the same phases. 

 Hence, in such a grating all diffracted waves depart from points in 

 the slits in equal phase, and consequently arrive in equal phase 



