450 Transactions of the Society. 



the co-operation at B of the first lateral spectrum is thus that the 

 angle of diffraction do not exceed the semi- angular aperture a. 

 By elementary theory we know that the sine of the angle of 

 diffraction is X/e, so that the action of the lateral spectrum requires 

 that e exceed \j sin a. If we allow the incidence upon the grating 

 to be oblique, the limit becomes £A,/sin a, as in (1). 



We have seen that if one spectrum only illuminate B, the field 

 shows no structure. If two spectra illuminate it with equal 

 intensities, the field is occupied by ordinary interference bands, 

 exactly as in the well known experiments of Fresnel. And it is 

 important to remark that the character of these bands is always 

 the same, both as respects the graduation of light and shade, and 

 in the fact that they have no focus. When more than two 

 spectra co-operate, the resulting interference phenomena are more 

 complicated, and there is opportunity for a completer representa- 

 tion of the special features of the original grating.* 



While it is certain that the image ultimately formed may be 

 considered to be due to the spectra fooussed at S , S x . . ., the 

 degree of conformity of the image to the original object is another 

 question. From some of the expositions that have been given 

 it might be inferred that if all the spectra emitted from the grating 

 were utilised, the image would be a complete representation of the 

 original. By considering the case of a very fine grating, which 

 might afford no lateral spectra at all, it is easy to see that this 

 conclusion is incorrect, but the matter stands in need of further 

 elucidation. Again, it is not quite clear at what point the utilisa- 

 tion of a spectrum really begins. All the spectra which the 

 grating is competent to furnish are focussed in the plane S S x ; 

 and some of them might be supposed to operate partially even 

 although the part of the image under examination is outside the 

 geometrical cone defined by the aperture of the object-glass. For 

 these and other reasons it will be seen that the spectrum theory, f 



* These effects were strikingly illustrated in some observations upon gratings 

 with (J000 lines to the inch, set up vertically in a dark room and illuminated by- 

 sunlight from a distinct vertical slit. The object-glass of the Microscope was a 

 quarter inch. When the original grating, divided upon glass (by Nobert), was 

 examined in this way, the lines were well seen if the instrument was in focus, but, 

 as usual, a comparatively slight disturbance of focus caused all structure to dis- 

 appear. When, however, a photographic copy of the same glass original, made with 

 bitumen, was substituted for it, very different effects ensued. The structure could 

 be seen even although the object-glass were drawn back through \h in. from its 

 focussed position ; and the visible lines were twice as close, as if at the rate of 12,000 

 to the inch. The difference between the two cases is easily explained upon Abbe's 

 theory. A soda flame viewed through the original showed a strong central image 

 (spectrum of zero order) and comparatively faint spectra of the first and higher 

 orders. A similar examination of the copy revealed very brilliant spectra of the first 

 order on both sides, and a relatively feeble central image. The case is thus approxi- 

 mately the same as when in Abbe's experiment all spectra except the first (on the 

 two sides) are blocked out. 



t The special theory initiated by Prof. Abbe is usually called the " diffraction 

 theory," a nomenclature against which it is necessary to protest. Whatever may be 



