388 Transactions of the Society. 



light to a focus we shall at the focal point have an image of the 

 source of light 250 times as bright as the image which could be 

 obtained in that position from the full aperture. It is not only 

 at this point that the grating brightens up the diffraction pattern 

 of the full aperture. Take, for example, the axis along which 

 each of the open facets would transmit three half-sets of phases, 

 two positive half-sets, say, and one negative. Then the blocked- 

 out facets will suppress the alternate triplets containing each two 

 negative half-sets and one positive. Thus, along this axis the 

 positive half sets will be in the proportion of 2 : 1 as compared 

 with the negative half-sets, and there being 250 of them added 

 together at the new focal point they will make up a conspicuous 

 image in the shadow of the full aperture. 



It is, however, to be noted that the increase of light within the 

 geometrical shadow of the aperture is paid for by a diminution of 

 light within its optical projection. For the blocking-out of half 

 the facets formed upon the face of the aperture will have pro tanto 

 diminished the directly transmitted light and so reduced the 

 brightness at the geometrical focus by one-half. In like manner 

 the diffracted beams, which being diffracted at very small angles 

 come to focus very near to the geometrical focus, will suffer each 

 in its own proportion ; in fact every beam which has a maximum 

 aperture value greater than the diameter of the full aperture will 

 be reduced in brightness by the placing of a diffraction grating 

 across it. The other beams, having a less aperture value than this, 

 will be brightened or darkened or left unaffected as the case may 

 be, according to the numerical relation between their aperture 

 values and the aperture values of the transparent zones in the 

 grating. 



This, in outline, is the theory of diffraction and the diffraction 

 grating. I have troubled you with it thus at length because it is 

 of essential importance that it should be in your minds when you 

 proceed to the discussion of the Helmholtz theory. But it will, of 

 course, be understood that Helmholtz himself does not develop 

 the theory of diffraction in his paper. On the contrary, lie takes 

 it all for granted, and writes as abstrusely about it as the most 

 hardened mathematician. He does not even pause to prove that 

 polyphase fronts, as we have seen, are propagated, refracted, 

 focussed, and reflected precisely like wave-fronts. These things 

 are clear enough to the mathematician who is certain that a par- 

 ticular formula accurately expresses a particular phenomenon. 

 But to readers with more turn for the physics than for the mathe- 

 matics of the explanation, the proof is grateful or even necessary. 



So far we have considered only the law of diffraction from 

 plane wave-fronts, but in the Microscope and all other image- 

 producing instruments we have to deal with spherical wave-fronts, 

 and the law of diffraction as applied to them becwhes of paramount 



