Dr. G. J. Stoney on Microscopic Vision, 437 



of the objective field, or from either of its " standard images/' 

 Draw what we have called the axial rays of these. They are 

 lines radiating from the middle of the objective field, and 

 each perpendicular to the waves of its own beam. Take a 

 group of these axial rays which lie within a definite cone, 

 then the corresponding beams may be called a sheaf of beams; 

 and where the cone within which the axial rays are confined 

 is a very acute one, the corresponding beams may be called 

 an elementary sheaf of beams. The whole of the beams 

 emitted by the objective field, or from either of its standard 

 images, may obviously be conceived of as divided up into 

 elementary sheafs of any required degree of minuteness. 



If we only have to deal with an image of limited extent 

 like standard image No. 1, or standard image No. 2 (which 

 are the same size as the objective field), then we are justified 

 in substituting a single beam travelling along the axis of the 

 cone for each elementary sheaf of beams. This may be proved 

 as follows : — 



Let U be one of the beams whose axial ray lies within the 

 elementary cone, and let 6 be the angle between that axial 

 ray and the axis of the cone. The cone, of course, has its 

 vertex at the centre of the objective field. Let now V be an 

 equivalent beam whose axial ray lies along the axis of the 

 cone, and let the phase of V be such that U and V are in the 

 same phase at the centre of the objective field, Then, as in 

 § 15, let —V mean the same beam as + V, only with it added 

 to all its phases. Accordingly, if + V and — V are simulta- 

 neously present they cancel one another absolutely. We may 

 therefore add both of these to the elementary sheaf of beams 

 without altering it. Now — V and U would produce a ruling 

 which will be the coarser, i. e. with its luminous bands more 

 widely spaced, the smaller the angle 6 is. Moreover, since 

 + V and U are in the same phase at the vertex of the cone, 

 which is also the centre of the objective field, it follows that 

 one of the minima of illumination of the ruling produced by 

 — V and U will occupy that position. Now by making 

 sufficiently small, the spacing of this ruling may be made so 

 many times larger than the objective field that there is no 

 appreciable illumination anywhere within the limits of the 

 objective field. If this be so, we may suppress the beams 

 — V and U without producing appreciable change within the 

 limits of the objective field. When this is done, the elemen- 

 tary sheaf of beams differs from what it was at first by having 

 -f-V now in the place of U. By a similar process we may 

 substitute V, V", &c. travelling along the axis of the elemen- 

 tary cone for the other beams whose axial rays lie within the 



Phil. Mag. S. 5. Vol. 12. No. 258. Nov. 1896. 2 I 



