Dr. Gr. J. Stoney on Microscopic Vision, 335 



5. The following important optical theorem may now be 

 enunciated, which in its generality compares with Fourier's 

 Theorem, of which it is, in fact, in ultimate analysis, an 

 extension. 



Proposition 1. 



However complex the contents of the objective field, and 

 whether it or parts of it be self-luminous or illuminated in any 

 way, however special, the light which emanates from it may be 

 resolved into undulations each of which consists of uniform 

 plane waves ; on the hypothesis that each point of the object 

 emits continuously the same light : an hypothesis the suffi- 

 ciency of which will appear in Part II. of this memoir. 



By an undulation is meant a succession or train of waves, 

 and by a uniform wave is meant one which is at each instant 

 alike in every part of each wave surface. 



6. To prove this theorem we proceed very much in the 

 same way as in dealing with Fourier's Theorem. We begin 

 by positing repetitions of the objective field. For this pur- 

 pose let a plane be drawn through some point of the objective 

 field, and preferably perpendicular to the line of sight. This 

 plane may be called the Objective Plane. Let a square be 

 drawn in this plane which may be of any size, provided that 

 it shall include within it the projection upon the plane, from 

 the point of view of the observer, of the contents of the ob- 

 jective field : in other words, the square is to be large enough 

 for the whole of the objective field — the whole of what the 

 observer can see — to fall within that square, and preferably 

 well within it. Divide the whole plane up into squares of 

 this size by two systems of equidistant parallel lines, and 

 imagine an exact repetition of the contents of the objective 

 field to occupy the position relatively to each of these except 

 the first, which is the same as the position actually occupied 

 by the contents of the real objective field in reference to the 

 first square. Next suppose light to be emitted from every 

 point of each of these replicas, which is at each instant similar 

 in every respect — i. e. the same in direction, intensity, phase, 

 and position of transversal — as is the light from the cor- 

 responding point of the original objective field at that 

 instant. 



Under these circumstances a point p in the original objec- 

 tive field, along with the corresponding points p ! p" &c. in 

 the replicas of the objective field, form a system of points 

 equally spaced over a plane which is parallel to the objective 

 plane. Now it is known, from the theory of diffraction 

 gratings (see the figure on p. 340), that such a system of 



