The TJicory of Optical Images. By Lord Rayleigh. 477 

 sin 2 u , s in 2 (u -f- |- tr) , sin 2 (w - -l tt) , 



•or by 



4- cos-u 



tt) 2 (w-1tt)* (w + |tt) 2 



The value of the first series has above been shown to be unity, 

 and by a like method the same may be proved of the second. 

 The illumination for all values of u is thus equal to 2. That it 

 should be twice as great as before might have been expected. 



But my principal object at present is to consider the problem, 

 suggested by Mr. Gordon, of a dark line of finite width upon a 

 uniformly bright ground. The problem assumes two forms accord- 

 ing as the various parts of the ground are supposed to be self- 

 luminous or to give rise to waves which are all in one phase. 

 The latter is the case of an opaque wire or other linear obstacle 

 upon which impinge plane waves of light in a direction parallel 

 to the axis of the instrument (telescope or Microscope), and as it 

 is somewhat the simpler we may consider it first.* 



In (28) we have the expression for the resultant amplitude at 

 any point u due to a series of points or lines, whose geometrical 

 images are situated at u = 0, u = ± v, u = ± 2 v, &c. If all 

 values of u are equally geometrical images of a uniformly bright 

 .ground of light, we have to consider 



r 



+ x sin u 



u 



du = tt (78) 



At present we suppose that the bright ground is interrupted at 

 points corresponding to u = a, n — — a, so that 2 a represents the 

 width of the geometrical image of the dark obstacle. The amplitude 

 .at u is the same for a given numerical value of u, whether u be 

 positive or negative. It will suffice therefore to suppose u positive. 

 If u < a, we have 



a <->-/:: ! t i .*-x 



du 



u 



_ p^sinu^ (?9) 



J o u 



* It should be remarked that in point of fact the field is limited through the 

 operation of a cause not taken into account in the formation of (28). It is there 

 assumed that equality of phase in the light emitted from the Tarious points of the 

 object carries with it a like equality of phase at the geometrical images of these 

 points. This will hold good only near the centre of the field. At a moderate dis- 

 tance out the illumination is destroyed by the phase-differences here neglected. 



Aug. 19th, 1903 2 i 



