with Special Reference to the Microscope. 183 



which includes the direction of primary vibration than in that 

 which is perpendicular to it. The result for a double point 

 illuminated by borrowed light would be a better degree of 

 separation when the primary vibrations are perpendicular to 

 to the line of junction than when they are parallel to it. 



Although it is true that complications and uncertainties 

 under this head are not without influence upon the theory of 

 the microscopic limit, it is not to be supposed that any con- 

 siderable variation from that laid down by Abbe and Helm- 

 holtz is admissible. Indeed, in the case of a grating the 

 theory of Abbe is still adequate, so far as the limit of 

 resolution is concerned ; for, as Dr. Stoney has remarked, 

 the irregularity of radiation in different directions tells only 

 upon the relative brightness and not upon the angular 

 position of the spectra. And it will remain true that there 

 can be no resolution without the cooperation of two spectra 

 at least. 



In Table II. and fig. 4 we have considered the image of a 

 double point or line as formed by a lens of rectangular 

 aperture. It is now proposed to extend the calculation to 

 the case where the series of points or lines is infinite, con- 

 stituting a row of points or a grating. The intervals are 

 supposed to be strictly equal, and also the luminous intensities. 

 When the aperture is rectangular, the calculation is the same 

 whether we are dealing with a row of points or with a grating, 

 but we have to distinguish according as the various centres 

 radiate independently, viz., as if they were self-luminous, or 

 are connected by phase-relations. We will commence with 

 the former case. 



If the geometrical images of the various luminous points 

 are situated at u = 0, u= ±v, u= +2v, &c, the expressions 

 for the intensity at any point u of the field may be written as 

 an infinite series, 



T/ shA sin 2 (?/ + v) sin 2 (?i — v) 

 {H) " ~S~ + (u + vf + (u-v)* 

 , sin> + 2i;) sin*(ti-2t?) 

 + (u + Zv)* * (u-2v)* ■*" ^V. 



Being an even function of u and periodic in period v, (22) 

 may be expanded by Fourier's theorem in a series of cosines. 

 Thus 



2irit t 2irru ,«.*»% 



l(u) = I + I x cos^- + . . . . +I,cos —+....; (23) 



and the character of the field of light will be determined when 



