462 Transactions of the Society. 



perpendicular 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 micro- 

 scopic limit, it is not to be supposed that any considerable variation 

 from that laid down by Abbe and Helmholtz 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 posi- 

 tion of the spectra. And it will remain true that there can be no 

 resolution without the co-operation of two spectra at least. 



In Table II. and fig. 120 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, constituting 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 

 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 = + 2 v, &c, the expressions for 

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

 infinite series, 



T , x sin 2 u , sin 2 (u + v) , sin 2 (it — v) 

 u z (u 4- vy (u — vy 



sin 2 (u + 2 v) , sin 2 (u — 2 v) , 99 x 



+ {u + 2 vf~ + (11 - 2 vy + '"• w 



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 



t/\ tit 2ttu . lT 2'irru , /00 \ 



I (■it) = I + I x cos + . . . . + I cos — —.+.»..: (2o) 



v 7 v 



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

 the values of the constants I , I 1} &c, are known. For these we 

 have as usual 



I s i pi ( U ) du, I = 2 - I"" I (u) cos 2 _JLUi du ; (24) 



V J V J V 



and it only remains to effect the integrations. To this end we 

 may observe that each term in the series (22) must in reality make 



