456 Transactions of the Society. 



When the aperture is rectangular, of width a parallel to x, and 

 of width b parallel to y, the limits of integration are from — ^ a 

 to + \ a for x, and from — £ b to + | ; b for y. Thus 



C = a b sfaQrgfl/V) 3in(Triy 5/ X/) ^ ^ 



7r ga / \f irrjb / \f 



and by (9) the amplitude of vibration (irrespective of sign) is 

 C f\f. This expression gives the diffraction pattern due to a single 

 point of the object whose geometrical image is at £ = 0, rj = 0. 

 Sometimes, as in the application to a grating, we wish to consider 

 the image due to a uniformily luminous line, parallel to ?;, and this 

 can always be derived from integration from the expression applic- 

 able to a point. But there is a distinction to be observed according 

 as the radiations from the various parts of the line are independent 

 or are subject to a fixed phase-relation. In the former case we 

 have to deal only with the intensity, represented by I 2 or C 2 /A 2 / 2 ;. 

 and we get 



+ x 12 f1 „ - a2h sin 2 (7rga/y ) (U) 



by means of the known integral 



f + -_5H^<fo,= f + ~^, !.,: = „. . (15) 



J —CO •'" J _ oo X 



This gives, as a function of £, the intensity due to a self-luminous 

 line whose geometrical image coincides with £ = 0. 



Under the second head of a fixed phase-relation we need only 

 consider the case where the radiations from the various parts of 

 the line start in the same phase. We get, almost as before, 



;. 



i r**ai,-o*i!r&g>. . . (16) 



for the expression of the resultant amplitude corresponding to £. 



In order to make use of these reults we require a table of the 

 values of sin uju, and of sin 2 u/u 2 . The following (Table I.) will 

 suffice for our purposes. 



When we have to deal with a single point or single line only, 

 this table gives directly the distribution of light in the image, u 

 being equated to tt f a/\f. The illumination first vanishes when 

 u = ir, or |// = \/a. 



On a former occasion * it has been shown that a self-luminous 

 point or line at u = — ir is barely separated from one at u = 0. 

 It will be of interest to consider this case under three different 

 conditions as to phase-relationship ; (i) when the phases are the 



* Phil. Mag., vol. viii. p. 'JGG, 1879. 



