with Special Reference to the Microscope. 



Ill 



rj = 0. Sometimes, as in the application to a grating, we wish 

 to consider the image due to a uniformly luminous line, 

 parallel to rj, and this can always be derived by integration 

 from the expression applicable 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 

 we get 



the intensity, represented by I 2 or C 2 /X 2 / 2 ; and 



I 



+« 



12 j n - °L^ sin 2 (7r|a/V) 



by means of the known integral 



%J — 00 *J — 00 



sin x 



dx=ir. 



(14) 



(15) 



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

 nous line whose geometrical image coincides with f = 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, 



~r^Cd V = a Sin ^ f) (16) 



J J —00 



ir^ajXf 



for the expression of the resultant amplitude corresponding 

 to£. 



In order to make use of these results we require a table of 



the values of sin u/u, and 

 suffice for our purposes : — 



of sin 2 u/u 2 . The following will 



Table I. 



4w 



S111M 



sin 2 u 



4a 



sin u 



sin 2 u 



7T 



u 



u* ' 



it 



u. 



u 2 ' 







+ 1-0000 



1-0000 



9 



+•1000 



•0J00 



1 



•9003 



•8105 



10 



•1273 



•0162 



2 



•6366 



•4053 



11 



•0818 



•0067 



3 



•3001 



•0901 



12 



•0000 



•0000 



4 



•0000 



•0000 



13 



-0692 



•0048 



5 



- -1801 



•0324 



14 



-•0909 



•0083 



6 



- -2122 



•0450 



15 



- -0600 



•0036 



7 



- -1286 



•0165 



16 



•0000 



•oooo 



8 



•0000 



•0000 









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



