2M 



Lord Rayleigh's Investigations in Optics, 



In order to obtain a more precise idea of the character of the 

 image of a luminous line, we must study the inarch of the 

 function ?f~ 2 sin 2 ?/. The roots occur when u is any multiple 

 of 7r, except zero. The maximum value of the function is 

 unity, and occurs when u = Q. Other maxima of rapidly dimi- 

 nishing magnitude occur in positions not far removed from 

 those lying midway between the roots. The image thus con- 

 sists of a central band of half width corresponding to u = ir, 

 accompanied by lateral bands of width 77-, and of rapidly dimi- 

 nishing brightness. The accompanying Table and diagram 

 (Plate VII. fig. 1) will give a sufficient idea of the distribution 

 of brightness for our purpose. 



Table I. 



u. 



u~ 2 sin 2 u. 



u. 



u— 2 sin 2 «. 







1-0000 



IT 



•0000 



**■ 



•9119 



I 77 " 



•0324 



T* 



•810(3 



1-77 



•0427 



i™ 



•6839 



i# 



•0450 



\n 



•4053 



\* 



•0165 



*» 



•1710 



27T 



•0000 



** 



•0901 



i* 



•0162 



** 



•0365 



37T 



•0000 



The curve ABCD represents the values of u~ 2 sin 2 u from 

 w=0tow = 37r. The part corresponding to negativevaln.es 

 of u is similar, O A being a line of symmetry. 



Let us now consider the distribution of brightness in the 

 image of a double line whose components are of equal strength 

 and at such an angular interval that the central line in the 

 image of one coincides with the first zero of brightness in the 

 imao'e of the other. In fig. 1 the curve of brightness for one 



o o © 



component is A B C D, and for the other A! C; and the 

 curve representing half the combined brightnesses is E'BE F. 

 The brightness (corresponding to B) midway between the two 

 central points A, A! is *8106 of the brightness at the central 

 points themselves. We may consider this to be about the 

 limit of closeness at which there could be any decided appear- 

 ance of resolution. The obliquity corresponding to u=ir is 

 such that the phases of the secondary waves range over a com- 

 plete period, i. e. such that the projection of the horizontal 

 aperture upon this direction is one wave-length. We conclude 

 that a double line cannot be fairly resolved unless its compo- 

 nents subtend an angle exceeding that subtended by the wave- 

 length of light at a distance equal to the horizontal aperture* '. 



* In the spectroscope the angular width of the slit should not exceed a 

 moderate fraction of the angle defined in the text, if full resolviug-power 

 be wanted. 



