M P RN Q 



i I ' I 



t 



! I : 



,.- v 



DISTRIBUTION OE ENERGY IN FLUORESCENCE SPECTRA. 1 83 



so as to obtain the benefit of the contrast field formed by the Lummer- 

 Brodhun cube, all the light passing through C is used in illuminating the 

 field, and the color is that resulting from mixing all the wave-lengths 

 present. When the instrument is set to a match we therefore have equality 

 between the total luminosity of the rays passing through C from A , and the 

 total luminosity of the rays passing through C from B. 



An expression for the total luminosity of the rays from A may be found 

 as follows: 



Let MN (Fig. 1 78) be the aperture at the focus of the telescope. It will be 

 convenient to express the width 2c of this aperture in terms of wave-length. 

 2c is therefore not a constant, even if the actual width 

 of the aperture is invariable, but depends upon the 

 dispersion in the region of the spectrum where the 

 observations are made. The widths of the slits A and 

 B, denoted by 2a and 2b, respectively, will also be 

 expressed in terms of wave-length. We shall consider 

 first the case where a<c. M' d' p" /?/v'Q' 



Let the center of C (00' in the diagram) correspond p iff 8 



to the wave-length X. That image of the slit A which 

 is formed by light of wave-length X will have its center coincident with 

 the center of the aperture. The image formed by light of wave-length 

 \-\-x will be displaced by the distance x, so that its central line falls at 

 RR'. For values of x lying between + (c a) and (c a) all of the light 

 forming the image of a will pass through the aperture. The total lumi- 

 nosity reaching the eye from such images will therefore be 



i=-7TJ f(\+x)dx=.-- {f(\ + x)+f(\-x)}dx 



where d\ is the distance of the source Si from the slit and m is a factor, de- 

 pending upon the dispersion of the prism, such that ma is proportional to the 

 width of the slit in scale divisions. It is clear that 2am/ d 2 measures the 

 intensity of the light entering the slit, the intensity being unity when the 

 source is at unit distance and the slit width one division. 



For values of x lying between c a and c-\-a only a part of the image v/ill 

 be transmitted, the transmitted fraction being 



x-\-a c 



1 



2a 



The total transmitted luminosity from these images that are partially 



transmitted will therefore be 



20m f c+a ( '_ x-\-a c y 



( 'Yl-*^)j/(X+:v)+/(X-*)jrf* 

 J c - a \ 2a / 



2am r c+a a-\-c x 



f l^-- \ \f(\+ x )+f(\- x )\dx 



J c - a I 2tf 2a I 



d<2, 

 Upon expanding/ (X+.t) and/ (X x) we have 



^(X4-.v)+/(X-A-)=/(X)+.v/'(X) + :^:2 //, (X) + . .+/(X)-.r/'(X) + ^ r (X)- . . , 



2 2 



= 2/(X)-f-* 2 /' , (A)+... 



For any ordinary case the terms in higher powers of x may be neglected 



