62 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 68 



vations indicate, as the angle of incidence decreases, a much more 

 uniform reflective power for dififerent wave-lengths, even with such 

 rough surfaces. Accordingly the assumption of a comparatiA/ely 

 uniform scattering power at nearly normal incidence (about 3° in the 

 present case) for poHshp^ "surfaces is not unreasonable. 



In order to compute the relative amount of true and scattered 

 radiation at any prismatic deviation an analytical expression for the 

 observed spectrum in terms of these quantities will be built up and 

 then modified to suit actual computations. The first term of this 

 equation should represent, for any deviation, the energy due to 

 the ordinary dispersion of the prism, the second term that lost from 

 the region by scattering and the third that gained by scattering". 



Let 6 be the deviation taken as zero at 1.8 fx. Let S{6) represent 

 the function which is the analytical expression of the complete, sym- 

 metrical curve of scattering of approximately monochromatic light, 

 the central and right-hand portion of which curve is shown as a^'h^'c^ 

 in figure 18. Let the true energy curve of the lamp be E{6) , and the 

 observed energy curve Eq{0). 



The first term of the equation for Ef^(O) is evidently E(6), the 

 energy proper to the deviation. 



The second term represents the energy properly belonging to this 

 deviation, but scattered elsewhere. To a close degree of approxima- 

 tion, it is equal to the area of the curve of scattering taken so that 

 its maximum ordinate is equal to the true intensity in the spectrum. 

 With the reservation stated in the second succeeding paragraph, which 



also applies to the third term, it is therefore equal to S(.v)d.v 



multiplied by the ratio of the ordinate of the true energy curve, E{9), 

 at this deviation to that at 1.8 /x, E{do), where 6^ is taken to sig- 

 nify O — o. 



The third term represents the energy belonging to other deviations 

 but scattered here. To a close degree of approximation it is equal 

 to a sum of successive infinitessimal areas determined by S(0), the 

 distance of each area, infinitely narrow in abscissae, from the maxi- 

 mum of S(6) being determined by the distance (d + .v) of the devia- 

 tion contributing the scattering. The height of the area is so taken 

 that the height of the corresponding maximum, S(On) , shall be equal 

 to the true height of the energy curve at the deviation from which 

 this scattered portion is derived. Analytically this is equal to S(x)dx 

 multiphed by the ratio of E{0 + .v) to E(0,,) and integrated over all 

 deviations. 



