﻿Widening of Spectrum Lines. 277 



Again, the relation between the original wave-length A and 

 the actual wave-length X, as disturbed by the motion, is 



t =i 4 v 



c denoting the velocity of light. The intensity of the light 

 in the interference bands, so far as dependent upon the 

 molecules moving with velocity f_, is by (2) 



rfI = 2{l + cos 2 ^(l + ^)},-^, . . (8) 



and this is now to be integrated with respect to f between 

 the limits + co. The bracket in (8) is 



X 27rXf . 2ttX . 27rXf 



1 + cos — i — cos - — sin — t — sin 



A Ac A iyc 



The third term, being uneven in f, contributes nothing. 

 The remaining integrals are included in the well-known 

 formula 



£ 



■ a: x 2 cos(2r.v)(Lv=^ r - e 



IT 



a 



Thus 



VT, , _2ttX _, / „- 2 X 2 



1 = 



a /^L 1+cos ""X-- Ex p(~ow)]- • (9) 



The intensity I 2 at the darkest part of the bands is found by 

 making X an odd multiple of %\, and I 2 the maximum 

 brightness by making X a multiple of \. 

 Thus <2V o T T 



Ex H-^)=ra= v ' • • • (10) 



where V denotes the " visibility " according to Michelson's 

 definition. Equation (10) is the result arrived at in my 

 former paper, and /3 can be expressed in terms of either the 

 mean velocity v, or preferably of the velocity of mean 

 square v' *. 



The next question is what is the smallest value of V for 

 which the bands are recognizable. Relying on photometric 

 experience, I estimated that a relative difference of 5 per 

 cent, between 1 ± and L would be about the limit in the ease 

 of high interference bands, and I took X = '0'lb. Shortly 

 afterwards! I made special experiments upon bands well 



* See also Proc. Eoy. Soc. vol. lxxvi, A. p. 440 (1905) ; Scientific 

 Papers, vol. v. p. 261. 



t Phil. Mag. vol. xxvii. p. 484 (1889) ; Scientific Tapers, vol. iii. 

 p. 277. 



