294 ON THE WIDENING OF SPECTRUM LINES [39 and this is now to be integrated with respect to £ between the limits ± oc The bracket in (8) is 1 4- cos -^j— cos A Ac — sin- —— sin —T— A Ac The third term, being uneven in £, contributes nothing. The remainrn integrals are included in the well-known formula cos \lir a Thus 7 = The intensity Ja at the darkest part of the bands is found by making X a odd multiple of iX, and /2 the maximum brightness by making X a mnltip where F denotes the "visibility" according to Michelson's definition. Equi tion (10) is the result arrived at in my former paper, and /9 can be expresse in terms of either the mean velocity v, or preferably of the velocity of mea square v'*. 'The next question is what is the smallest value of Ffor which the ban( are recognizable. Relying on photometric experience, I estimated that a rel; tive difference of 5 per cent, between 7a and 72 would be about the limit i the case of high interference bands, and I took F = '025. Shortly afterwards I made special experiments upon bands well under control, obtained by meai of double refraction, and I found that in this very favourable case the bant were still just distinctly seen when the relative difference between 7i and , was reduced to 4 per cent. It would seem then that the estimate F = 'OS can hardly be improved upon. On this basis (10) gives in terms of v ~y 9/> /> (12) as before. In terms of v' by (6) X = V3.c A j /o As an example of (12), let us apply it to hydrogen molecules at 0° < Here v' = 1839 x 102 cm./sec.J, and c = 3 x 1010. Thus J/A = T222 xlO6............................(13) * See also Proc. Roy. Soc. Vol. LXXVI A. p. 440 (1905); Scientific Papers, Vol. v. p. 261. t Phil. Mag. Vol. xxvii. p. 484 (1889); Scientific Papers, Vol. in. p. 277. t It seems to be often forgotten that the first published calculation of molecular velocities w that of Joule (Manchester Memoirs, Oct. 1848, Phil. Mag. ser. 4, Vol. xiv. p. 211).889, but with a few slight changes of notation.