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298 ON THE WIDENING OF SPECTRUM LINES Ml be encounters of a milder kind where the free vibrations are inilueneed but yet not in such a degree that the vibrations after the encounter have no relation to the previous ones. And in the case; of flames there is another question to be faced: Is there no distinction in kind between encounter* first of two sodium atoms and secondly of one sodium atom and an atom say of nitrogen f The behaviour of soda flames shows that there is. Othenvi.Hr if MM-IUK impossible to explain the great effect of relatively very small additions of n.rtla in presence of large quantities of other gases. The phenomena snn#'M that the failure of the least coloured flames to give so high an interfrrenn- us is calculated from Doppler's principle may be due to encounters with oth«T giiws, but that the rapid foiling oft' when the supply of soda is inrivasi-il is due t.. something special. This might be of a cjnasi-ehemieal diameter, e.<(, temporary associations of atoms; or again to vibrators in dost; proximity putting one another out of time. In illustration of such effects unilrulatimi ha* bri-n given in the previous paper*. It is in accordance with (his virw fhut, an Gotiy found, the emission of light tends to increase as the .KIJUHJV r«»«»t **f the amount of soda present. We come now to cause (iv). Although it in certain that this ran**- muni operate, wo are not able at tho present time to point to any e\jirriun-nt«I verification of its influence. Asa theoretical illustration " w<« may ei»n*i«l«'r the analysis by Fourier's theorem of a vibration in which thfamplitutli- i'uliuwft an exponential law, rising from zero to a maximum and afterward* falling again to zero. It is easily proved that e-a^ cos rx = . 1. 2a v TT in which the .second member expresses an aggregate of trains «>f wavr*. «<Hrh individual train being absolutely homogeneous. If a he wunll in ruiujmriwirt with r, as will happen when the amplitude on the left varies but. slowly, e-(u+r) »/•«»» may })(3 neglected, and e-<«-r>*<'-wa in Hennible <»tily wtu-u u i* wry nearly equal to r "f. An analogous problem, in which the vibration i« reprewntetl l»y r? *' xin lttt has been treated by GarbaasoJ. I preHumo that the funtt (j,u«»ti«<l r«-lUtt*« t« positive values of t and that for negative values of t it JH to be rojikwl by zero. But I am not able to confirm Garbasao's formula^. As regards the fifth cause of (additional) widening <muincrnt«ii nt thi' beginning of this paper, tho case is somewhat similar ti» that »f tlu« fmtrth. It must certainly operate, and yet it does riot appear to be important in practice. In such rather rough observations as 1 have made, it wcnm t« rrinke rt<* * Phil. Mac/, suprlk, p. 20f). [This volume, Art. 85)0.] t Phil. Man. Vol. xxxiv. p. 407 (1892); Scientific Papm, Vol. iv. j>. Ifl. t Ann. der Plnjsik, Vol. xx. p. 848 (1906). § Possibly the sign of a is supposed to change when t p&mm throngh w-ro. Btjj, men what are perhaps misprints would need correction.on with soda ap to require the presence of oxygen (Mitcherlich, Smithells).