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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 othT 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 rt **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-ntI verification of its influence. Asa theoretical illustration " w< may ein*il'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 ly r? *' xin lttt has been treated by GarbaasoJ. I preHumo that the funtt (j,uti<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 <muincrntii 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).