of Approximately Homogeneous Light, 409 



character of the luminous vibration itself. In the case of a 

 mathematical spectral line, the waves are regular to infinity, 

 and the bands are formed without limit and with maximum 

 visibility throughout. Again, in the case of a double line 

 (with equal components) the waves divide themselves into 

 groups with intermediate evanescences, and this is also the 

 character of the interference bands. Thirdly, if the spectral 

 line be a band of uniform brightness, and if the waves at 

 the origin be supposed to be all in one phase, the actual 

 compound vibration will be accurately represented by the 

 corresponding interference bands. But this law is not 

 general for the reason that in one case we have to deal with 

 amplitudes and in the other with intensities. The accuracy of 

 correspondence thus requires that the finite amplitudes in- 

 volved be all of one magnitude. A partial exception to this 

 statement occurs in the case of a spectral line in which the 

 distribution of brightness is exponential. 



Another question related to the effect of the gradual loss 

 of energy from communication to the ether upon the homo- 

 geneity of the light radiated from freely vibrating molecules. 

 In illustration of this we may consider the analysis by Fourier's 

 theorem of a vibration in which the amplitude follows the 

 exponential law, rising from zero to a maximum, and after- 

 wards falling again to zero. It is easily proved that 



x = -J— f du cosiiff { e -(*-r)W + e -(«+r)W} 

 2a^/7rJ 



in which the second member expresses an aggregate of trains 

 of waves, each individual train being absolutely homogeneous. 

 If a be small in comparison with r, as will happen when the 

 amplitude on the left varies but slowly, e -i u + r ) 2 l^ 2 mav be 

 neglected, and e~ {u ~ r)2 /4a2 is sensible only when u is very 

 nearly equal to r. 



As an example in which the departure from regularity con- 

 sists only in an abrupt change of phase, let us suppose that 



yjr(x) = +sin (2tt#/Z), 



the sign being reversed at every interval of ml, so that the 

 + sign applies from to ml r 2 ml to 3 ml, 4 ml to 5 ml, &c, 

 and the negative sign from ml to 2 ml, 3 ml to 4 ml, <fec. As 

 the analysis into simple waves we find 



. , . _ ^ 2 cos (27rnx/2ml) 

 ^ [X) ~ * mir(l-n*l±m*) > 



the summation extending to all odd values 1, 3, 5, ... of n. 

 The fundamental component cos {2irxj2ml) and every odd 



e~ a ~*~ cos r 



