11 



differential equations of motion are reduced to a very simple 

 form ; and it is manifest that the assumptions themselves 

 involve no real limitation of the problem. The well known 

 expressions for the component displacements are deduced 

 by the integration of these equations. The following is that 

 in the direction of the axis of x : 



%— a cos (tit — Xs-f-a) ; 



in which 



_2tt 2tt 



T A 



r being the period of vibration, and A the length of the wave. 

 These quantities are connected by a relation given by the 

 method of integration. 



The preceding formula, however, is not the most general 

 form of the expression for the displacement. It is found 

 that in certain cases the integral becomes 



%, — ae~ Ass cos (ut— gz+a). 

 From this expression it follows that the amplitude of the 

 displacement, and therefore the intensity of the light, de- 

 creases in geometrical progression, as the distance in- 

 creases in arithmetical progression ; and as the constant h is 

 in general a function of a, or of the colour, the differently 

 coloured rays will be differently absorbed. The complete 

 value of £ being the sum of a series of terms similar to the 

 preceding, it is manifest that we have here a satisfactory ac- 

 count of the apparently irregular distribution of light in the 

 absorbed spectrum. To explain the absolute deficiency of 

 the light at certain points, it is only necessary to admit that 

 the function h varies in certain cases rapidly with moderate 

 changes in u, and becomes very great for certain definite 

 values of that quantity. 



The preceding integral has been already obtained by 

 M. Cauchy, in a valuable memoir recently printed in litho- 

 graph. The method employed by the author seems, how- 



