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: 
E=acos(ut—kz+a); 
in which 
7 being the period of vibration, and X 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 
E=ae—* 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 his 
in general a function of w, 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 / 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- 
