386 Royal Irish Academy, 



quiries suggest themselves naturally. Thus the axes of symmetry 

 of the medium are taken as the axes of coordinates, and the direc- 

 tion of propagation is assumed to coincide with one of these axes. 

 By these suppositions the 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 di- 

 rection of the axis of x ; 



^ = a cos {ut — kz •\- (i)\ 

 in which 



2»r ; 2* 

 T A 



r being the period of vibration, and X the length of the wave. These 

 quantities are connected by a relation given by the method of in- 

 tegration. 



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 



J = ae~^^ cos {ut — gz -{■ a). 



From this expression it follows that the amplitude of the displace- 

 ment, and therefore the intensity of the light, decreases in geome- 

 trical progression, as the distance increases in arithmetical progres- 

 sion ; and as the constant h is in general a function of m, 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 account 

 of the apparently irregular distribution of light in the absorbed 

 spectrum. To explain the absolute deficiency of the light at cer- 

 tain points, it is only necessary to admit that the function h varies 

 in certain cases rapidly with moderate changes in w, 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 lithograph. I'he method 

 employed by the author seems, however, to be fundamentally dif- 

 ferent from that of M. Cauchy ; and in fact he was led to this form 

 of the integral by other considerations before he was aware that he 

 had been preceded in the deduction. 



The remainder of the present communication is taken up with 

 the discussion of the relation between the coefficients u and A;, which 

 expresses the law of dispersion. Following M. Cauchy*, the au- 

 thor has transformed this relation by converting the triple sums into 

 triple integrals ; and he has found that, by applying this transfor- 

 mation at an earlier stage of the investigation, the resulting relation 

 is deduced with great simplicity. 



♦ Nouveaux Exercices dc Mathematiqucs Livraison 7"*. 



