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XXXI. An Abstract of the essential Principles ofM, Cauchy's 

 View of the Undnlatory Theory^ leading to an Explanation 

 of the Dispersion of Light ; with Remarks. By the Rev, 

 Baden Powell, M,A., F.R.S.^ Savi Han Professor of Geo- 

 metfy, Oxford. 



[Continued from p. 113.] 



TN the motions expressed by equation (^O.)* we may observe 

 ^ that the displacements and velocities depend on the sole 

 variables g and t ; and at the end of the time /, therefore, they 

 are the same for all molecules situated at the same distance § 

 from the plane (16.) to which it is perpendicular. 



We have thus far obtained expressions forf>)f, the re- 

 solved parts of the actual displacement of a molecule m in the 

 directions of three rectangular axes in terms of »' a'' «'", which 

 represent three distinct absolute motions or displacements in 

 the directions of three lines at right angles in space, deter- 

 mined by the circumstance of their coinciding with the axes 

 of a given ellipsoid, and having determinate inclinations to a 

 given plane dependent on the values of the arbitrary quantities 

 which enter into the expressions. We have also general ex- 

 pressions for the velocities in those directions ; and in general 

 some of the molecules may take each one of the three motions 

 thus defined. 



Now, if at the commencement of the motion the displace- 

 ment of all the molecules take place in directions parallel to 

 one of the three axes of the ellipsoid just referred to, and the 

 whole velocities are consequently to be estimated in those 

 lines, then the initial values, or tlie functions '^ {g) n{g) ex- 

 pressed by equations (36.) and (37.), will vanish for two of 

 the values of s. And consequently for any time /, the dis- 

 placement a determined by equation (40.) will also vanish for 

 the same two values of s : or, in other words, two of the dis- 

 placements of the molecules will likewise always vanish, or 

 the whole motion will continue always parallel to the same 

 axis of the ellipsoid. We will take a as that one value which 

 does not vanish. Except so far as the remark above made ex- 

 tends, viz. that the motions are the same at the same time for 

 all molecules situated at the same distance from the plane 

 (16.)5 the expression above given for the value « (40.) is not 

 of such a nature that we can directly infer from it the actual 

 conditions of the sort of displacement which a molecule un- 

 dergoes, or the consequences which result ; but we may ar- 

 rive at some conclusions of this kind if we can suppose the 



