296 Prof. K. Pearson, On plane waves of the [May 25, 



(2) Note on Prof. Kowland's paper on spherical waves of light 

 and the dynamical theory of diffraction (Phil. Mag. Vol. xviil, 

 June, 1884). By E. T. Gwyther, M.A. Communicated by Prof. 

 J. J. Thomson. 



(3) On plane waves of the third order in an isotropic elastic 

 medium, with special reference to certain optical phenomena. By 

 Prof. K Pearson. 



1. The object of the present paper is to consider a simple 

 case of wave motion in an isotropic elastic medium, when the 

 displacements are not considered so small that the cubes of the 

 space variations may be neglected. In general the three body- 

 equations for an isotropic solid each contain upwards of 80 terms 

 when we do not neglect the squares of small quantities and over 

 200 when we retain the cubes, so that they appear to be ex- 

 tremely unmanageable. In the simple case we are about to con- 

 sider we shall find that it is necessary to retain cubes in order to 

 introduce any change into the equations for wave motion. 



2. Let u, v, w be the displacements of the point xyz of the 

 solid parallel to the axes. 



Then the strain on a very small element of the solid at (xyz) 

 is fully determined by the quantities s i} s 2 , s 3 , a, /3, 7, where if PQ 

 be any elementary line drawn from xyz, and having projections 

 Sx, Sy, hz on the axis which becomes PQ' after distortion, 

 PQ' 2 = PQ' + 2 Sl Bx 2 + 2s 2 8y 2 + 2s 3 Sz 2 



+ 2aSydz + 2/3Bzdx + 2<ySx8y. 

 It is easily shewn that 



8 1 = u x + 1 (u* + v* + w x % 



a = w tt + v z + u v ii z + v y v s + w y w z , 

 /3=u z + w x + u z u x + v z v x -1- w,w x , 

 r y = % + u y + u x ii y + v x v y + w x w y . 

 Further, W the work will be a function of the variables 

 s 1} s 2 , s 3 , a, 0, 7 =F(s 1 , s 2 , s 3 , a, /3, 7), say. 



3. Let us consider the case of a plane wave in which the 

 vibrations are in the face, and suppose this face parallel to the 

 plane of xy. Furthermore let us assume the same vibrations are 

 taking place at every point of the face and that there is no 



