82 Mr. Gwyther on a 



Instead of trying to satisfy this condition, let us imagine 

 that a displacement directed along the axis of 7, and propa- 

 gated along that of x^ is broken up at the origin, and 

 geometrically represented at a distance from the origin by 

 a secondary spherical wave of the form which has been 

 considered in this paper. Let us imagine also that this 

 secondary wave has travelled till its displacements lie on a 

 sphere of radius r which is large compared with X, so that the 

 first terms are the only terms which are sensible, and then 

 imagine the velocities reversed, so that, since the forces 

 acting in the medium depend upon displacements and not 

 on velocities, the wave will reverse its course tracing out 

 its previous history in reversed order, and let us examine 

 the nature of the resulting motion at the origin. 



In general terms we know that we shall obtain a wave 

 converging to the origin and then diverging outwards again. 

 Our notation being the same as before, imagine the sphere 

 of radius c to be drawn, and that the displacements upon it 

 at a point {x.y.z) are given by 



each affected by 



sin/|/<r + /„)-r + ^| 



where c^bt^ and a,^.b„ have been replaced by An.B„ to avoid 

 confusion in what follows. 



Call the point at which this displacement exists Q, and 

 the point at which we are to consider the displacement P, 

 of which the co-ordinates are pcosa, psinacos(3, psinasin^, 

 where ^ is to be small compared with X and still more so 

 compared with c. 



Take new axes of which the axis of x passes through 

 P as before, let the co-ordinates of Q relative to these 



