

the Theory of Vibratory Motions. 59 



grations, all within finite limits and without indetermination ; 

 they can be carried out rigorously ; and we can thus demon- 

 strate that/(#,y,2)* is found again as its value. 



If we express in this way the initial functions, and reduce 

 them all to the single function corresponding to one and the 

 same element co, they will be found to be functions of a single 

 coordinate a,v-\-/3y + <yz, which is the abscissa of any point 

 whatever, counted in a direction parallel to the fixed direction 

 OH ; moreover they will have values different from only 

 within a restricted region : for example, the abscissa must be 

 comprised between ±(i, attributing to the volume V the form 

 of a sphere of radius p and placing the origin at the centre. 



Now, in this case the equations of the motion are integrated 

 immediately, the excursions are functions solely of the same 

 abscissa, and the simple motion which results is as easy to find 

 as if it were represented by a cosine ; only it is of quite a differ- 

 ent nature and composed of a limited plane wave. Let us call 

 that plane which was at first carried through the origin per- 

 perdicular to OH the middle plane ; and suppose it to be dis- 

 placed parallel, with a constant velocity S. The disturbed 

 region will at each instant be bounded by two planes parallel 

 to that plane, taken on both sides at the distance jjl. 



The total motion results from the superposition of these 

 plane waves ; their middle planes form, at the end of the time 

 t, the whole of the tangent planes to one and the same interior 

 envelope, which is the surface of the waves ; the motion is 

 sensible only within a small thickness on both sides of the 

 envelope ; and it can be ascertained by a very simple reason- 

 ing that the prolongations of the plane waves outside of this 

 limited region interfere with one another, at least at a notable 

 distance from the centre of disturbance. 



Either the complete integrals given by Poisson for an 

 isotropic medium, or, in the case of a crystallized medium, the 

 law of the motion for a point at a distance from the origin is 

 also found with sufficient facility. The latter is found to de- 

 pend on the radius of curvature of the apparent contour of the 

 surface of the waves upon any plane which passes through one 

 of its normals. Now it is remarkable that, if a secant plane 

 parallel to the former be carried through the same normal, the 

 product of the radius of curvature of the section into that of 

 the contour is constant ; besides, the radii of curvature of the 

 surface of the waves are determined by those of the surface of 

 the sixth order which represents the velocities; so that the for- 

 mulae no longer contain any thing unknown. 



* For further details see the Memoires de la Societe de Physique dt 

 Geneve, annee 1880. 



