24 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



where the correct values of the terms in question were obtained at once by the theorem of 

 Art. 14. 



26. It appears from Arts. 11 and 12, that in the wave of distortion the density of the 

 medium is strictly the same as in equilibrium ; but the result obtained in Art. 23, that the dis- 

 placements in this wave are transversal, that is, perpendicular to the radius of the wave, is only 

 approximate, the approximation depending upon the largeness of the radius, r, of the wave 

 compared with the dimensions of the space T, or, which comes to the same, compared with 

 the thickness of the wave. In fact, if it were strictly true that the displacement at due to 

 the original disturbance in each element of the space T was transversal, it is evident that the 

 crossing at of the various waves corresponding to the various elements of T under finite, 

 though small angles, would prevent the whole displacement from being strictly perpendicular 

 to the radius vector drawn to O from an arbitrarily chosen point, O lt within T. But it is not 

 mathematically true that the disturbance proceeding from even a single point 0,, when a dis- 

 turbing force is supposed to act, or rather that part of the disturbance which is propagated 

 with the velocity b, is perpendicular to 00j, as will be seen more clearly in the next article. 

 It is only so nearly perpendicular that it may be regarded as strictly so without sensible 

 error. As the wave grows larger, the inclination of the direction of displacement to the wave's 

 front decreases with great rapidity. 



Thus the motion of a layer of the medium in the front of a wave may be compared with 

 the tidal motion of the sea, or rather with what it would be if the earth were wholly covered 

 by water. In both cases the density of the medium is unchanged, and there is a slight increase 

 or decrease of thickness in the layer, which allows the motion along the surface to take place 

 without change of density : in both cases the motion in a direction perpendicular to the surface 

 is very small compared with the motion along the surface. 



27. From the integral already obtained of the equations of motion, it will be easy to 

 deduce the disturbance due to a given variable force acting in a given direction at a given 

 point of the medium. 



Let 0, be the given point, T a space comprising 0,. Let the time t be divided into 

 equal intervals t; and at the beginning of the n th interval let the velocity tF(ht) be com- 

 municated, in the given direction, to that portion of the medium which occupies the space T. 

 Conceive velocities communicated in this manner at the beginning of each interval, so that the 

 disturbances produced by these several velocities are superposed. Let D be the density of the 

 medium in equilibrium; and let F(nr) = (2)7')" 1 /(»t), so that t/(«t) is the momentum 

 communicated at the beginning of the n th interval. Now suppose the number of intervals t 

 indefinitely increased, and the volume T indefinitely diminished, and we shall pass in the limit 

 to the case of a moving force which acts continuously. 



The disturbance produced by given initial velocities is expressed, without approximation, 

 by equation (29), that is, without any approximation depending on the largeness of the 

 distance O0 a ; for the square of the disturbance has been neglected all along. Let 00i = r ; 

 refer the displacement at to the rectangular axes of x, y, % ; let I, m, n be the direction- 



