PROFESSOR STOKES, ON THE DYNAMICAL THEORY OP DIFFRACTION. 23 



The first term in the right-hand member of the first of these equations is got from (32) by 

 putting 1=1, introducing the function /„, and replacing at in the denominator by r, which 

 may be done, since at differs from r only by a small quantity depending upon the finite 

 dimensions of the space T. The second term is derived from the first by the theorem of 

 Art. 14, and u is of course got from (• by differentiating with respect to t. Had t been re- 

 tained in the denominator, the differentiation would have introduced terms of the order t~ 2 , and 

 therefore of the order r -2 , but such terms are supposed to be neglected. 



The wave of dilatation will have just passed over O at the end of the time a~ l (r + p.^). 

 The medium about O will then remain sensibly at rest in its position of equilibrium till the 

 wave of distortion reaches it, that is, till the end of the time 6 _1 (r - p x ). During the 

 passage of this wave, the displacements and velocities will be given by the equations 



,fAbt-r)+-±-fi(bt-r), 



(37). 



4nrbr " 47rr" 



U = °' * ' j£? fJ (6 ' " r) + ^r f "" ibt ~ r) ' 



471-r 47tf 



After the passage of the wave of distortion, which occupies an interval of time equal to 

 b' 1 (p\ + pi), the medium will return absolutely to rest in its position of equilibrium. 



25. A caution is here necessary with reference to the employment of equation (30). If we 

 confine our attention to the important terms, we get 



e- r^ ff l ( d ^r) ^+-^7 /y#-'£l *& ■ ( 38 )- 



4>irat JJ \dr) at 4,TrbtJJ\dr drf bt v ' 



Now the initial displacements and velocities are supposed to have finite, but otherwise arbitrary, 

 values within the space T, and to vanish outside. Consequently we cannot, without unwarrant- 

 ably limiting of the generality of the problem, exclude from consideration the cases in which 

 the initial displacements and velocities alter abruptly in passing across the surface of T. In 

 particular, if we wish to determine the disturbance at the end of the time t due to the initial 

 disturbance in a part only of the space throughout which the medium was originally disturbed, 

 we are obliged to consider such abrupt variations ; and this is precisely what occurs in treating 

 the problem of diffraction. In applying equation (38) to such a case, we must consider the 

 abrupt variation as a limiting case of a continuous, but rapid, variation, and we shall have 



to add to the double integrals found by taking for -£- and —2 the finite values which refer 



to the space T, certain single integrals referring to the perimeter of that portion of the plane 

 P which lies within T. The easiest way of treating the integrals is, to reserve the differen- 

 tiation with respect to t from which the differential coefficients just written have arisen until 

 after the double integration, and we shall thus be led to the formulae of the preceding article, 



