20 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



Substituting in the second member of equation (31), and writing down for the present only 

 the terms involving « , we obtain 



!SS{t?+*k ?)<"><•< 



^dy 



, . , . & y da. 



which, since = — — , becomes 



dx r 3 dy r 3 



filly- ^*»**** or Jf &)*"**'' 

 Treating the terms involving w in the same manner, and substituting in (31), we get 



fff(u,-S lq „ ) d -I-ff^)^ + ff(^)^>ff{^)^ ! ,. 



Now the integration is to extend from r = bt to r = SO. The quantities once integrated 

 vanish at the second limit, and the first limit relates to the surface of a sphere described round 

 as centre with a radius equal to bt. Putting dS or b 2 t 2 dtr for an element of the surface of 

 this sphere, we obtain for the value of the second member of the last equation 



- (bt)-*ff(lu + mv + nw )JdS, or - ffl \q a ) bt da ; 



and therefore the triple integral in equation (28) destroys the second part of the double integral 

 in the same equation. Hence, writing down also the terms depending upon the initial 

 displacements, we obtain for P the very simple expression 



d 



f = he ffM*** + ^ ' tf^» d °- 



This expression might have been obtained at once by applying the formula (7) to the first 

 of equations (18), which in this case take the form (6), since h = 0. 



23. Let us return now to the general case, and consider especially the terms which alone 

 are important at a great distance from the space to which the disturbance was originally con- 

 fined ; and, first, let us take the part of £ which is due to the initial velocities, which is given 

 by equation (29). 



Let the three parts of the second member of this equation be denoted by £ a , £ 6 , £ c , respec- 

 tively,and replace da by (at)~ Q dS or (bt)~ 2 dS, as the case may be; then 



t-'^iff'toW- (32) " 



Let Oi be a fixed point, taken within the space T, and regarded as the point of reference for 

 all such points as 0. Then when is at such a distance from O y that the radius vector, 

 drawn from O, of any element of T makes but a very small angle with 00 15 we may 

 regard I as constant in the integration, and equal to the cosine of the angle between OO t and 

 the direction in which we wish to estimate the displacement at O. Moreover the portion of 

 the surface of a sphere having O for centre which lies within T will be ultimately a plane per- 



