18 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



may be at once written down by observing that these integrals express the components of the 

 attraction of a spherical shell, of radius bt and density 1, having Q for centre, on a particle 

 situated at 0. Hence if w, y', z' be the co-ordinates of Q, measured from O, and r = OQ, 

 the integrals vanish when r < bt, and are equal to 



^{btfx r'~\ i,r(btyy'r'- 3 , ^{btf z'r'- 3 , 



respectively, when r>bt. Hence we get from (26), omitting the accents, which are now no 

 longer necessary, since we have done with the point P, 



& = £fff(<»°"y-«>o'*)^(r>bt) (27). 



Now 



, dw dv „ du dw „, dv du 



2w = -- — ; 2w =— — ; 2a> = - — , 



ay dz dz dx dx dy 



du 

 Substituting in (27), and adding and subtracting x — — under the integral signs, we get 



dx 



t rrr\ f d d d\ j du dv dw \\ dV 



^rj}I\-{ X Tx + y Ty^d7.) U ^{ V Jx- + y Jx +Z ^))^ r>ht) - 



d d d d 



But x— - + y-— + z— is the same thine as r -— , and we get accordingly 

 dx * dy dz 8 dr 8 ° J 



The second part of £ a is precisely the expression transformed in the preceding article, 

 except that the sign is changed, and b put for a. Hence we have 



^ = —ff(u -lq ) bt d<r-—fJf(u -3lq )~(r>bt). . . (28). 



19. Adding together the expressions for £, and £>, we get for the disturbance due to the 

 initial velocities 



l m ^}J l (lo)atdv + ^ffao ~ l 1oXt d <r + — fff( 3l 9° ~ u o) -^( bi<r <«*>• ( 2 9>- 



The part of the disturbance due to the initial displacements may be obtained immediately 

 by the theorem of Art. 14. Let £ , ij , £ be the initial displacements, p the initial dis- 

 placement resolved along a radius vector drawn from 0. The last term in equation (2.9), it 

 will be observed, involves t in two ways, for t enters as a coefficient, and likewise the limits 

 depend upon t. To find the part of the differential coefficient which relates to the variation of 

 the limits, we have only to replace dV hy r'drda, and treat the integral in the usual way. 

 We get for the part of the disturbance due to the initial displacements 



?=s//H ( M4:)-E-b»s//H^'f-'(v.*»f)l'« 



dV 



+ ^fjJ(* l f*-fa-jr(i>t<r<oi) (30). 



