16 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



conceive to extend to all space, provided we regard the quantity to be integrated as equal to 

 zero when PQ\ (or as it may now be denoted PQ, Q being a point taken generally,) lies 

 beyond the limits at and at + h, as well as when the point Q falls outside the space T, to 

 which the disturbance was originally confined. Now perform the first of the two triple inte- 

 grations on the supposition that Q remains fixed while P is variable, instead of supposing P 

 to remain fixed while Q is variable. We shall thus have F constant and r variable, instead 

 of having F variable and r constant. This first triple integration must evidently extend 

 throughout the spherical shell which has Q for centre and at, at + h for radii of the interior 

 and exterior surfaces. We get, on making h vanish, 



dS being an element of the surface of a sphere described with Q for centre and at for radius. 

 Now if OQ = r, the integral ffr~ l dS, which expresses the potential of a spherical shell, of 

 radius at and density unity, at a point situated at a distance r from the centre, is equal to 

 ■iirat or 47rr'" 1 a 2 £ 2 , according as r <> at. Substituting in (22), and omitting the accents, 

 which are now no longer necessary, we get 



+ = -^~fff F - dV ( r<at) -rJff-r- dV ( r>at) ■ • • (23) - 



where the limits of integration are defined by the imparities written after the integrals, as will 

 be done in similar cases. 



16. Let u , v , w , be the initial velocities; then 



du dv dw 

 dx dy dss 



Substituting in the first term of the right-hand member of equation (23), and integrating 

 by parts, exactly as in Art. 5, we get 



-^affI F - dr ^ <at) -^ff^ U ^- (U ^^ dX ' 



where the 2 denotes that we must take the sum of the expression written down and the two 

 formed from it by passing from x to y and from y to as, and the single and double accents 

 refer respectively to the first and second point in which the surface of a sphere having O for 

 centre, and at for radius, is cut by an indefinite line drawn parallel to the axis of x, and in 

 the positive direction, through the point (o, y, z). Treating the last term in equation (23) in 

 the same way, and observing that the quantities once integrated vanish at an infinite 

 distance, or, to speak more properly, at the limits of the space T, we get 



- -hSIi-r dnr>at ^^ jfifiw-- w.l d y d * 



dV 



- — fff( u o a! + %y + "><>*) -5- ( r > °0- 



The double integrals arising from the transformation of the second member of equation 

 (23) destroy one another, and we get finally 



