94 





manner . 





9i=~ 



K d 



4.7T c//3 



Lord Rayleigh on the 



^^{h^-K-^.^dxdydz. . . (56) 



In the case of a small homogeneous sphere, whose centre 

 is taken as origin of coordinates, these formulae lead to fairly 

 simple results. The triple integral in (55), (56) may readily 

 be exhibited in its real character of a surface-integral. Thus 



Jffs^*-') 5 ? <^=-AK-.jJ fel^d* (57) 



where dS is an element of the surface whose radius is c. This 

 applies to a sphere of any size; but we have now to introduce 

 an approximation depending on the supposition that kc is small. 

 As far as the first power of kc, 



in which the double integral is the common potential of matter 

 distributed over the spherical surface with density (z + ikzx). 

 Calling this for the moment V, we have (Thomson and Tait, 

 ' Nat. Phil. 5 § 536) at any internal point (a, |3, 7), 



Y = 47Tc(y + -J iky at) ; 

 so that 



= -47rAK-V'*"(7+i%«). . (58) 

 Thus by (55), (56), 



/ 1= iKAK-»iV", <ft=0 (59) 



We are now prepared to calculate f 2 ,(?2 fr° m (52), (53). 

 These formulas apply to both directions along the axis of z; 

 but in what follows it will be convenient to suppose that it is 

 the positive direction which is under consideration. In this 

 case, if p denote the distance from the centre of the sphere, 

 r r =p—y and e~ ikr ' = e~ ikp {l+iky) approximately; so that 



f 2 = * Y2^p J J J lh Y( l + **y) dct d & dry 



