and Electric Waves upon Small Obstacles. 45 



gtfar—^ |] 10 primary potential is AirVy; so that in (1) we are 

 to take m=0, v = iirV, 10= 0. Hence by (20) a, /3, 7 for 

 the disturbance are given by 



a = d-ty/dx, /3 = dty/dy, 7 = d-yfr/dz, 



where 



+— *- v &&2 (99) 



In like manner /, g, h are derivable from a potential ■%. 

 The primary potential is s simply, so that in (1), u = 0, v = 0, 

 w=l. Hence by (20) 



K'-K a?z 



x^—Kr+m-r* • • • • (10 °) 



from which /, g, li for the disturbance are derived by simple 

 differentiations with respect to x, y, z respectively. 



Since/, g, h, a, ft, 7 all satisfy (92), the values at a distance 

 can be derived by means of (41). The terms resulting from 

 (99), (100) are of the second order in spherical harmonics. 

 When r is small, 



r- 1 e- ikr f 2 {ikr)=-3/k*r*, 



and when r is great 



r~ l e- ikr f 2 {ikr) = r~ l e~ ikr ; 



so that, as regards an harmonic of the second order, the value 

 at a distance will be deduced from that in the neighbourhood 

 of the origin by the introduction of the factor — lk 2 r 2 e~ ikr . 

 Thus, for example, /in the neighbourhood of the origin is 



J dx K' + 2K r 5 ' • ' ( Wi ) 



so that at a great distance we get 



/_ ~K 7 +2K ? * * * * (LU " } 



In this way the terms of the second order in spherical 

 harmonics are at once obtained, but they do not constitute 

 the complete solution of the problem. We have also to 

 consider the possible occurrence of terms of other orders in 

 spherical harmonics. Terms of order higher than the second 

 are indeed excluded, because in the passage from r small to 

 r great they suffer more than do the terms of the second 

 order. But for a like reason it may happen that terms of 

 order zero and 1 in spherical harmonics rise in relative 



