242-246] Spherical Harmonics 213 



by replacing each unit charge of the original system by a doublet of unit 

 strength parallel to the axis of x. Thus all harmonics of type 



d s+t+u 



(cf. 236) can be regarded as potentials of systems of doublets at the origin, 

 and, as we have seen ( 239), it is these potentials which give rise to the 

 rational integral harmonics. 



244. For instance in finding a system to give potential * 2 ( - j , we may replace the 



charge in fig. 73 by a charge at distance 2a from and - at 0. The charge at <7 

 may be similarly treated, so that the whole system is seen to consist of charges 



E, -2E, E, 



at the points x= -b, 0, b where 6 = 2a, and Z? 2 =p. 



A system of this kind placed along each axis gives a charge - 6E at the origin and 

 a charge E at each corner of a regular octahedron having the origin as centre. The 

 potential 



*/i\. a/i\ a*A\ 



a*,- 2 \r) "" fy 2 \r) * dz* \r) 

 = 0, 

 so that such a system sends out no lines of force. 



245. The most important class of rational integral harmonics is formed 

 by harmonics which are symmetrical about an axis, say that of x. There is 

 one harmonic of each degree n, namely that derived from the function 



These harmonics we proceed to investigate. 



LEGENDRE'S COEFFICIENTS. 



246. The function 



_L = ........................... (151) 



Va 2 - 2ar cos 6 + r 2 



can, as we have already seen (cf. equation (144)), be expanded in a convergent 

 series in the form 



...... (152) 



if a is greater than r. Here the coefficients 7?, J%, ... are functions of cos 0, 

 and are known as Legendre's coefficients. When we wish to specify the 

 particular value of cos 6, we write P n as P n (cos 0). 



