Mr. J. J. Thomson on the Electric and Magnetic 





dydz. 



(P 1 Y 



Now -= „ - = ., 2 , wliero Y 2 is a surface harmonic of the second 

 ax p p l 



order. And when p > R, 



1 - 1 + R + R2 o + 



and when p < R, 



+ TD2 Ql + B3 Q2 + • ■• 9 \ 



1 1 P n , 9" 



PQ 



where Q 1? Q 2 , <fcc. are zonal harmonics of the first and second 

 orders respectively referred to OP as axis. 



Let Y' 2 denote the value of Y 2 along OP. Then, since 

 ^Y n Q m dsj integrated over a sphere of unit radius, is zero 



when n and m are different, and- z Y' when n = m, Y f n 



zn + 1 



being the value of Y n at the pole of Q n , and since there is no 



electric displacement within the sphere, 



7 



MP v 4ttY 2 

 4tt X 5 



_P e Pyr ($_ a 2 ) 

 ~ 5 2 \j5R 2RV> 



{l>+ff} 



or, as it is more convenient to write it, 



_/^p/5R 2 _a 2 \ d 2 1 

 5 V6 2/d^R' 



Bj symmetry, the corresponding values of G and H are 



B = f iep(5B? a 2 \ d 2 1 

 5 V 6 2 J da: As R' 



These values, however, do not satisfy the condition 



d¥ c]G cm =Q 



dx dy dz 



If, however, we add to F the term iyrC j this condition will 



be satisfied ; while, since the term satisfies Laplace's equation, 

 the other conditions will not be affected : thus we have finally 



