143, 144] FORMATION OF HARMONICS. 397 



A 5 d d 

 A<r^r- err 



/W 2 _ 3m 2 j/ 2 _ Zl^xy _ 3n 2 yz\ 



m i\ r s r 5 ,.5 5 j 



/t^ 3^02 _ 3J^^ _ 8m, 



"* {ft* r * r s r& 



- n^ (x 



+ 3 (?!? 2 ^ 2 4- m^m^y 2 ) 

 + (mj^ + m^yz + (w^ 2 + Wg^J^a?)}. 



The coefficients are of course subject to the relations 



144. Zonal Harmonics. If all the axes of the harmonic 

 coincide, we may conveniently take the axis for one of the coordinate 

 axes, and write 



97) Vn^Ar^+i (-}. 



VP\r) 



It is evident that this will contain only powers of g and r, so 

 that the surface harmonic will be simply a polynomial in 



= cos (rz). 



The equation Y n {cos(rs)} = may be shown to have n real roots 

 lying between 1 and 1, and hence represents n circular cones of 

 angles whose cosines are these roots, intersecting the surface of a 

 sphere in n parallels of latitude which divide the surface into zones. 

 The harmonics are therefore called Zonal Harmonics. The polynomial 

 in cos(r^) which constitutes the zonal surface harmonic, when the 

 value of the constant A is determined in the manner to be given 

 in 106), is called a Legendre's Polynomial, and denoted by 



