188 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



= _ AT* fo 1 + 3 + Ml |. ) f^ 



\ 8# 9?/ 8,37 V 



\ 

 ) 



r 3 r 5 r 5 r* 



/m 2 _ 3m 2 y 2 3l 2 xy 

 mi ~ ~~ 



**- 



F 2 = A {- ( 



+ 3 ( 



+ (m^ + m^j) 2/5 + n, 2 + ?i 2 ^a?. 

 The coefficients are of course subject to the relations 



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

 coincide, we may conveniently take the axis for one of the coor- 

 dinate axes, and write 



(14) V n 



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

 that the surface harmonic will be simply a polynomial in 



zfr = cos (rz). 



The equation T n (cos (rz)) = 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 (rz) which constitutes the zonal surface 

 harmonic, when the value of the constant A is determined in the 

 manner to be shortly given, is called a Legendre's Polynomial, and 

 denoted by 



99. Harmonics in Spherical Coordinates. We have 

 transformed Laplace's Operator into spherical coordinates in 

 88, and A V = becomes 



,8 / 9F\ d . ,aF\ 9 



C5) 



