5H 



Prof. John Perry on a 



In a great number of practical cases V is symmetrical 

 about an axis, and a symmetrical spherical harmonic is said 

 to be a Zonal Spherical Harmonic. Taking the axis of sym- 

 metry as the axis of z, V is a function of z and \/x^+y 2 . 

 Or, in polar coordinates, V is a function of r and 0. 



Let be a point in the axis, the origin of coordinates ; 

 let the distance of any point P from be called r, let the 

 angle between P and the axis be called 0, then in any dis- 

 tribution which has an axis of symmetry we need only to 

 know r and 0. And over any spherical surface whose centre 

 is 0, the distribution will be a function of 0. Any zonal 

 surface spherical harmonic is then merely a function of 0, and 

 I give a table of values of these for values of differing by 

 1° from 0° to 90°, up to the harmonic of the seventh degree. 

 These are indicated by P , P 1? P 2 , &c. P 7 . 



The surface harmonic of no degree is 1, and is indicated 



by P . 



The student is referred to Mathematical treatises for the 

 proof that, if /j, be written to represent cos 0, then 



Po=l, 



p _V-l 



1 2— S J 



3 ~ 2 ' 



p _35//-30 / u, 2 -i-3 

 n " 8 



P* = 



68^ 5 -7(V + 15 / u, 



8 



p _ 231^-315^ + 105/* 2 -5 

 ^ 6 ~ ~W 



P 7 = 



42 V -693/* 5 + 315^-35/* 



16 



Any function of 6 may be expanded in terms of P , P 1? P 2 , 

 &c.j that is any symmetrical function V may be expanded in 

 a series of Zonal Spherical Harmonics. Take, for example, the 

 powers of cos 0, it may be shown that 



