Table of Zonal Spherical Harmonics. 521 



For the sake of ease of calculation let 27rC = l, and let 

 a = l centim. 



Then V=l- 



Vz 2 + 1 



We can expand this in powers both of z and of -, and we have 

 either 



v - 1 - + i'-s' + n'-S' +to - • • (2) 



or 



T7 1 1 3 15 1 35 1 k ,_. 



Now, if we can find V as a function of r and which is correct 

 along the axial line, then it must be correct everywhere. 

 [This is Green's theorem, assuming that the axial line to 

 infinity is a cylindric surface of no lateral dimensions.] But 

 it is obvious that 



V = 1 - rP, + ± ,*P, - | tT, + ^ r'P, - g »- 9 P 9 + &c. . (4) 

 becomes (2) when = ; and 



V " 2 r» 8 r 4 + 16 r 6 128 r 8 + ^°* " * ^ 5j 



becomes (3) when 0=0. 



Hence these express the potential everywhere. The first 

 of these is useful for calculation only when r is less than 1. 

 The second is useful only when r is greater than 1. 



And this shows a defect of the Spherical Harmonic method. 

 For if r is nearly 1 we cannot easily calculate V from either 

 of the series, having to use too many terms. It will, however, 

 be found, even in this case, that when we use the harmonics 

 up to Pn for any value of 6 we can plot V on squared paper 

 from r = to r = # 9, and from r=l\L to values of r as 

 large as we please. If the intermediate part of the curve from 

 ?' = 09 to r=l'l be drawn with a little judgment, it is astonish- 

 ing how quickly and accurately the equipotential surfaces 

 may be drawn. The result may be compared with the lines 

 of force as worked out by the elliptic integral method of 

 Sir William Thomson. 



Any such Electromagnetic solution is also the solution of a 

 Hydrodynamic problem. 



The principle adopted in this example is very useful. It is 



