85-86] ZONAL HARMONICS. 123 



where the constant factor has been adjusted so as to make 

 P n fa) = 1 for //, = !. The formula may also be written 



1 rJn 



p &amp;lt;^=^-v n ............... &amp;lt; 2 &amp;gt;* 



The series (1) may otherwise be obtained by development of 

 Art. 83 (6), which in the case of the zonal harmonic assumes the 

 form 



As particular cases of (2) we have 



The function P n (p} was first introduced into analysis by 

 Legendref as the coefficient of h n in the expansion of 



(1 - 2ph + h*)-*. 



The connection of this with our present point of view is that if 

 &amp;lt;/&amp;gt; be the velocity-potential of a unit source on the axis of x at 

 a distance c from the origin, we have, on Legendre s definition, 

 for values of r less than c, 



= ( C 3 _ 2/icr + r 2 )~i 



Each term in this expansion must separately satisfy V 2 = 0, and 

 therefore the coefficient P n must be a solution of Art. 85 (1). 

 Since P n is obviously finite for all values of /it, and becomes equal 

 to unity for /* = 1, it must be identical with (1). 



For values of r greater than c, the corresponding expansion is 



* The functions P 1 ,P. 2 ,...P 7 have been tabulated by Glaisher, for values of /j. 

 at intervals of -01, Brit. Ass. Reports, 1879. 



+ &quot; Sur 1 attraction des spheroides homogenes,&quot; Mem. des Savans Etrangers, t. x., 

 1785. 



