we find 



Ui,, = .f A" (f„ (8) 



whereas for (f„, as the simplest expression, we must take: 



Assigning to C the value: 



1 

 (4n— l)(4n— 3)...(2/i-hl) 

 the zonal harmonic function, as given in (5), is again found also for 

 the limits 1 and 0. 



In the case of uneven poljnomia 



U2n+1 = C A" ,'C'^>'+^ {.ü' — I)" (9) 



which for 



1 



r — 



(4/i + l)(4n— 1). . . (2/i + 3) 

 again leads to the expression (5). 



Giving C the value , we obtain from (8) as well as from (9' 



the zonal harmonic function in the form as commonly used. 



No more as for ^he limits 1 and 0, the development (7) for the 

 limits 00 and leads to new expressions ; we have to put 



ffn = C .^2"+l e-^^ 

 for even as well as for uneven functions, and b}^ the formulae 



(-1)" d , '. 



(2n 4-1)2» dx^ 



(—l)n ' ' ' \ f 



Lhn+i = —^ e—"' (A — 2)« .r2«+i 



we find the same expression as in § 3 for </„ of formula (6), but by 

 an abridged calculation. 



5. The problem, which form of development is the tittest for 

 frequencies of a quantity which assumes the form of a function of 

 one variable, moving between Ihe limits 1 and or go and 0, but, 

 as a matter of fact, must be considered as a function of two vari- 

 ables, is not solved satisfactorily in § 5, at least if we are not satisfied 

 b}' a merely formal representation. 



A graphical representation of such a function is given by the 

 distribution of points in a plane about a given origin, die element 

 of integration is then, not civ, but ^ttBcIR and the question must 

 be put as follows : to find a polynomium such that 



