64 Proceedings of the Poyal Society of Edinburgh. [Sess. 
where Y” is the spherical surface harmonic of degree n and is of the general 
form 
i — n 
i = 0 
where A and B are the 2U+1 arbitrary constants to be determined, and 
where 
t nM = V-l - ’ 
P'" being the zonal spherical harmonic of degree n of one variable. 
In Frank Neumann’s method 2^(p + l) values of the function are 
taken, namely. 
Ah , 0 ) , ... . , (‘ip - 1 ) 
f(p, , 0 ) , ,^y ... . .f{p, , {ip - 1 ).^) 
./V.+1 , 0) , , {ip - 1)^) 
where /xi . . . /x^+i are the (p + 1) roots of the equation 
P^+V) = 0. 
For each root the values of C and S are defined by 
v^2'p-l 
= 2 
where is unity, except when i = 0 or p, when the value is 2. 
If now we write 
it may be shown that the constants A and B are determined by the values 
given in (I) below.] 
Take then the expressions for the constants sought, namely, 
X=j0+1 
Ant= 2 ^ni{P\)Ci{f^x) 
X = l 
• ( 1 ) 
