Let {
p n (
t)}
n=0 t8 be a system of algebraic polynomials orthonormal on the segment ?1, 1] with a weight
p(
t); let {
x n,ν (p) }
ν=1 n be zeros of a polynomial
p n (
t) (
x x,ν (p) = cos
θ n,ν (p) ; 0 <
θ n,1 (p) <
θ n,2 (p) < ... <
θ n,n (p) <
π). It is known that, for a wide class of weights
p(
t) containing the Jacobi weight, the quantities
θ n,1 (p) and 1 ?
x n,1 (p) coincide in order with
n ?1 and
n ?2, respectively. In the present paper, we prove that, if the weight
p(
t) has the form
p(
t) = 4(1 ?
t 2)
?1{ln
2(1 +
t)/(1 ?
t)] +
π 2}
?1, then the following asymptotic formulas are valid as
n → ∞:
$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$