The paper may be considered as a supplement to the book by M. Krein and M. Naimark. Problem: find the number of real solutions of an algebraic system f₁(x, y) = 0, f₂(x,y) = 0 satisfying an algebraic condition g(x, y) > 0. We give a review of Hermite's method, which permits one to solve the problem using a finite number of algebraic operations on the coefficients of f₁, f₂, g. The method reduces the problem to an investigation of the quadratic form in x₀,..., x_N-1: ∑ j=1 N(α_j, β_j)[x₀ + x₁β_j+⋯+x_n-1β_j ^N-1]²: where (α_j, β_j) is a solution of the system, and N = deg f₁ deg f₂. Methods involved: elimination theory, the theory of symmetric functions, and the theory of Hankel quadratic forms. Applications: analogue of Sturm series in R², calculation of the Kronecker-Poincaré index of an algebraic field with respect to an algebraic curve in R², conditions for the sign-definiteness of a homogeneous higher-order polynomial of three variables. We discuss also the possibility of extension of the method to R².