Harnack's Theorems

Harnack's first theorem states that a real irreducible curve of order $n$ cannot have more than

$${\textstyle{1\over 2}}(n-1)(n-2)-\sum s_i(s_i-1)+1$$

circuits (Coolidge 1959, p. 57).

Harnack's second theorem states that there exists a curve of every order with the maximum number of circuits compatible with that order and with a certain number of double points, provided that number is not permissible for a curve of lower order (Coolidge 1959, p. 61).

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, 1959.