  In this paper, we derive the exact formula of Klein's fundamental 2-form of
second kind for the so-called $C_{ab}$ curves. The problem was initially solved
by Klein in the 19th century for the hyper-elliptic curves, but little progress
had been seen for its extension for more than 100 years. Recently, it has been
addressed by several authors, and was solved for subclasses of the $C_{ab}$
curves whereas they found a way to find its individual solution numerically.
The formula gives a standard cohomological basis for the curves, and has many
applications in algebraic geometry, physics, and applied mathematics, not just
analyzing sigma functions in a general way.
