Abstract: For the solution of the downline problem with a given descent time function，the problem is transformed into an Abel integral equation solution problem.For the integral equation on infinite interval,this paper introduces Abel's method and the process of solving integral equation with Laplace transformation,and gives the solution formula.For the limited range of integral equation,this paper uses the Abel integral transformation method to solve the integral equation,uses the iterated integral to exchange integral sequence,and obtains the solution formula from a definite integral identity.The solution formula of the integral equation is applied to the isochronous curve problem.The isochron is proved be an inverted cycloid with the solution of  the differential equation of the isochronous curve problem.

Key words: downline problem, Abel integral equation, Abel integral transformation, isochronous descending line, differential equation, cycloid