On the Yamazaki-Hori solution of the Ernst equation

A family of solutions to the Ernst equation is presented, which, in a certain limit, recovers the Yamazaki-Hori solution - an extension of the Tomimatsu-Sato solutions for all integer values of the deformation parameter $\delta$. Our solution also recovers, as a special case, the formulation given by Vein, which is equivalent to the Yamazaki-Hori solution. Furthermore, our construction establishes a connection to a nonlinear differential equation proposed by Cosgrove, which is also associated with the Ernst equation.