A function $G(\xi ,\eta ,\tau )$ is expanded as a tri-variate power series, and recursion formulas are used to obtain a general inversion algorithm for $G(\xi ,\eta ,\tau ) = 0$ which as an analytical and numerical tool may be specialized to solve a variety of problems. Various special cases are examined, including a two-dimensional generalization of the Schlomilch–Cesaro formula for the high order total derivatives of a function of three variables when two of the variables are in turn functions of the third, the expression of the solution x of the equation $f(x) = h(t)$ as a power series in t, the recursive inversion of power series, and a recursive form for the one-dimensional Schlomilch-Cesaro algorithm. A brief review of concise formulas for the high order derivatives of composite functions due to Faa di Bruno and Duboshin is also presented.