In two recent papers the authors have obtained a number of first integrals for similarity solutions of nonlinear diffusion and of general high-order nonlinear evolution equations. Such integrals exist only for special parameter values and are obtained via integration of the ordinary differential equation, which results when the functional form of the solution is substituted into the governing partial differential equation. In this paper we show that these special parameter values also occur in a natural way when we utilize the first order partial differential equation instead of the explicit functional form and we ask under what conditions can a first integral with respect to either of the independent variables x or t be deduced. This simple procedure generates all previous results and presents the idea of similarity solutions in an entirely new light. That is, the significant features of similarity solutions for partial differential equations are not necessarily the explicit functional form and subsequent reduction to an ordinary differential equation but rather that the solutions sort are common to two partial differential equations. The process is illustrated with reference to an extensive number of examples including nonlinear diffusion, general diffusion equations containing a number of parameters and high-order nonlinear evolution equations. In addition a new exact solution for nonlinear diffusion is obtained which is illustrated graphically.