The labour of the approximation in proceeding to a high order, when conducted according to the method of the former paper whether we employ the function φ or ψ, depends in great measure upon the circumstance that the two equations which have to be satisfied simultaneously at the free surface are both composed in a rather complicated manner of the independent variables, and in the elimination of y the length of the process is still further increased by the necessity of expanding the exponentials in y according to series of powers, giving for each exponential a whole set of terms. This depends upon the circumstance that of the limits of y belonging to the boundaries of the fluid, one instead of being a constant is a function of x, and that too a function which is only known from the solution of the problem.

If we convert the wave motion into steady motion, and refer the fluid to two independent variables of which one is the parameter of the stream lines or a function of the parameter, and the other is x or a quantity which extends with x from -∞ to + ∞, we shall ensure constancy of each independent variable at both its limits, but in general the equations obtained will be of great complexity.