It might be thought that the "futures" of different months, being substitutes in proportion to their temporal proximity to one another, should vary together exactly; but it would seem to be a sufficient reply that as they are not perfect substitutes they are in some slight degree independent variables.
Since the two expressions (9) are the partial differential-coefficients of a single function E of the independent variables v and 0, we shall obtain the same result, namely d 2 E/d0dv, if we differentiate the first with respect to v and the second with respect to 0.
These variables represent the whole assemblage of generalized co-ordinates qr; they are continuous functions of the independent variables x, y, 1 whose range of variation corresponds to that of the index r, and of 1.
For example, in a one-dimensional system such as a string or a bar, we have one dependent variable, and two independent variables x and t.
To determine the free oscillations we assume a time factor e~1 the equations then become linear differential equations between the dependent variables of the problem and the independent variables x, or x, y, or x, y, 1 as the case may be.