into a function where is the independent variable, having SECT. 2. Functions of Two or more Variables. Let u be a function of two variables, x and y, so that du dx u = f (, ); du then to express and in terms of two new variables dy r and 0, of which x and y are functions given by the If r and be given explicitly in terms of x and y, we have at once For the successive differentials we proceed in the same manner; and if there be more than two independent variables, the only difference is that the expressions become more complicated. Such cases however seldom occur. If the independent variables enter into multiple integrals, we cannot substitute directly the values of the original diffe rentials in terms of the new variables, because one is supposed to vary while the others are constant. To introduce this condition we proceed as follows. as follows. Let for example there be a double integral ffVda dy, and let Since a is to vary when y is constant and vice versâ, we must make dy = 0 when we wish to find da, and dæ = 0 when we wish to find dy. Taking the latter condition, we have the two simultaneous equations From this it follows that when dy = 0, dr = 0. Hence we have If we had three variables x, y, ≈ to be transformed into three others p, q, r, we should have three equations of the form dz = P2dp + Q2dq + R2dr; and we should determine da by supposing dy = 0, and dx = 0, and then eliminating two of the three quantities dp, dq, dr. Supposing we eliminate the last two we have da Mdp, M being a function of p, q, r. From this it follows that when dx = 0, dp = 0. Hence supposing y to vary while a and ≈ are constant we have dy = Q1dq + R1dr, 0 = Q2dq + R2dr; and eliminating dr between these we have dy = Ndq, N being a function of p, q, r. It follows that when dy = 0, dq = 0, and therefore if we suppose to vary while x and y are constant, we find dx = R2dr, so that finally The general expression for M is complicated, and it is of little use to give it here, as the consideration of the particular conditions of any given transformation will usually give us its value more readily than a substitution in the general formula.* • Lagrange, Mémoires de Berlin, 1773, p. 121. Legendre, Mémoires de l'Académie des Sciences, 1788, p. 454. |