then that generally to effect the proposed elimination we should have to proceed as far as the partial differentials of z of the fifth order, and should arrive at three partial differential equations of this order, between x, y, z. But if we establish the relations X(c) = $'(c), 4(c) = $ ̋(c), that is, if the equations are of the form ƒ {x, y, z, c, p(c), p′(c), $ ̈(c)} = 0, F {x, y, z, c, $(c), $'(c), $'(c)} = 0 : then, if we proceed as far as differentials of the second order, we shall have, in all, twelve equations, between which we may eliminate the eleven quantities the result being a single equation between x, y, z, involving partial differentials of z only to the second order. Ex. Eliminate the arbitrary functions from the equations x $(c) + y x(c) + z ¥(c) = 1. ..(1), x p'(c) + y x(c) + z f'(c) = 0............(2). Differentiating (1), with respect to x, we have Similarly, differentiating (1) with respect to y, From (7) and (8), we get, as our final result, (8). Elimination of arbitrary Functions when the number of 62. In the preceding considerations respecting the elimination of arbitrary functions, we have always supposed that there are only two independent variables. We will now proceed to develop the theory of elimination when the number of independent variables is any whatever. For the sake of simplicity we shall confine ourselves to the case where it is not necessary to proceed to partial differentials beyond the first order. Let ...... x, y, z, .. being (m + 1) independent variables, and u the dependent variable. The quantity c is supposed to be an arbitrary function of a, ß, y,.... which are m known functions of x, y, z,. .. .and u. G Taking partial differentials with regard to x, y, z, succession, we get Taking these differential equations conjointly with the proposed equation, we shall have altogether m + 2 equations involving the m + 1 arbitrary functions These arbitrary functions may therefore be eliminated, when we shall arrive at a partial differential equation of the first order of partial differentials. Ex. To eliminate the arbitrary function from the equation Multiplying these differential equations in order by x, y, z, and adding, we get, attending to the proposed equation, and that, when a particular value x is assigned to x, f(x) and F(x) both become zero: the value of (x) will, for such a value of x, present itself under the indeterminate form. We proceed to investigate a rule which is often useful for the determination of the true value of (x). We have generally $ (x + 8x) = £ (x) + 8ƒ (x) and, for the particular value of x of x, 0 x and dx being supposed to be replaced by x and Sx。 in the expressions for Sf(x) Ex SF (x) бх which are generally functions of x and dx. Passing to the limit, when dx, becomes less than any assignable quantity, we it being supposed that, in the expressions for the functions ƒ'(x) |
