This equation, together with the particular equation of Monge, and the equation Rr + Ss + Tt + U (33 — rt) =0, both which though falling under Ampère's general form possess peculiarities demanding special notice, I propose to consider in this Chapter. I shall in conclusion make some observations on the theory of partial differential equations of the second order with more than two independent variables. Monge's method, and Ampère's in so far as it is an extension of Monge's, consists in a certain procedure for discovering either one or two first integrals of the form u= f (v) ....... ..(4), u and v being determinate functions of X, Y, Z, P, and 9 ; and f being an arbitrary functional symbol. From these first in . tegrals, singly or in combination, the second integral involving two arbitrary functions is obtained by a subsequent integration. Now this procedure involves the assumption that the proposed equation admits of a first integral of the form (4). But such is not always the case. There exist primitive equations involving two arbitrary functions, from which by proceeding to a second differentiation both functions may be eliminated and an equation of the form (2) obtained, but from which it is impossible to eliminate one function only so as to lead to an intermediate equation of the form (4). Especially this happens if the primitive involve an arbitrary function and its derived function together. Thus the primitive $ (y + x) + 4(y - 3) - x {$'(y + 2) - (y — 2)}... (5), leads to the partial differential equation of the second order 2= but not through an intermediate equation of the form (4). It is necessary therefore, not only to consider the case in which the assumed condition is satisfied, but also to notice what has been done in those cases which do not at present fall under the dominion of any known method. Genesis of the Equation. 2. PROP. I. A partial differential equation of the first order of the form u=f(v), or its symmetrical equivalent, F(u, v) = 0, in which u and v are any functions of x, y, z, p, q, always leads to a partial differential equation of the form Rr + S8 + Tt + U (s* – rt) = V. For, differentiating the proposed first integral with respect to x, and with respect to y, we have and then eliminating dF IF dp dp du du idol do do + p+ dq which, on effecting the multiplication, gives (du du (dv) du do dp) dv du do + dx) dy) dq dq dy + du du du du dx) t dq a result which, since u and v are by hypothesis given functions of x, y, z, p, q, is seen to be a particular case of the general form (3) We may hence deduce also the conditions under which particular forms included in the general form (3) arise. Thus, in order that the equation urf (v) may give rise to a partial differential equation of the second order of Monge's form Rr + Ss + Tt = V, it is necessary that the condition du du du do = 0 dq dp dp da should be identically satisfied. This requires, by Chap. II. U Art. 1, that u and v, considered as functions of p and 9, should not be independent. 3. The geometrical relations of the equation (3) are also remarkable. It may in particular be shewn that an equation of this form will be satisfied by the equation of any surface which constitutes the envelope of any system of surfaces formed by the variation of three parameters in subjection to two arbitrary conditions. For let the common equation of the enveloped surfaces be z=f(x, y, a, b, c) (8), the parameters a, b, c varying in subjection to the conditions , $. (a, b, c) = 0, $, (a, b, c) = 0, conditions which, determining b and c as functions of a, may be reduced to the form b=$ (a), c=(a).. (9). Now the values of p and q being the same for any point P in the envelope as for the same point in the generating surface, we have for all such points df (x, y, a. b. c) df (x, y, a, b, c) p= (10). dx q= dy These two equations in conjunction with (9) enable us to determine a, b, c as functions of X, Y, Z, P, q. Let these values be a=u, b=v, Then substituting in (9) we have v=$ (u), wrz (u), equations which hold for all such points. These are then the partial differential equations of the first order of the envelope. Now each of these equations is of the general form (4); whence by Prop. I. the partial differential equation of the second order is of the form (3), as was to be proved. ). C= W. = Let us actually construct this equation. Differentiating the first of the equations (10) with respect to x and to y, and regarding therein a as a function of those variables, and b and c as functions of a, we have = d'f d'f d'f db + + d'f dc da dcdc da) da' d’f d’f dc da + dbdoc d'f db S = + Proceeding in the same way with the second equation of the system (10) we have Comparing this with the general form (3) we have the equations |