constants, and general forms of singular solutions containing fewer than n arbitrary constants.. A solution y=(x) of unknown origin being given, we construct a differential equation for determining dy, and, solving it, form the expression for y+dy, and from the number of infinitesimal arbitrary constants it contains, determine the nature of that general value of y of which the given value is a particular case. Now we are not to infer from this that the form of y+dy will be the same as the general value of y in question. But we may infer that it will be a form to which that general value is reducible. And the actual reduction will be effected by expressing the general solution (as is always possible) in a form permitting its expansion in ascending powers of the arbitrary constants, and in the expansion making these constants infinitesimal, and rejecting all powers of them above the first. In fact, if y=f(x, α1, α2, ... αr) be any general form of solution which, when we assign to a, a,... a, particular values (e. g. make them vanish) reduces to the brackets denoting that after differentiation we make a1, a,... α, vanish. a, This is that limiting form of the solution which Legendre's method enables us to construct by the solution of a linear differential equation; and the ground of the sufficiency of his method consists in this, that the infinitesimal quantities which are in fact the arbitrary constants of that solution, are equal in number to the arbitrary constants of the general unlimited solution, the nature of which is thus made known. 2ndly. Legendre's tests for differential equations of the higher orders are in kind and effect analogous to the tests for differential equations of the first order. They enable us to decide whether a solution possesses singularity, not whether it possesses the envelope species of singularity. The completion of Legendre's theory would consist in the discovery of those further tests dependent upon integration which correspond to the test of Euler and Cauchy for differential equations of the first order. C CHAPTER XXIV. ADDITIONS TO CHAPTER XIV. [Art. 1 was intended to follow Chap. XIV. Art. 2.] 1. As the condition of dependence of functions of two variables is of fundamental importance in connexion with the theory of ordinary differential equations, so the generalized condition of dependence of functions of any number of variables forms a fundamental part of the theory of partial differential equations. This is contained in the following proposi tion. PROP. I. If u1, u,... un are functions of x, xq,... Xn, but are as such so related that some one of them is expressible as a function of the others, or more generally that there exists among them some identical equation of the form F(U1, U2, ... Un) = 0, .......................................................(1), so that as functions of x,,,,... x, they are not mutually independent, then, adopting the notation of determinants, the condition is identically satisfied. Conversely, if the above condition be identically satisfied, the functions u,, U,... Un are not mutually independent in the sense above explained. First let it be noticed that the Proposition is but a generalization of that of Chap. II. Supposing U and u to be two functions of x and y, the condition of their dependence is affirmed to be i. e. it is the result of eliminating dx, dy, from the equations as expressed in Chap. II. We proceed to the general demonstration. Let the first member of (1), considered as a function of u1, u2,... u, be represented for brevity by F; then differentiating, we have from which it follows that if du,, du,,... du, are equal to 0, then is du, equal to 0; or, since u1, u,,... u, are functions of xn, that if ... ... dun-1 dx ... dun dx1 + dxs ... Thus the last n equations, linear with respect to dx1, dx, ....dxn are not independent, and therefore by the theory of linear equations the determinant of the system vanishes identically. Now this is expressed by the condition (2). It remains to prove the converse, viz. that if the condition (2) be identically satisfied, the functions u,, u,, ... u will not be mutually independent. First, the n-1 functions u1, u,, ... u are either mutually independent or not mutually independent. If not, then the n functions u,, u,,... u, are not mutually independent, and the Proposition to be proved is granted. n-1 n-1 If u1, u,,... U are mutually independent as functions of x1, x2,..., they may be made to take the place of n-1 of these quantities, e. g. x1, x2, ... X, in the expressing of u„, i. e. we may, by means of the expressions for u,, u,,... Un-1? eliminate from that of u, the quantities x,, x,,... 1, and so Xn-12 express u as a function of u,, u,,... u and x. Suppose. this done, then the system (3), (4) will be converted into du1 = 0, du,= 0, .............. du2-1 = 0, Now, the determinant (2) vanishing, the equations of the linear system (3), (4) are not independent; therefore those of the transformed system, as written above, are not independent; therefore the last equation of that system must be a consequence of the others which manifestly are independent. But from the form of that last equation we see that such cannot be the case unless we have |