d d to dx' dy' &c. and linear with respect to x, y, &c. The following is one of the results. If, assuming a partial differential equation can be presented in the form on the assumption that and operate only on the subject u, d.x dy then it can be expressed in the form F(T,, T,) u = 0, independently of such restrictive hypothesis. It might be added, that all such equations are reducible to equations with constant coefficients, by assuming To the above might be added many other special deductions, isolated now, but destined perhaps, at some future time, to be embraced in the unity of a larger theory. reduced directly by Prop. I. to a form integrable by Prop. 1, or, by assuming (m+2) 0=20', converted into a particular case of Art. 7 in the Chapter. which includes the above, is integrable in finite terms if i being a positive whole number or 0). (Malmsten, Cambridge Mathematical Journal, 2nd Series, Vol. v. p. 180.) Verify this. 7. As an illustration of the theory of disappearing factors, integrate the equation d'u (x2+qx3) dx2 + {(a+3) qx2 + (b − i + 1) x} ? du dx grable in finite terms in the following three cases; viz. x2 1.2... nx+2.3... (n + 1) x2 ... +p (p + 1) ... (p+n-1) ∞3 is integrable in finite terms if n is an integer. Apply the method of Art. 13 to reduce the symbolical equation to a bino can be integrated in finite terms, whatever function of x is represented by Q. (Curtis, Cambridge Mathematical Journal, Vol. IX. p. 280.) Let e Qdæ u=v; then compare the resulting form with Ex. 8 of the Chapter. 15. Shew generally that, if we can integrate the equation dx we can integrate f+Q) u + $ (x) u = X. 16. We meet the equation d'y 1-3c2 dy 1 φ in the theory of the elliptic functions (Legendre's modular equation). Shew that it is not integrable in finite terms, but is integrable in the form y = A + B log c, where A and B are series expressed in ascending even powers of c. 17. Prove the following generalization of Prop. III. CHAPTER XVIII. SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS BY 1. THE solution of linear differential equations by definite integrals was first made a direct object of inquiry by Euler. His method consisted in assuming the form of the definite integral, and then, from its properties, determining the class of equations whose solution it is fitted to express. Laplace first devised a method of ascending from the differential equation to the definite integral. And Laplace's is still the most general method of procedure known. Its application is however not wholly free from difficulties, due partly to the present imperfection of the theory of definite integrals, partly to an occasional failure of correspondence in the conditions upon which continuity of form in the differential equation and continuity of form in its solution depend. Indeed it ought never to be employed without some means of testing the result a posteriori, e. g. by comparison with the solution of the proposed differential equation in series. Frequently indeed it is possible to deduce the solution in definite integrals from the solution in series without employing Laplace's method at all. Laplace's method is applied with peculiar advantage to equations in the coefficients of which a enters only in the first degree, and of which the second member is 0. Expressing any such equation in the form T being a function of t, the form of which, together with the limits of integration, must be determined by substituting the |