Now as every equation involving only p and q may be considered as representing a developable surface, it may be satisfied by the equation to a plane in which the arbitrary constants are afterwards supposed to vary. Hence assuming = ar + By typy we find p = a, q = B, and therefore B = C aa - ba; so that a particular integral of (3) is z = ax + (C = aa - ha) y + y. To deduce the general integral we must take for y an arbitrary function of a, and then join with the equation to the plane its differential with respect to a, so that the system of equations = ax + (C + aa- ja”) y + P(a), () = v - (a + a)y + p' (a), , is the general integral of (3), and a particular integral of (1). A different form of should be taken for each sign of a, so that this system is equivalent to two. The equation (15) (1 + 9°)? – 2pqs + (1 + p') t = 0, belongs to those surfaces in which the principal radii of curvature are equal but of opposite signs. On assuming 9 = f (P), we have 1 + {f()} – 2pf(P)/' (1) + (1 + p) {f'(p)}? = 0. 9 = ap + (-1 - a")"; from which we have z = 0 (x + ay) + y (- 1 - a) as the particular integral of the given equation. It is easy to see that this must represent a plane, as that is the only developable surface which has its principal radii of curvature equal and of opposite signs. From the difficulties attending the integration of ordinary differential equations of a high order it will readily be understood that the integration of partial differential equations of the second and higher orders is a problem in the solution of which still less progress has been made. The subject has much occupied the attention of mathematicians, and processes have been given for integrating various classes of these equations, but they are unfortunately exceedingly long and complex, and the solutions are frequently given in a form which renders thenı practically useless. I shall therefore not give any examples of them here, but shall content myself with referring the reader to the original memoirs: such as those of Laplace, Mémoires de l'Académie, 1773 ; Legendre, Ib. 1787; Ampére, Journal Polytechnique, Cahiers xvii, et XVIII ; and Cardinali, Sul Calcolo Integrale dell' equazioni di differenze partiali. Some examples of the application of Definite Integrals to express the integrals of partial Differential Equations will be found at the end of Chap. XI. Sect. 1. Linear Differential Equations with Constant Coefficients. The solution of any number of simultaneous equations of this class may always be reduced to the principles of the elimination of the same number of linear algebraical equations. For the symbol of differentiation may be treated exactly like any constant involved in the equation, and therefore the rules for eliminating, when the variables are involved along with constants, may be applied to equations in which they are involved, along with symbols of differentiation. Let there be two simultaneous equations involving two variables, dic + 6 x = 0. dt d To eliminate y, operate on the first equation with and dt multiply the second by a; we have then Es. (1) dy Subtracting the second of these from the first, y disappears, and we have dx - abx = 0. It might at first appear that as we might obtain an equation involving y alone, similar to the resulting one in w, there must be four arbitrary constants, and not two. But the second pair can always be determined in terms of the other two, and are therefore not arbitrary. This remark applies to such equations generally: and it is best to avoid the introduction of the superfluous constants by deducing (as we have done in this example) the other variables from the first without integration. The real number of arbitrary constants is always equal to the sum of the highest indices of differentiation in the different equations. dx (2) Let + ax + by = 0, dt dy +0,0 +b, y = 0, dt be two simultaneous equations. Operate on the first with d + b ; ing, y disappears and we have d d + a + bi ldt dt where h and k are the roots of the equation x - (a + b) x + ab, - a, b = 0. x = Ce-"+ Cie-kt; k b a - a Ce-he + Cie |