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----- ( -- + q -- ) - ( ------ + q ------ ) -- = 0, \dx dz/ dq \dy dz/ \ dy dz / dq moreover obtainable by omitting the term in df/dp in [p-[psi], f] = 0. Let x0, y0, z0, q0, be values about which the coefficients in this equation are developable, and let [zeta], [eta], [omega] be the principal solutions reducing respectively to z, y and q when x = x0. Then the equations p = [psi], [zeta] = z0, [eta] = y0, [omega] = q0 represent a characteristic chain issuing from the element x0, y0, z0, [psi]0, q0; we have seen that the aggregate of such chains issuing from the elements of an arbitrary chain satisfying dz0 = p0dx0 - q0dy0 = 0 constitute an integral of the equation p = [psi]. Let this arbitrary chain be taken so that x0 is constant; then the condition for initial values is only dz0 - q0dy0 = 0, and the elements of the integral constituted by the characteristic chains issuing therefrom satisfy d[zeta] - [omega]d[eta] = 0. Hence this equation involves dz - [psi]dx - qdy = 0, or we have dz - [psi]dx - qdy = [sigma](d[zeta] - [omega]d[eta]), where [sigma] is not zero. Conversely, the integration of p = [psi] is, essentially, the problem of writing the expression dz - [psi]dx - qdy in the form [sigma](d[zeta] - [omega]d[eta]), as must be possible (from what was said under _Pfaffian Expressions_). System of equations of the first order. To integrate a system of simultaneous equations of the first order X1 = a1, ... Xr = ar in n independent variables x1, ... xn and one dependent variable z, we write p1 for dz/dx1, &c., and attempt to find n + 1 - r further functions Z, X_r+1 ... Xn, such that the equations Z = a, Xi = ai,(i = 1, ... n) involve dz - p1dx1 - ... - pndxn = 0. By an argument already given, the common integral, if existent, must be satisfied by the equations of the characteristic chains of any one equation Xi = ai; thus each of the expressions [Xi Xj] must vanish in virtue of the equations expressing the integral, and we may without loss of generality assume that each of the corresponding 1/2r(r - 1) expressions formed from the r given differential equations vanishes in virtue of these equations. The determination of the remaining n + 1 - r functions may, as before, be made to depend on characteristic chains, which in this case, however, are manifolds of r
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