(x0 + dx0, y0, z0 + dz0)
and then to (x0 + dx0, y0 + dy0, Z0 + d1z0), ought to lead to the same value of d1z0 as do the operations of passing along the surface from (x0, y0, z0) to (x0, y0 + dy0, z0 + δz0), and then to
(x0 + dx0, y0 + dy0, z0 + δ1z0),
namely, δ1z0 ought to be equal to d1z0. But we find
| a0dx0 + b0dy0 + dx0dy0( | db | + a0 | db | ), |
| dx0 | dz0 |
and so at once reach the condition of integrability. If now we put x = x0 + t, y = y0 + mt, and regard m as constant, we shall in fact be considering the section of the surface by a fixed plane y − y0 = m(x − x0); along this section dz = dt(a + bm); if we then integrate the equation dx/dt = a + bm, where a, b are expressed as functions of m and t, with m kept constant, finding the solution which reduces to z0 for t = 0, and in the result again replace m by (y − y0)/(x − x0), we shall have the surface in question. In the general case the equations
dxj = cijdx1 + ... crjdxr
similarly determine through an arbitrary point xº1, ... xºn Mayer’s method of integration. a planar manifold of r dimensions in space of n dimensions, and when the conditions of integrability are satisfied, every direction in this manifold through this point is tangent to the manifold of r dimensions, expressed by ωr+1 = x0r+1, ... ω_ = xºn, which satisfies the equations and passes through this point. If we put x1 − xº1 = t, x2 − xº2 = m2t, ... xr − xºr = mrt, and regard m2, ... mr as fixed, the (n − r) total equations take the form dxj/dt = c1j + m2c2j + ... + mrcrj, and their integration is equivalent to that of the single partial equation
| dƒ/dt + Σ | n | (c1j + m2c2j + ... + mrcrj) dƒ/dxj = 0 |
| j=r+1 |
in the n − r + 1 variables t, xr+1, ... xn. Determining the solutions Ωr+1, ... Ωn which reduce to respectively xr+1, ... xn when t = 0, and substituting t = x1 − xº1, m2 = (x2 − xº2)/(x1 − xº1), ... mr = (xr − xºr)/(x1 − xº1), we obtain the solutions of the original system of partial equations previously denoted by ωr+1, ... ωn. It is to be remarked, however, that the presence of the fixed parameters m2, ... mr in the single integration may frequently render it more difficult than if they were assigned numerical quantities.