n of the r equations of a
Jacobian system to that of a single equation in n - r + 1 variables
may be explained in connexion with a geometrical interpretation which
will perhaps be clearer in a particular case, say n = 3, r = 2. There
is then only one total equation, say dz = adz + bdy; if we do not take
account of the condition of integrability, which is in this case da/dy
+ bda/dz = db/dx + adb/dz, this equation may be regarded as defining
through an arbitrary point (x0, y0, z0) of three-dimensioned space
(about which a, b are developable) a plane, namely, z - z0 = a0(x -
x0) + b0(y - y0), and therefore, through this arbitrary point [oo]^2
directions, namely, all those in the plane. If now there be a surface
z = [psi](x, y), satisfying dz = adz + bdy and passing through (x0,
y0, z0), this plane will touch the surface, and the operations of
passing along the surface from (x0, y0, z0) to

(x0 + dx0, y0, z0 + dz0)

and then to (x0 + dx0, y0 + dy0, Z0 + d^1z0), ought to lead to the same
value of d^1z0 as do the operations of passing along the surface from
(x0, y0, z0) to (x0, y0 + dy0, z0 + [delta]z0), and then to

(x_ + dx_ , y_ + dy_ , Z_ + [delta]^1z_ ),
0 0 0 0 0 0

namely, [delta]^1z0 ought to be equal to d^1z0. But we find

d^1z0 = a0dx0 + b(x0 + dx0 , y0, z0 + a0dx0)dy0 =

/db db \
a0dx0 + b0dy0 + dx0dy0( --- + a0--- ),
\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

dx_j - c_1j dx1 + ... c_rj dx_r

similarly determine through an arbitrary point x1^0, ... xn^0 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 [omega]_r+1 = x_r+1^0, ... [omega]_n = x_n^0,
wh