Let us consider the differential equation (1) g(z, Y) f b , Y) dY = 0 and suppose that Po(zo , yo) is a singular point ( f ( z o , yo) = g(ro , yo) = O), more precisely a saddle point, i.e., that It is well known that, in this case, there whic.h pass through the point Po and separate 0.
exist two integral curves of (1) a neighborhood of Po into four regions. These solution curves, through the saddle singularities, are, in many cases, very important in fixing the qualitative character of all the solution curves; they enable us to separate the z,y-plane into regions R , in such a way that an integral curve which has one point in Ri lies entirely in Ri.'
If we substitute for equation (1) the system
(where t is the independent variable), the solution (x = zo , y = yo] of (3) is represented, in t,z,y-space, by the straight line z = xo , y = yo , parallel to the t-axis, and the integral curves of (l), through P, , become the orthogonal projections onto the z,y-plane of the integral curves of the system (3).
The plane region R, is the orthogonal projection of a space region R: (bounded by a cylindrical surface) such that if R: contains any point { t o , x(t,), y(to) 1 of an integral curve, [ z ( l ) , y ( t ) } of (3), R: contains all its points. *Lecture at the Seminar of the Institute for Mathematics and Mechanics, November, 1950, N.Y.U.