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By M. Sion

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This proves our previous statement that the first-order partial differential equation Ω(. ) = 0, considered in the space x, y, z, p, q, is satisfied by every two-dimensional point manifold z = f, p = fx , q = fy provided that the equation z = f represents in the three-dimensional space x, y, z a surface satisfying both equations F1 = 0 and F2 = 0. 39. If the unboundedly integrable system F1 = 0, F2 = 0 in the space x, y, z has only ∞4 integral surfaces z = ϕ(x, y, a, b, c, d) the result is trivial.

In the x, y, z1 , z2 space there are obviously infinitely many curves, namely ∞∞ , satisfying both Monge equations Φ1 = 0, Φ2 = 0. If we take an arbitrary integral curve of the equations Φ1 = 0, Φ2 = 0, it will contact a certain characteristic curve of the equation Ω = 0 in every point. The ∞1 characteristic curves thus obtained generate a two-dimensional point manifold which furnishes an integral manifold of the equation Ω = 0 as well as of the involutory system. This reduces integration of the involutory system F1 (x, y, z1 , z2 , p1 , q1 , p2 , q2 ) = 0, F2 = 0, F3 = 0 to the simplest operations.

Vzm ) = 0. We claim that this system of first-order partial differential equations is semilinear. Moreover we assert that a solution z1 = ϕ1 (x1 , . . , xn ), z 2 = ϕ2 , . . , z m = ϕm of the original unboundedly integrable system F1 = 0, . . , Fq = 0 provides, in my sense, a solution of the system Ω1 = 0, Ω2 = 0, . . 50. For the proof, we note that the equations z1 = ϕ1 , . . , zm = ϕm represent in the space x1 , . . , xn , z1 , . . , zm an n-dimensional manifold. According to my general theory, the elements x1 , .

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