Let 7 be a domain of rationality, and let yi, , Yn be a set of indeterminates. Then the set of prime ideals in the ring of polynomials j7 [yi, , yn] satisfies a divisor-chain condition for decreasing sequences as well as for increasing sequences. That is, a sequence of prime ideals 11, 12, in 7[yi, , y.] must be of finite length not only if 2;i1 properly includes 2i for every i, but also if 2;i1 is properly included in 2i for every i.t However, if the domain of rationality 7 is a set of functions meromorphic in an open region Xf, and if 7 is closed to differentiation (in other words, if j7 is a field, in the terminology of algebraic differential equationst), and if 71 yl, , Yn } is the differential-ring consisting of all forms in yi, , Yn with coefficients in 7, then the set of prime differential-ideals in 7 { Yi, , Yn satisfies a divisor-chain condition for increasing sequences,? but does not satisfy such a condition for decreasing sequences. That is, we can have an infinite sequence 21, 12, * of prime differential-ideals such that 2i+l is properly included in Ii. In the set-theoretic sense the sequence 1 2, converges to a limiting set 2 which is the intersection of the Ti. If ?'i is the manifold of 1j, then the sequence S7YC1, S7YC2, is a monotonically increasing sequence converging in the set-theoretic sense to a set N which is the union of the S7yC. However, while the limiting set 2 is a prime differential-ideal, the limiting set N not only is not the manifold of 2, but is not a manifold at all. We are concerned in this paper with the relation between N and the manifold S7YC of 2. In the terminology of ADE, what we are considering is an infinite sequence 1 2, of closed irreducible systems in yi, , yn such that ?Jyrj, the manifold of 1j, is a proper part of the manifold of 1j+j, (i= 1, 2, ).