Let G be a 1-dimensional complex linear algebraic group. In [7] Horrocks has shown that when G is the additive group C of complex numbers acting regularly on a normal (complex) projective variety, say X, or an algebraic variety X which is algebraically locally factorial, then the closure C of each G-orbit of is nonsingular. Moreover, Mabuchi [9] has shown that, if C touches a 1codimensional component in X of the set X of fixed points, then C is nonsingular and it intersect X transversally for any complex manifold X. On the other hand, when G=C*, the multiplicative group of complex numbers, Horrocks [7] showed that on a variety X as above the closure of each orbit is locally irreducible. If x is any point of X, this is equivalent to saying that either xeX, or 0(X)/oo(x), where 0(x) = limt^0t(x) and oo(x) = lim f_> 0 0/(x) for teC*. Furthermore, this fact was generalized as follows (cf. [3], [4]). Let X be a complex manifold on which G = C* acts biholomorphically and meromorphically. Then a sequence of points xί9 9xs9 s>l, of X is said to generate a quasi-cycle if xte X—X for each /, and co(xt) and 0(^ί+1) are contained in one and the same connected component of X for ^defining the action extends to a meromorphic map Px X^> X with respect to the natural inclusion G<—»P, where P denotes the complex projective line. The purpose of this note is then to generalize the above results in the following two theorems: