The J^* test which was previously proposed by the present author for the detection of a trend in a time series does not depend on any quantitative assumptions, but in the case of a polynomial trend it depends on its degree; if this degree is too high, the test cannot be applied. The author finds a bound of the significance level at which the test can be applied when the sample size, as well as a bound of the degree of the trend, are given. Asymptotic results are used only when we trust the asymptotic distribution of J^* under the null hypothesis.