In this paper an asymptotic expansion is derived for the power function of the likelihood ratio criterion for testing independence between two sets of variates for the case when the population canonical correlation coefficients are small. The method used can theoretically give the expansion up to any order of $N$ where $N$ is the sample size. Here the expansion is given up to $N^{-3}$ and is an extension of an expansion obtained independently by Sugiura [10] using a different method. The theorem in Section 3 summarizes the final result. In Section 4 the expansion is compared numerically with a different approximation obtained by Sugiura and Fujikoshi [11] and with exact results obtained by Pillai and Jayachandran [9].