Saddlepoint approximations are derived for the conditional cumulative distribution function and density ofwhereis the sample mean ofni.i.d. bivariate random variables andg(x, y) is a non-linear function. The relative error of orderO(n–1) is retained. The results extend the important work of Skovgaard (1987), and are useful in conditional inference, especially in the case of small or moderate sample sizes. Generalizations to higher-dimensional random vectors are also discussed. Some examples are demonstrated.