The present work concerns the second order ancillary determined by the tangent directions V = ∂y/∂θ|{y0; θ̂(y0)} at the observed data y 0 for fixed pivotal value, by using the structural approach of Fraser (1986). We show that for scalar parameter case, such an approximate ancillary is well defined and can be described as a rotation. These second order ancillaries are particularly useful in prov...