The problem under consideration is that of determining the seven elements (the vector ) characterizing the relative three-dimensional motion of the components of a binary star, using observations of the two-dimensional projection on the sky of the relative positions of those components as a function of time. Sooner or later a differential correction process is employed; therefore, this correction procedure will be reviewed first.

The basic approach is to derive a first estimate of the orbital parameters, form residuals from each of the observations, and assume these residuals represent a normally distributed random error, plus systematic errors due to an incorrect initial orbit. Expressing the residuals as functions of the orbital elements, these functions can be expanded in a Taylor series about the initial orbit and truncated after the first-order terms, giving the following approximate linear relation: