Wants to use method of indeterminate coefficients to integrate the differential equations of the planets' perturbations. [Originally written to W. Fairfax.]

Needs advice on integrating equations. Praises JH's book [Prelim. Discourse]. Says James Mackintosh and [Henry] Hallam also praise it. Note from William Somerville, which needs a reply, is enclosed.

Explains to MS P. S. Laplace's method of indeterminate coefficients in mathematics. Thanks MS for her praise of his book [Prelim. Discourse], which JH values more highly than newspaper reviews.

Sends preliminary observations that will be prefixed to her book [Mechanism of the Heavens]. Wants JH to make any criticisms of it before it is published.

Returns proofs and sends a few remarks concerning them. Discusses difficulty of measuring Eta Coronae. Anxious to see her [Mechanism of the Heavens]. Has another daughter [Isabella].