Let, now, the sun’s longitude EH be denoted by Z (considered as a flowing quantity); then, (sin. Z)² being = ½-½ co-s. 2 Z, we shall have _k_ × co-s. E × (sin. EH)² = ½_k_ × co-s. E × 1-co-s. 2 Z. But the angle described about the axe of rotation P_p_, in the time that the sun’s longitude is augmented by the particle Ż, will be = (_T/t_) × Ż. Therefore (by the general proposition) we have, as 1: ½_k_ × co-s. E × 1-co-s. 2 Z ∷ (_T/t_) × Ż : ½_k_ × (_T/t_) × co-s. E × Ż - Ż co-s. 2 Z, the true regress of the equinoctial point E, during that time: whose fluent, ½_k_ × (_T_/_t_) × co-s. E × (Z- ½ sin. 2 Z), will consequently be the total regress of the point E, in the time that the sun, by his apparent motion, describes the arch HE or Z; which, on the sun’s arrival at the solstice, becomes barely = ½_k_ × (_T_/_t_) × co-s. E × an arch of 90°: the quadruple whereof, or ½_k_ × (_T_/_t_) × co-s. E × 360° (= (3_t_/4_T_) × ((OA²-OP²)/OA²) × co-s. E × 360°) is therefore the whole annual precession of the equinox caused by the sun. This, in numbers (taking OP/OA = 229/230) comes out (3/(4 × 366¼)) × (2/230½) × 0.917176 × 360° = 21´´ 6´´´. The very ingenious M. Silvabelle, in his essay on this subject, inserted in the 48th volume of the Philosophical Transactions, makes the quantity of the annual precession of the equinox, caused by the sun, to be the half, only, of what is here determined. But this gentleman appears to have fallen into a twofold mistake. First, in finding the _momenta of rotation_ of the terrestrial spheroid, and of a very slender ring, at the equator thereof; which _momenta_ he refers to an axis perpendicular to the plane of the sun’s declination, instead of the proper axe of rotation, standing at right angles to the plane of the equator. The difference, indeed, arising from thence, with respect to the spheroid (by reason of its near approach to a sphere) will be inconsiderable; but, in the ring, the case will be quite otherwise; the equinoctial points thereof being made to recede just twice as fast as they ought to do. This may seem the more strange, if regard be had to the conclusions, relating to the nodes of a satellite, derived from this very assumption. But, that these conclusions are true, is owing to a second, or subsequent mistake, at Art. 27; where the measure of the sun’s force is taken the half, only, of the true value; by means whereof the motion of the equinoctial points of the ring is reduced to its proper quantity, and the motion of the equinoctial points of the terrestrial spheroid, to the half of what it ought to be.