218 Proceedings of Philosophical Societies. [Sept. 



ing, by means of induction and calculation, from phenomena to 

 laws, and from laws to the powers themselves." 



The author proceeds from these researches to the considera- 

 tion of the motion of the system constituted by the earth and 

 moon. He shows that, neglecting the action of the sun, the 

 ascending node of the lunar orbit on the invariable plane of this 

 system, always coincides with the descending node of the earth's 

 equator; and that these nodes have an uniform retrograde 

 motion, the planes of the lunar orbit and of the equator preserv- 

 ing the same constant inclinations upon the invariable plane of 

 this system. 



The action of the sun modifies the preceding results, and 

 impresses such motions upon the nodes of the lunar orbit and 

 the plane of the maximum of the areas, that the two planes 

 always meet at the equator ; the plane of the maximum of the 

 areas dividing the angle formed by the equator and the lunar 

 orbit into two angles, the sines of which have a constant ratio 

 to one another. The retrograde motion of the nodes of the 

 moon combined with the action of that satellite upon the terres- 

 trial spheroid occasions the nutation ; and the reaction of this 

 spheroid upon the moon, produces the two lunar inequalities 

 which depend on the flattening of the earth at the poles. These 

 inequalities, examined by Messrs. Burg and Buckhardt, by some 

 thousands of observations, agree in giving -^-^ for the flattening 

 of the earth, which differs very little from -^^ or -g-f^, the flatten- 

 ing deduced from the measurement of terrestrial degrees. " But 

 if, on the one hand, the irregularities occurring in these measures 

 are considered ; and, on the other hand, the agreement between 

 the two lunar inequalities, and also the immense number of 

 observations by which their coefficients have been determined, 

 it will appear that these inequalities offer the surest method of 

 determining the true figure of the earth." 



At this place begin the analytical calculations, by which it is. 

 proved that there exists in every spheroid covered with a fluid a 

 certain axis around which the system of the spheroid and the fluid 

 can revolve uniformly, the axis of rotation remaining invariable ; 

 that in the respective motions of the terrestrial equator and of 

 the lunar orbit, the two planes preserve a common intersection, 

 and constant inclinations upon the invariable plane, and that this 

 intersection has a secular, retrograde, and uniform motion ; 

 lastly, that the inequality known under the name of nutation 

 produces a correspondent inequality in the inclination of the 

 lunar orbit upon the ecliptic by means of the reaction of the 

 terrestrial spheroid upon the moon. 



The second memoir is entitled, " On the Influence of the 

 great Inequality of Jupiter and Saturn on the Motions of the 

 Bodies composing the Solar System." This great inequality, 

 whose period is nine centuries, amounts to one-third of 



