1819.] Royal Academy of Sciences. 221 



very ingenious machine, but this cannot give the least idea of 

 its quantity. Indeed analysis itself only shows an approxima- 

 tion to this ; and for a long time there will be only astronomical 

 observations to determine the motion of the precession with 

 sufficient exactness. 



On the Libration of the Moon, by M. Poisson. — " Accord- 

 ing to the laws of this phenomenon, discovered by D. Cassini, 

 and confirmed by the beautiful calculation of M. Lao-ran^e the 

 moon revolves upon her axis in the same time that she accom- 

 plishes her mean revolution round the earth : her equator preserves 

 a constant inclination upon the ecliptic, and the descendino- 

 node of this equator coincides with the mean ascending node of 

 the lunar orbit. M.. de Laplace proved that these results are not 

 affected either by the secular equation of the mean motion of the 

 moon, nor by the secular displacing of the ecliptic ; it is also 

 certain that they are not altered by the secular equation that 

 affects the mean motion of the moon's node ; but these results 

 agree only with the mean velocity of rotation, and with a mean 

 state of the lunar equator ; and theory shows that this velocity of 

 rotation, the inclination of the equator, and the distance of its 

 node from that of the orbit, are subject to periodical inequalities, 

 whose maxima depend upon the ratio among the momenta of the 

 moon's inertia. M. Lagrange gave the expression of the princi- 

 pal inequalities of the velocity of rotation ; so that in order to 

 render the theory complete there remained only to determine the 

 inequalities of the inclination and of the node. This is what I 

 propose to determine, by taking up afresh the solution of the 

 problem, and by carrying on the approximation unto the terms 

 of the second order in respect to the elements of the lunar orbit, 

 which terms contain the inequalities in question. I shall confine 

 myself to giving the formulae I have discovered, and shall 

 suppress the details of the calculations which led me to them, 

 and which are only a development of the calculation of M. La- 

 grange." 



The latter end of this paragraph does not mean that no 

 formulae are to be found in the memoir, they being, on the 

 contrary, indispensable to show the changes which take place in 

 the expressions by the introduction of terms of the second order. 

 The author considers successively the different inequalities of 

 the longitude of the node : the second is known, it is about a 

 fifty-fifth part of the mean inclination. He shows that the first 

 is less than a twenty-seventh part of the same inclination. Two 

 similar inequalities are to be found in the distance of the node of 

 the equator from that of the orbit. By the second, the two 

 nodes are separated from one another by more than a degree ; 

 the maximum of the first does not surpass two degrees. 



M. Bouvard found that the distance of these nodes is 2° ; 

 Mayer found it to be 4°, but in a contrary way. The difference 

 betwixt these two results may be partly attributed to the errors. 



