330 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [44 



Both Damoiseau and Plana, following the example of Laplace, start 

 from differential equations in which the Moon's longitude is taken as the 

 independent variable; and after the equations have been integrated, they 

 obtain the values of the Moon's coordinates in terms of the time by 

 reversion of series. An important innovation, however, was introduced by 

 Plana in the mode of conducting the investigation and exhibiting the results. 

 The values of the Moon's coordinates being developed in series of sines and 

 cosines of angles which vary uniformly with the time, the coefficients of 

 the several terms of these series will depend on the eccentricities of the 

 orbits of the Sun and Moon, the inclination of the Moon's orbit to the 

 plane of the ecliptic, the ratio of the mean motions of the Sun and Moon, 

 and the ratio of their mean distances from the Earth. Now Damoiseau, 

 in common with all previous writers, having assumed certain values of the 

 quantities just mentioned as given by observation, contented himself with 

 determining the numerical values of the coefficients. Although this is all 

 that is required for the construction of tables, yet, from a theoretical point 

 of view, it leaves the mind unsatisfied, inasmuch as any coefficient in its 

 numerical form shews no trace of its composition, that is of the manner 

 in which its value depends on the value of the assumed elements. The 

 several coefficients are far too complicated functions of the elements to be 

 represented analytically, except in the form of infinite series, and Plana, 

 accordingly, developes these coefficients in such series, proceeding by powers 

 and products of the eccentricities, the tangent of the inclination, the ratio 

 of the Sun's mean motion to that of the Moon, and the ratio of the 

 Moon's mean distance to that of the Sun, all these quantities being assumed 

 to be small, and the last mentioned ratio, which is much smaller than the 

 others, being considered as a quantity of the second order. 



In this mode of development, the numerical factor which enters into 

 any term of the coefficient of any of the lunar inequalities is an ordinary 

 fraction which admits of being determined not merely approximately, but 

 with absolute accuracy. It is easy to see what great facilities are afforded 

 by this circumstance for the verification of the work by a comparison of 

 the results obtained by different methods. The greater or less degree of 

 approximation will thus depend on the greater or less number of terms 

 taken into account in the several series. 



The numerical values of the several elements are not substituted in 

 the formulae until the work is completed, and this is attended with the 

 important advantage that when a comparison of the theory with observation 



