332 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [44 



Meantime, M. Hansen had undertaken a completely new investigation of 

 the lunar theory, by a remarkable method peculiar to himself and explained 

 in his Fundamenta nova investigations orbitce verce quam Luna perlustrat, 

 which appeared in 1838. 



In applying the method described in this work to the case of the 

 Moon, M. Hansen throughout employs numerical values of the elements of 

 the Moon's orbit, and consequently the coefficients of the lunar inequalities 

 as obtained by him are also purely numerical. The process is one of 

 successive approximations, which are repeated again and again until the 

 values of the inequalities which are found from the last approximation 

 sensibly coincide with those which were assumed in entering upon that 

 approximation. 



The numerical values of the coefficients thus finally obtained are un- 

 doubtedly very exact. The slight corrections which these coefficients still 

 require are probably chiefly due to the small corrections required by the 

 numerical elements on which the calculations are based, and in the method 

 employed no provision is made for taking into account the effect of these 

 corrections. 



From his formulas, M. Hansen constructed tables of the Moon, which 

 were published in 1857, at the expense of the British Government; and 

 these tables, having been found far superior in accuracy to all others, are 

 now exclusively employed in the calculation of ephemerides. 



A detailed account of the calculations leading to M. Hansen's last 

 approximation, was given by him in the two parts of his Darlegung der 

 Theoretischen Berechnung der in den Mondtafeln angeivandten Stomngen, 

 which severally appeared in 1862 and 1864. 



After the great works, to which we have thus briefly referred, had 

 been either completed or were in progress, it might have been supposed 

 that the matter was exhausted. 



Our Associate M. Delaunay, however, was not of this opinion. Having 

 devised, so long ago as 1846, a perfectly original and singularly beautiful 

 method of integrating the differential equations of the Moon's motion, he 

 determined to apply this method to the complete re-investigation of the 

 theory, and to carry on the approximation to a much greater extent than 

 had been done by his predecessors. The principal fruits of his labours, to 

 which he has devoted himself with almost unexampled perseverance for so 

 many years, are contained in the magnificent volumes which the Imperial 



