336 ADDRESS ON PRESENTING THE GOLD MEDAL OF THE [44 



Now let us imagine, for a moment, that the disturbing function con- 

 tained no periodic terms, but was reduced simply to its non-periodic part. 

 Consequently the partial differential coefficients taken with respect to the 

 angular elements would all vanish, and therefore the three conjugate elements 

 would be all constant, as well as the major semi-axis, the eccentricity and 

 inclination, of which those elements are functions. Hence, again, the partial 

 differential coefficients taken with respect to the conjugate elements would 

 be functions of those elements, and would therefore be constant. Hence 

 each of the angular elements would consist of an arbitrary constant and a 

 term proportional to the time, the multiplier of the time in each case being 

 a known function of the three constant elements. 



The object of M. Delaunay's method is, by means of a series of changes 

 of the variables, to cause all the more important periodic terms to disappear 

 from the disturbing function, one by one, while the differential equations 

 continue to retain their canonical form, so that after each transformation 

 we approach more nearly to the conditions of the ideal case which has 

 just been considered. 



In order to effect any one of these transformations, M. Delaunay 

 supposes, for the moment, that the disturbing function is reduced to its 

 non-periodic part, together with one of the periodic terms selected from 

 among those which have the greatest influence in producing the lunar 

 inequalities. With this simplified form of the disturbing function, the 

 equations admit of being easily integrated. The elements with which we 

 start may thus be expressed in terms of three new angular elements which 

 vary uniformly with the time, and three new constant elements. M. Delaunay 

 shews how the constant elements may be so chosen that they may be 

 considered as respectively conjugate to the three new angular elements, so 

 that, in fact, the quantities which are multiplied by the time in the 

 expressions of these angular elements are respectively equal to the partial 

 differential coefficients of a function of the new constant elements taken 

 with respect to these elements. 



Having thus found the relations between the old set of elements and 

 the new ones by means of the simplified form of the disturbing function. 

 M. Delaunay now restores the complete value of that function, and chooses 

 new elements which are connected with the old ones by exactly the same 

 relations as in the case just considered. Of course the three new angular 

 elements will no longer vary uniformly with the time, and the three elements 

 respectively conjugate to these will no longer be constant. 



