44] ROYAL ASTRONOMICAL SOCIETY TO M. CHARLES DELAUNAY. 337 



When, by means of the proper formulae of transformation, the new 

 variables have been substituted for the old ones in the disturbing function 

 and in the expressions of the Moon's coordinates, M. Delaunay shews that 



1st. One of the important terms of the disturbing function disappears, 

 viz., the periodic term which was selected in the preliminary investigation. 



2nd. Various inequalities corresponding to this term are introduced into 

 the values of the three coordinates of the Moon. 



3rd. The values of the six new variables in terms of the time are 

 determined by differential equations of exactly the same form as those which 

 determined the values of the six variables for which they have been sub- 

 stituted. 



One of the periodic terms having been in this manner caused to dis- 

 appear from the disturbing function, a new operation of exactly the same 

 kind causes another term of this function to disappear ; similarly a third 

 term may be taken away by means of a third operation, and so on to any 

 number of terms. 



In this way, after a suitable number of operations of this kind have 

 been effected, the disturbing function will have been simplified by the 

 removal from it of its most important periodic terms, after which the 

 further process of integration becomes simple enough to be treated in the 

 same manner as if we were concerned with the perturbations of a planet 

 or of the Sun. 



The whole difficulty in the determination of the lunar inequalities is 

 caused by the great magnitude of the disturbing force of the Sun. M. De- 

 launay has therefore at first confined his attention to the investigation 

 of the irregularities which are produced by this disturbing force, and the 

 two magnificent volumes before us are entirely occupied with this investi- 

 gation. Thus he has provisionally left out of consideration the very small 

 inequalities due to some secondary causes, such as the attraction of the 

 planets and the figure of the Earth ; and, besides, he has omitted to consider 

 the perturbations of the Sun's apparent motion about the Earth, intending 

 in a supplementary volume to take into account the effects due to these 

 several causes. 



By means of repeated applications of the beautiful method of trans- 

 formation which I have above attempted to describe, M. Delaunay proceeds 

 to get rid of all the periodic terms of the disturbing function due to the 

 Sun's disturbing force, which are capable of producing inequalities in the 



A. 43 



