500 THE LUNAR INEQUALITIES DUE TO THE [62 



be formed when it is stated that the development so obtained contains 

 one constant term accompanied by 121 periodic terms. 



The next process is by a series of transformations of the variables 

 involved gradually to remove these periodic terms from the disturbing 

 function, so that it is at length reduced to the form of a constant term. 



The number of such operations required to effect this reduction amounts 

 to 103, although each operation is individually sufficiently simple. 



By the essential principle of Delaunay's method the differential equa- 

 tions throughout these transformations always preserve their canonical form, 

 and therefore when the disturbing function has been reduced to the above- 

 mentioned simple form, the integrals are at once obtained. 



In the next place the transformations indicated in the 103 operations 

 above mentioned are also made in Delaunay's expressions for the three 

 coordinates of the Moon, so that finally the values of these coordinates 

 are found in terms of three arbitrary constants and three angles, each of 

 which consists of a term proportional to the time joined to an arbitrary 

 constant. 



The coordinates thus expressed are the longitude, the latitude, and the 

 reciprocal of the radius vector. As this last quantity is only intended to 

 be employed in finding the Moon's parallax, it is given by Delaunay with 

 much less precision than the other two coordinates, a circumstance which 

 is to be regretted as an imperfection from a theoretical point of view. 



The expressions thus found are purely analytical, that is the coefficients 

 are expressed in series of powers and products of Delaunay's constants 

 m, e, e', y, each term also involving as a factor a constant quantity which 

 depends on the figure of the Earth. 



In order to make his work more complete, Mr Hill determines the 

 numerical value of this last-mentioned factor by a very elaborate discussion 

 of the results of numerous pendulum experiments. 



Finally, by the substitution of the known values of the constants 

 employed, the numerical expressions for the perturbations of the Moon's 

 coordinates produced by the figure of the Earth are obtained. 



It will be remarked that comparatively few of the coefficients so found 

 amount to an appreciable quantity, by far the larger number being utterly 

 insensible. 



