46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 363 



with one and the same theory, however great may be the length of time 

 over which the observations extend. In order to satisfy this condition, he 

 develops the whole of his formulas algebraically, leaving in a general 

 symbolical form all the elements which vary with the time, such as the 

 eccentricities, the inclinations, and the longitudes of the perihelia and nodes. 

 He treats in the same way the masses which are not yet sufficiently known. 



All the work is given in full detail, and is divided as far as possible 

 into parts independent of each other, so that any part may be readily 

 verified. 



All the terms which are taken into account are clearly defined, so that 

 if it should ever be necessary to carry on the approximations still further, 

 it will be easy to do so without having to begin the investigation afresh. 



The whole work is presented with such clearness and method as to 

 make it an admirable model for all similar researches. 



After the development of the disturbing functions, and the formation of 

 the differential equations on which the variations of the elements depend, 

 the first step to be taken is to determine by integration of these equations 

 the periodic inequalities of the elements of the orbits of Jupiter and Saturn 

 which are of the first order with respect to the masses. As we have already 

 said, the expressions of these periodic variations of the elements are given 

 with such generality that, in order to obtain their numerical values at any 

 epoch whatever, it is sufficient to substitute the secular values of the elements 

 at that epoch. The calculation of the various terms under this general form 

 is very laborious, and it requires great and sustained attention in order to 

 avoid any error or omission of importance. On the other hand, by substi- 

 tuting from the beginning the numerical values of the elements at a given 

 epoch, the calculation is rendered much shorter and admits much more readily 

 of verification ; but the result thus obtained only holds good for the given 

 epoch, and is thus entirely wanting in generality. 



In the determination of the long inequalities of Jupiter and Saturn, the 

 approximation is carried to terms which are of the seventh degree with 

 respect to the eccentricities and the mutual inclination of the orbits. 



In the next place the terms of the first order in the secular variations 

 of the elements of the orbits are determined. 



After this the periodic inequalities of the second order with respect to 

 the masses are considered. These are determined in the same form as the 



462 



