viii INTRODUCTION. 



of the differential variations of the elements at a particular epoch, it follows that 

 they afford us no knowledge respecting the ultimate condition of the planetary 

 system, or even a near approximation to its actual condition at a time only com- 

 paratively remote from the epoch of the elements on which they are founded. But 

 aside from any considerations connected with the immediate needs of practical 

 astronomy, the study of the secular inequalities is one of the most interesting and 

 important departments of physical science, because their indefinite continuance in 

 the same direction would ultimately seriously affect the stability of the planetary 

 system. The demonstration that the secular inequalities of the planets are not 

 indefinitely progressive, but may be expressed analytically by a series of terms de- 

 pending on the sines and cosines of angles which increase uniformly with the time, 

 is due to La Grange and La Place. It therefore follows that the secular inequali- 

 ties are periodic, and differ from the ordinary periodic inequalities only in the 

 length of time required to complete the cycle of their values. The amount by 

 which the elements of any planet may ultimately deviate from their mean values 

 can only be determined by the simultaneous integration of the differential equa- 

 tions of these elements, which is equivalent to the summation of all the infinitesi- 

 mal variations arising from the disturbing forces of all the planets of the system 

 during the lapse of an infinite period of time. 



The simultaneous integration of the equations which determine the instantaneous 

 variations of the elements of the orbits gives rise to a complete equation in which 

 the unknown quantity is raised to a power denoted by the number of planets, whose 

 mutual action is considered. La Grange first showed that if any of the roots of 

 this equation were equal or imaginary, the finite expressions for the values of the 

 elements would contain terms involving arcs of circles or exponential quantities, 

 without the functions of sine and cosine, and as these terms would increase inde- 

 finitely with the time, they would finally render the orbits so very eccentrical that 

 the stability of the planetary system would be destroyed. In order to determine 

 whether the roots of the equation were all real and unequal, he substituted the 

 approximate values of the elements and masses which were employed by astrono- 

 mers at that time in the algebraic equations, and then by determining the roots he 

 found them to be all real and unequal. It, therefore, followed, that for the parti- 

 cular values of the masses employed by La Grange, the equations which determine 

 the secular variations contain neither arcs of a circle nor exponential quantities, 

 without the signs of sine and cosine ; whence it follows that the elements of the 

 orbits will perpetually oscillate about their mean values This investigation was 

 valuable as a first attempt to fix the limits of the variations of the planetary 

 elements ; but, being based upon values of the masses which were, to a certain 

 extent, gratuitously assumed, it was desirable that the important truths which it 

 indicated should be established independently of any considerations of a hypothetic 

 character. This magnificent generalization was effected by La Place. He proved 

 that, whatever be the relative masses of the planets, the roots of the equations 

 which determine the periods of the secular inequalities will all be real and un- 

 equal, provided the bodies of the system are subjected to this one condition, that 

 they all revolve round the sun in the same direction. This condition being satisfied 



