202 RECENT RESEARCHES ON THE 



cieutly near the truth to be admissible iu most astronomical investiga- 

 tions during the comparatively short period of time over which astro- 

 nomical observations or human history extends; but since the values 

 of these variations are derived from the equations 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 plane- 

 tary system, or even a near approximation to its actual condition at a 

 time only comparatively remote from the epoch of the elements on which 

 tliey 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 direc- 

 tion 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 depending 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 inequalities 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 i)lanet may ultimately deviate from their mean values can only be 

 determined by the simultaneous integration of the differential equations 

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

 nitesimal 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 indefinitely with the time, they would finally render the orbits 

 so very eccentrical that the stability of the planetary system would be 

 destroyed. Iu order to determine whether the roots of the equation 

 were all real and unequal, he substituted the approximate values of the 

 elements and masses whi(;h were enq)loyed by astronomers 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 particular values of the masses employed by La Grange, the equa- 

 tions which determine the secular variations contain neither arcs of a 

 circle nor exponential quantities, without the signs of sine and cosine; 

 Avhence it follows that the elements of the orbits will perpetually oscil- 

 late about their mean values. This investigation was valuable as a 

 first attemjit to fix the limits of the variations t)f the planetary elements; 



