14 THE ORBIT OF NEPTUNE. 



1. Those produced by their direct attraction on each other. 



2. Those produced by the displacement of the Sun by the attraction of the 

 disturbing planet. The co-ordinates of the disturbed planet being counted from 

 the centre of the Sun, the displacement of the Sun not only changes the value 

 of the co-ordinates by changing their origin, but also by modifying the attraction 

 of the Sun itself. 



The perturbations of both classes may be included in the same formulas, and 

 the total perturbations computed in the same way that those of the first class 

 are, by a very simple modification of those functions of the ratio of the mean 

 distances which enter into the different values of h. But in the case of the action 

 of an inner on an outer planet more than twice as far from the Sun, this method 

 will be subject to this serious inconvenience; that the perturbations of the 

 elements are many times greater than those of the co-ordinates. Referring to 

 formulas (4) and (5), it will be remembered that L, E, and W really express per- 

 turbations of the mean longitude, perihelion, and eccentricit} r , and it will be seen 

 that the perturbations of the true longitude fe are expressed as a function of the 

 perturbations of those elements. Now, having in this way computed the perturb- 

 ations of any co-ordinate which depend upon the different terms of the perturbative 

 function, when we collect those coefficients which are multiplied by the sine or 

 cosine of identical angles, we shall frequently find that their sum will nearly vanish, 

 as has been already remarked. As this circumstance depends on a theorem of 

 some importance, which will furnish a valuable check on the developments we 

 shall presently give, it is worth while to trace it to its origin. 



The elements of a planet depend on its position and its velocity at a given epoch ; 

 each element is a function of the co-ordinates, their differential coefficients, and 

 the time, or, representing an element by a, and putting, for shortness, 



t dx _ dy dz 



t-dt^-di^-dt' 



we have six equations of the form 



a n =f(x,y, z, , >?, , f) (12) 



When we express the co-ordinates as a function of the elements and the time, we 

 have 



x, y, or z f (a l} a. 2 , a 3 , a 4 , a 6 , a e , f) (13) 



Substituting for the elements the values just given, , y, and must vanish iden- 

 tically in the value of each co-ordinate. If, now, the changes in , >y, and ^ are 

 of a higher order of magnitude than those in x, y, and z, the co-ordinates will he 

 subject to smaller variations than the elements. 



Suppose, now, that one of the co-ordinates is affected with an inequality of 

 which the period is very short compared with that of the revolution of the planet. 

 Represent it by 



c sin (pnt -f- e ) . 

 Its differential coefficient will be 



pnccos 



