THE ORBIT OF NEPTUNE. 



15 



Since the elements contain tliis coefficient, and therefore include terms in which 

 the hirge number p multiplies the coefficient of the angle, their perturbations will 

 be much larger than that of the co-ordinate. But, in passing from the perturb- 

 ations of the elements to those of the co-ordinates, these large terms must destroy 

 each other. 



Let us apply this principle to the case under consideration. That portion of 

 the perturbative function which arises from the action of an inner planet on the 

 Sun may be developed in a series of terms of the form 



^cos(^';l' + a-f C); 



G representing a number, not a line. , 



It therefore becomes infinite when a is infinitely small. 



The second differential coefficient of the perturbation of any rectilineal co- 

 ordinate of the outer planet will be of the order of magnitude 



dR mc -.T 



-^-7 = -TT COS N. 

 da a" 



putting 



N— i'W -f a + G. 



If we integrate this difFerential, and develop the quantity .^ . according to 



t Jit — |— lit 



the powers of — = ~, the largest terms in the first differential coefficient of the 



co-ordinates will be of the form 



mc . ,^ 

 — - sin N. 



This also will become infinite when a is infinitely small ; and since the perturb- 

 ations of the elements contain these terms, it follows that they also will be infinite 

 in this case. Finally, by another integration, we shall have for the largest j)er- 

 turbations of the co-ordinate itself 



mcci ,_ 



—r^ cos N, 

 V 



which will vanish when a is infinitely small. Hence, in the case under consider- 

 ation, altlioutjli the j^erturhations of the elements hecome infinite, those of the co-ordi- 

 nates vanish. 



The co-ordinates referred to are linear. The order of magnitude of the angular 

 co-ordinates, or the logarithms of any linear co-ordinate, will be given by dividing 

 by a'. We shall, therefore, have for largest term in the perturbations 



hv, SB, or S log r zz mca M 

 ' ^ cos 



Hence, when we collect the perturbations of the co-ordinates due to the cause in 

 question, all terms of a higher order of magnitude than this ought to destroy each 

 other identically. 



