218 NEWTON S PKINCIPIA. 



its course.&quot; &quot; And if that body is continually disturbed 

 by the action of some foreign force, we may nearly know 

 its course by collecting the changes which that force 

 introduces in some points, and estimating the continual 

 changes it will undergo in the intermediate places from the 

 analogy that appears in the course of the series.&quot; This 

 method, which Newton only applies to determine the 

 general effect of any disturbing force, we can use, by the 

 aid of the differential calculus, to determine its effect to any 

 degree of accuracy. 



Let us conceive a planet to be describing an ellipse 

 round the sun in the focus, and let it be continually dis 

 turbed by the resistance of the medium in which the planet 

 moves. At the time t, let , e, w be the mean distance, 

 eccentricity, and longitude of the apse of the ellipse, and 

 let r, 0, v be the distance, longitude, and velocity of the 

 planet. The attraction of the sun and the resistance of 



the medium will be represented by ~ and K v 2 , where ju, 

 and K are certain constants. 



From the sixteenth proposition of the third section of 

 the first book it is easy to deduce by a known property of 

 the ellipse, that 



1 _ 2 __ V 



a r p 



But in the time dt, v is decreased by the resisting me 

 dium by 



dv KV 2 dt 



da 2v 



. . - = d v 



a? a 



