NEWTON S PRINCIPIA. 219 



we might express u, the velocity, in terms of the time by the 

 usual formula of elliptic motion ; this would not lead to 

 any lengthy calculations, but as e and x are very small, 

 there is no practical advantage in investigating more than 

 the principal terms in the series expressing the changes of 

 the elements. &quot;We may then put 



v = G?Z, 



where n is the mean angular velocity ; and hence 

 p = aW, 



... *? = - 2 x a? n 

 d t 



and x being very small, we may reject the variations of 

 the quantities on the right hand side, 



.*. a l = 2 x a 2 ?it) 



where a , a l9 are the values of the mean distance at the 

 beginning and end of the interval t. 



By the fourteenth proposition of the third section and 

 the first of the second, we can easily see that 



ft, a (1 e 2 ) = v 2 r 2 sin 2 , 



where a is the angle between the radius vector and tan 

 gent. Hence 



da ,. _ de _ . 9 e?u 



P TJ Q ~ e ) ~&quot; ^ pae -j-. = % v sm a r -r t 



= 2 x v 3 sin 2 a . r 2 

 Now as we retain only the principal terms, we have 



a = 90, 



r = a(l ecos w), 

 substituting, we get 



de 



-T- = 2 x a n cos w , 



rt 6 



.. ^ e = 2 x a . sin ?i # 3 

 where e ^j are the values of e at the beginning and end of 



