NEWTON S PRINCIPIA. 217 



it will require a very long time before r can be per 

 ceptibly changed. 



But regarding 8 r as indefinitely small, these expressions 

 will be accurate. Hence the whole time which it will take 

 the planet to arrive at the centre will be 



2 

 and the whole number of revolutions will be 



1 



These expressions cannot be integrated until the law of 

 density is known as a function of the distance. 



If we assume that the density varies inversely as the 

 distance from the centre, we have x r = D, a constant ; 

 performing the integrations, we have 



The number of revolutions is infinite, because the time 

 of a revolution becomes ultimately infinitely small. Com 

 pare the first of these formula with (4) and we learn that 

 if r l be the radius at any instant, r 2 the radius after one 

 complete revolution, the whole time of reaching the centre 

 of force will be to one revolution from radius T^ to radius 



3 3 _3 



r 2 in the ratio of r^ to r^ r 2 2 or f r t to r^ r 2 nearly. 

 In the case of the planets, r l ?* 2 is so small as to be 

 altogether insensible ; hence the above time is indefinitely 

 great. 



/3. In the seventeenth proposition of the third section of 

 the first book Newton remarks, that &quot; if a body move in a 

 conic section, and is forced out of its orbit by any impulse, 

 we can discover the orbit in which it will afterwards pursue 



