212 NEWTON S PRINCIPIA. 



It is needless to point out how different this is from 

 motion in vacuo, where the velocity would have gone on 

 increasing without any limit. 



One familiar instance of motion in a resisting medium 

 is the descent of rain. The drops descend then with a 

 uniform motion, the larger drops going quicker than the 

 smaller, and the velocity of descent increasing as the drops 

 grow in size. If the rain descended with the velocity due 

 simply to the action of gravity, a heavy shower of rain 

 would commit serious injury. A drop of rain falling from a 

 cloud a mile high would have acquired a velocity of about 

 576 feet a second. The actual velocity is perhaps less than 

 one five thousandth part of this. 



In the fourth section Newton discusses the motion of 

 a particle in a resisting medium when acted on by a feebly 

 resisting medium. He begins by considering the case in 

 which the density of the medium varies inversely as the 

 distance from the centre of force, and under peculiar con 

 ditions of the motion of the body extends his deductions 

 to any law of distance. We have followed his method, 

 with the exception that, as we have the powerful aid of 

 analysis, we can treat the question with greater generality. 

 But since Newton s time we have discovered much better 

 methods ; it has been thought not out of place to give a 

 very brief view of them, so far as they depend only on 

 first principles. 



6. OL. A particle moves in an equiangular spiral under the 

 action of a central force in the pole, in a medium whose 

 density varies as some function of the distance from the 

 pole. To determine the connexion between the law of density 

 and the law of force that this motion may be possible. 



The equiangular spiral, by definition, possesses the 

 property that the tangent at any point makes a constant 



