366 NEWTON S PRINCIPIA. 



The motion will therefore be the same as if the sphere 

 were resisted by a force x m n multiplied by the velocity, 

 and a mass xm were added to its centre, increasing the 

 inertia without affecting the weight. 



We are now enabled to account for many of our expe 

 rimental results, that is, such of them as relate to spheres. 

 We see that neither x nor x depend on the density of the 

 sphere, but only on the volume ; that both are greater for 

 small than large spheres. The resistance also is independ 

 ent of the roughness of the surface. One experiment of 

 Sabine showed that x remained the same when p was 

 reduced one half; this would seem to show that for the 

 same fluid, at the same temperature, the value of ja, the 

 coefficient of the friction, varies as the density. But since 

 the value of x was not the same as before, when hydrogen 

 was substituted for air, we see that in different media ft 

 depends on something else besides the density. 



We may apply these conclusions to the pendulum, and 

 obtain results which we may test by experiments. The 

 effect of the term depending on x will clearly be to alter 

 the time of the vibration, but not the arc of oscillation ; 

 the term depending on x (being multiplied by the small 



m 

 factor , whose square may be neglected) will not affect 



the time of the vibration, but will decrease the arc continu 

 ally, so that the successive arcs form a geometrical progres 

 sion. The time will be increased by a fraction of the time 

 equal to ^ x 8 nearly, and the common ratio by which the 



5* 3 



arcs decrease is s 2 nearly. The less the sphere the 

 greater are x and x ; the more, therefore, is the time altered 

 and the quicker does the arc of vibration decrease. 



If a sphere move uniformly in a fluid with friction, we 



