358 NEWTON S PRINCIPIA. 



of Newton, been considered equal to - . This for- 



m 



mula is founded on the presumption that the moving force 

 which the body undergoes, and which is denoted by m m 

 is confined to the mass m. But it must be distributed not 

 only over the moving body, but on all the particles of fluid 

 set in motion by that body, and consequently the denomi 

 nator of that expression denoting the accelerating force 

 must necessarily be greater than m. From some general 

 mathematical considerations he concludes that a fluid of 

 very small density surrounding a pendulum has no other 

 influence on the duration of the vibrations than that it 

 dimi nishes its gravity and increases the moment of in 

 ertia.&quot; 



The effect of the resistance of the air is then the same as if 

 that air was removed and a mass of air, equal to x times &quot;the 

 fluid displaced &quot; was attached to the centre of gravity of 

 the fluid displaced, which increases the inertia of the whole 

 without affecting its gravity. The effect of the buoyancy 

 of the air is the same as if a weight of air equal to that of 

 the fluid displaced were removed from the centre of gravity 

 of the fluid displaced, affecting the gravity but not the 

 inertia. Let Z and y be the distance of the centres of gra 

 vity of the pendulum and air displaced from the axis of 

 suspension, b s the ratio of the density of the fluid to the 

 mean density of the pendulum, and let i be the radius of 

 gyration of the pendulum about an axis through its centre 

 of gravity parallel to the axis of suspension. Then by 

 merely uniting the equation of motion of the pendulum, it 

 becomes evident that a pendulum affected as above stated 

 will oscillate in exactly the same manner that a simple 

 pendulum whose length is A does in vacuo, where 



I 2 + i* + y 2 



This manner of expressing the result will be presently 



