NEWTON S PRINCIPIA. 363 



the pendulum, after allowing for the buoyancy of the air, 

 is affected by two causes, the variations of pressure 

 over its surface, and the friction of the air along its sur 

 face. This friction is assumed to be proportional to the 

 difference of velocities of the air and surface resolved in 

 any direction along the surface. He now forms the dif 

 ferential equations of the motion of the fluid and pendu 

 lum, but finds that without some further limitation they 

 cannot be solved. 



In the case, however, in which the surface of the pen 

 dulum is spherical, and the rod so thin that we may 

 neglect the action between it and the air, the equations 

 can be integrated, and the motion both of the fluid and 

 sphere found. The motion is supposed to be given to the 

 body by moving the pendulum very slightly from its po 

 sition of rest, and then leaving it to the action of gravity, 

 without impressing on it any velocity. At the commence 

 ment of the motion the whole air is supposed to be at 

 rest homogeneous and boundless in all directions. 



The motion of the spherical pendulum is found to be 

 the same as that of a simple pendulum of a certain length 

 K oscillating in a medium resisting as the velocity. But it 

 is to be remarked that this resistance is found to arise 

 entirely from the friction of the air against the sphere. The 

 assumption that this friction exists is directly contrary to 

 the usual theory of Hydrodynamics. The very equations of 

 motion are founded on the supposition that it does not exist, 

 and therefore takes no account of the equal and opposite 

 friction of the sphere on the fluid. In testing the theory 

 we must omit this resistance, and the motion is therefore 

 the same as that of a simple pendulum oscillating in vacuo. 

 Theory, therefore, gives the arc constant. The value of X 

 thus found is 





comparing this with our former value of A, we see that 



