1 86 Rev. J. Challis on the Theory of the Ball- Vendulum. 



ployed in reducing to a vacuum, the effect of the motion of 

 the fluid not being taken into account. According to our 

 theory m v" is to be added, in consequence of the simultaneous 

 movement of the air with the pendulum. The reasoning led 

 to the conclusion that so little change of density takes place 

 at the surface of the ball, that we may consider the air to be 

 put in motion just as if it were incompressible. The influence 

 of the air carried by the suspending wire was neglected in the 

 theory. Also the surface of the ball was supposed to be per- 

 fectly smooth, so as to impress no motion on the aerial parti- 

 cles in the direction of a tangent plane. Hence the air in 

 contact with the ball will move in directions normal to its 

 surface, and consequently directed to a centre. Because the 

 density is very nearly unchanged, the velocity at a given in- 

 stant will very nearly vary at different points, in a radius 

 produced, inversely as the square of the distance from the 

 centre. 



These results being admitted, we may proceed to calculate 

 m v 2 . For conceive two straight lines to be drawn at any in- 

 stant through the centre of the ball, one in the direction of its 

 motion, the other in any direction making an angle with the 

 first. Let the plane of these two lines make an angle <p with 

 a plane through the centre of the ball, at right angles to the 

 suspending wire. The velocity of the air at the point where 

 the latter line meets the surface is v cos ; and at any point 

 P on the line, distant by r from the centre, the velocity is 



vb % 



— — cos 0, b being the radius of the ball. The mass of a fluid 



element at P, its density being 1, is 



d r x r d x r sin d <p 9 

 and the vis viva of the fluid in motion, or m v% is equal to 



fffC^r cos W sin & dr dQd<p. 



The integral with respect to $ is to be taken from to 2?r, 

 with respect to from to 7r, and with respect to r from b 

 to infinity. Hence 



b* tf/Tf^ cos 2 sin QdQd <p 

 >dr 



mv" 



= 2* V tfff^ cos 9 Q sin U & 

 3 J 7 r : 3 



