356 Professor Airy on a Correction requisite 



and to examine particularly whether, with the dimensions which 

 have been used or may probably again be used, the neglect of 

 the relative rotation of the ball and wire will introduce any 

 sensible error in the concluded length of the seconds' pendulum. 



The force which causes the ball to move horizontally is the 

 resolved part of the tension of the wire. Suppose for the sake 

 of simplicity the arc of vibration to be so small that the tension 

 of the wire may be considered constant. Then the motion of 

 translation of the ball will be occassioned by a force proportional 

 to the distance of the point of attachment of the wire from the 

 vertical : the motion of rotation will depend upon the whole 

 tension, and upon the angle made by the wire produced with 

 the diameter which was vertical (as the other -force which acts 

 on the ball, namely, its weight, ihay be considered as acting at 

 its center of gravity.) 



Let a be the length of the wire, from the point of suspension 

 to the point where it is attached to an inflexible part of the 

 ball or bob : r the distance of the ball's center of gravity from 

 the same point : k the distance of the center of gyration ; 6 the 

 angle made by the wire with the vertical : ^ the angle made 

 with the vertical by that line in the ball which in the position 

 of rest was vertical ; M the mass of the ball : which will also 

 represent the tension of the wire. Then the horizontal force 

 acting on the ball is M . sin & : and the equation for the motion 

 of translation gives 



T^(a sin e + r sin <p) M = - gM sin 6. 



The momentum of the force of tension about its center of gravity 

 is Mr sin (6 — (p), and the equation of rotation gives 



A-'M J^ = sMr sin {Q - cp). 



