646 TRANS. ST. I.OUIS ACAD. SCIENCE. 



M^K sin rt — J^l sin w I = 0, 



wiiere M represents the entire mass of the pendulum. This gives- 

 the well known value of /. 



M . K" 



The loci of the isodynamic lines in the disc pendulum are de* 

 termined from (3), which may be put into the following form : 



^ = r sin (fi-{-o.) ^ sin ^. (5) 



£■ am V 1 / I 



This expression represents the moment of the impressed force 

 dF per unit of weight at any point determined by the values 

 r, 0, and a. Making this value constant, = a, it gives the condi- 

 tion for an isodynamic line, which is, therefore, 



a = r sin (t) -}- a) ^ sin 0. (6) 



Let S be the origin of a system of rectangular coordinates, x 

 being the horizontal and j the vertical coordinate of dm; then, as 



r''-= ^2 -j- j2 and sin (« + «)= — , equation (6) becomes 



For a fixed value of H, and a varying value of a, this is the 

 equation of a series of concentric circles, the common center be- 

 ing on the horizontal through S at a distance ±-5 ■ — ^from S.. 



The radius of any circle is 



R'=N/^fr-^ a] (8) 



sm (y 1^4 sm H J 



If « = 0, we have the condition that the motion of a particle 

 is unaffected by its connection with the system. The radius of 

 this neutral circle is, therefore, 



R' = h-^. (9) 



sm 6' 



